The Chiral Lagrangian of CP-Violating
Axion-Like Particles

Luca Di Luzioa, Gabriele Levatia,b, Paride Paradisia,b

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

> We discuss the construction of the most general CP-violating chiral
> Lagrangian for an axion-like particle (ALP). Starting with an effective
> Lagrangian containing light quarks and gluons, we provide its matching onto
> a chiral effective Lagrangian at $\mathcal{O}(p^{2})$ described in terms of
> mesons and baryons, identifying the correspondence between the Jarlskog
> invariants of the two theories. After deriving the ALP interactions with
> mesons and baryons, we analyse a few relevant phenomenological implications
> such as the permanent electric dipole moments of nucleons and the CP-
> violating ALP and kaon decays. This work provides the necessary tools for
> further phenomenological analyses connecting low-energy observables with the
> couplings of the underlying ultraviolet complete theory.

###### Contents

  1. 1 Introduction
  2. 2 Construction of the CP-violating Chiral Lagrangian for ALPs
    1. 2.1 ALP interactions with Mesons
    2. 2.2 ALP interactions with Baryons
  3. 3 ALP interactions for $N_{f}=2$
    1. 3.1 ALP interactions with Mesons
    2. 3.2 ALP interactions with Nucleons
    3. 3.3 Summary and Jarlskog invariants
  4. 4 ALP interactions for $N_{f}=3$
    1. 4.1 ALP interactions with Mesons
    2. 4.2 ALP interactions with Baryons
    3. 4.3 Summary and Jarlskog invariants
  5. 5 Phenomenological applications
    1. 5.1 Proton and neutron EDM
    2. 5.2 CP-violating ALP decays
    3. 5.3 Kaon decays
  6. 6 Conclusions

##  1 Introduction

The lack for any convincing evidence of heavy new physics (NP) at the LHC has
motivated an increasing theoretical as well as experimental investigation of
scenarios with new light mediators. The most popular example is provided by
extensions of the Standard Model (SM) entailing light pseudoscalar bosons,
generically referred to as axion-like particles (ALPs) [1]. The emergence of
light ALPs can be naturally justified if they are pseudo-Nambu-Goldstone
bosons associated with the spontaneous breaking of an underlying global
symmetry. Remarkably, ALPs can address several open problems in particle
physics like the strong CP problem [2, 3, 4, 5], the evidence of dark matter
[6, 7, 8], as well as the flavour [9] and the weak-scale hierarchy [10]
problems.

Standard QCD axion models [11, 12, 13, 14], originally invoked as a solution
to the strong CP problem, predict a strict relation between the axion mass and
decay constant. However, since this relation depends on the specific
ultraviolet (UV) completion, a model-independent approach is commonly assumed
where ALPs are treated as a generalization of the QCD axion, with unrelated
mass and couplings to be probed experimentally. ALP interactions with SM
fermions and gauge bosons are then described via an effective Lagrangian
containing operators up to dimension-5 [15]. This approach provides the
opportunity to look for light ALPs with masses in the range between an MeV and
tens of GeV, whose couplings are not tightly constrained either by
cosmological and astrophysical bounds [16, 17, 18] or by collider searches
[19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. Moreover, low-energy
observables that are very sensitive probes of ALP interactions are given by
flavor-changing neutral-currents (FCNC) processes, both in the quark [31, 32,
33, 34, 35, 36] and in the lepton sectors [37, 38, 39]. Indeed, since there is
no fundamental motivation for ALP interactions to obey the SM flavor group,
ALPs can induce FCNC already at the tree level.

So far, most of the phenomenological analyses related to ALPs were inherent to
CP conserving phenomena (see, however, Refs. [40, 41, 42, 43, 44, 45, 46, 47,
48]). The CP symmetry is violated provided ALP couplings to photons (or
equivalently to gluons) include both $\phi F\tilde{F}$ and $\phi FF$
interactions (where $F$ is the QED field strength tensor, $\tilde{F}$ its dual
and $\phi$ denotes the ALP field) and/or if ALP couplings to fermions ($f$)
account for both $\phi\bar{f}i\gamma_{5}f$ and $\phi\bar{f}f$ interactions.
These requirements are fulfilled if the global shift symmetry, protecting the
ALP mass, as well as the CP symmetry are broken by the UV sector of the theory
and such a symmetry breaking is eventually transmitted to the infrared sector
by some dynamical mechanism. The necessary UV dynamics can naturally arise in
strongly-coupled theories, and in fact an explicit realization arises in the
case of the neutral pion in the SM through QCD and electromagnetic
interactions (see e.g. [49]). A strongly-interacting sector emerging at a
scale $\Lambda\gtrsim 1$ TeV which resembles the QCD dynamics is therefore
conceivable.

A CP-violating ALP is also motivated by relaxion models [10] addressing the
weak-scale hierarchy problem by introducing an ALP field, the relaxion, which
scans the Higgs boson mass in the early universe from a scale $M\gg 1$ TeV
down to the electroweak scale. In this case, the simultaneous presence of the
relaxion-Higgs mixing and the relaxion-photon/gluon couplings induces CP
violation.

From the experimental side, there is an outstanding program to improve the
current limits of permanent electric dipole moments (EDMs) of nuclei,
nucleons, atoms and molecules by few orders of magnitude [50]. This motivated
the analysis of Ref. [43] where the CP-violating Jarlskog invariants emerging
in the ALP effective field theory (EFT) were classified and the leading short-
distance effects on EDMs up to two loops were evaluated. However, Ref. [43]
assumed ALP masses above the GeV scale where QCD can be treated
perturbatively.

The main goal of the present work is to extend the analysis of Ref. [43] to
ALP masses in the well-motivated sub-GeV region where non-perturbative
methods, such as chiral perturbation theory ($\chi\text{pt}$), 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.
[15, 51, 52], as well as [53, 54, 55, 56, 57] for analyses beyond leading
order in $\chi\text{pt}$). In our work, we will generalise the above studies
by including both CP-odd and CP-even ALP interactions with mesons and baryons.
In particular, we will discuss in detail the matching of the perturbative
effective Lagrangian at low energies onto a chiral effective Lagrangian,
identifying the relations between the high-energy Jarlskog invariants and
those emerging in the chiral theory. The results of this work represent the
basis for phenomenological analyses related to a light CP-violating ALP,
connecting low-energy observables, such as EDMs and flavour-violating
processes entailing mesons and baryons, with the couplings of the underlying
UV complete theory.

The paper is structured as follows. In Section 2, we present the construction
of the most general CP-violating chiral Lagrangian for ALPs describing
interactions with mesons and baryons. In Section 3, we apply the general
results of Section 2 to a two-flavour setting, therefore obtaining explicit
expressions for ALP interactions with pions and nucleons. Section 4 is instead
relative to a three-flavour setup where explicit ALP interactions with pions,
kaons, and the eta meson as well as with baryons will be presented. Section 5
is dedicated to a short phenomenological analysis including the predictions
for the proton and neutron EDMs and CP-violating ALP and kaons decays.
Conclusions and an outlook for future possible studies, which can be obtained
starting from our results, are presented in Section 6.

##  2 Construction of the CP-violating Chiral Lagrangian for ALPs

In this section, we are going to build the most general chiral Lagrangian
describing CP-violating interactions of a light ALP with mesons and baryons.
First, let us consider the $SU(3)_{c}\times U(1)_{\text{em}}$ invariant
dimension-5 effective Lagrangian accounting for CP-violating interactions of
an ALP with photons, gluons and light quarks [43]:

$\begin{split}\mathcal{L}_{\text{ALP}}^{\text{QCD}}&=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}M_{\phi}^{2}\phi^{2}+e^{2}\frac{\tilde{C}_{\gamma}}{\Lambda}\,\phi\,F^{\mu\nu}\tilde{F}_{\mu\nu}+g_{s}^{2}\frac{\tilde{C}_{g}}{\Lambda}\,\phi\,G_{a}^{\mu\nu}\tilde{G}^{a}_{\mu\nu}+\frac{\partial_{\mu}\phi}{\Lambda}\bar{q}\,\gamma^{\mu}(Y_{S}+Y_{P}\gamma_{5})\,q\\\
&\quad+e^{2}\frac{C_{\gamma}}{\Lambda}\,\phi\,F^{\mu\nu}F_{\mu\nu}+g_{s}^{2}\frac{C_{g}}{\Lambda}\,\phi\,G_{a}^{\mu\nu}G^{a}_{\mu\nu}+\frac{v}{\Lambda}\,\phi\,\bar{q}\,y_{S}\,q+\mathcal{O}\left(\frac{1}{\Lambda^{2}}\right)\,,\end{split}$
(2.1)

that is valid above the GeV scale. Here, $\phi$ is the ALP field, while
$\Lambda\gg v\approx 246$ GeV is the EFT cutoff scale, where the electroweak
vacuum expectation value keeps track of the $SU(2)_{L}\times U(1)_{Y}$ origin
of the effective operator. Moreover, $q^{T}=(u,d,s)$ and $Y_{S}$, $Y_{P}$ and
$y_{S}$ are hermitian matrices. $G_{\mu\nu}^{a}$ and $F_{\mu\nu}$ are the QCD
and QED field-strength tensors, while
$\tilde{G}_{\mu\nu}^{a}=1/2\,\varepsilon_{\mu\nu\rho\sigma}G^{\rho\sigma}_{a}$
and $\tilde{F}_{\mu\nu}=1/2\,\varepsilon_{\mu\nu\rho\sigma}F^{\rho\sigma}$ are
their duals. Note that the interaction terms in the first line of Eq. (2.1)
preserve the ALP shift symmetry (up to non-perturbative effects in the case of
$\phi G\tilde{G}$), while the mass term, $M_{\phi}$, and the interaction terms
in the second line of Eq. (2.1) explicitly break it. See Ref. [47] for a more
detailed account of the shift-breaking orders in the ALP effective Lagrangian.

The derivation of the associated chiral Lagrangian below the GeV scale is
implemented in a few steps. In particular, it is convenient to remove the
gluonic scalar and pseudoscalar densities, $\phi GG$ and $\phi G\tilde{G}$
respectively, in favor of other quantities.

As far as the coupling $\phi GG$ is concerned, we exploit the trace anomaly
which allows to trade $\phi GG$ for the trace of the symmetrized QCD energy-
momentum tensor $\theta^{\mu\nu}$ [58]:

$\theta^{\mu}_{\mu}=\sum_{q}m_{q}\bar{q}q-\frac{\alpha_{s}}{8\pi}\,\beta^{0}_{\text{QCD}}\,G^{\mu\nu}_{a}G_{\mu\nu}^{a}-\frac{\alpha_{\text{em}}}{8\pi}\,\beta^{0}_{\text{QED}}\,F^{\mu\nu}F_{\mu\nu}\,,$
(2.2)

where $\beta^{0}_{\text{QCD}}=(11N_{c}-2n_{f})/3$ and
$\beta^{0}_{\text{QED}}=-4/3\sum_{q=\\{u,d,s\\}}Q_{q}^{2}N_{c}^{q}$ are the
leading-order beta function coefficients for QCD and QED with three active
light quarks, respectively. As a result, we obtain

$\begin{split}\mathcal{L}_{\text{ALP}}^{\text{QCD}}&=e^{2}\frac{C^{\prime}_{\gamma}}{\Lambda}\,\phi\,F^{\mu\nu}F_{\mu\nu}+e^{2}\frac{\tilde{C}_{\gamma}}{\Lambda}\,\phi\,F^{\mu\nu}\tilde{F}_{\mu\nu}+g_{s}^{2}\frac{\tilde{C}_{g}}{\Lambda}\,\phi\,G_{a}^{\mu\nu}\tilde{G}^{a}_{\mu\nu}-\frac{8\pi}{\alpha_{s}}\frac{g_{s}^{2}C_{g}}{\beta^{0}_{\text{QCD}}}\,\frac{\phi}{\Lambda}\,\theta^{\mu}_{\mu}\\\
&\quad+\frac{\partial_{\mu}\phi}{\Lambda}\bar{q}\,\gamma^{\mu}(Y_{S}+Y_{P}\gamma_{5})\,q+\phi\,\frac{v}{\Lambda}\bar{q}\,\mathcal{Z}\,q+\mathcal{O}\left(\frac{1}{\Lambda^{2}}\right)\,,\end{split}$
(2.3)

where, defining $M_{q}=\text{diag }(m_{u},m_{d},m_{s})$, we have:

$\begin{split}C^{\prime}_{\gamma}&=C_{\gamma}-\frac{\beta^{0}_{\text{QED}}}{\beta^{0}_{\text{QCD}}}\frac{g_{s}^{2}}{e^{2}}C_{g}\,,\\\
\mathcal{Z}&=y_{S}+\frac{1}{\beta^{0}_{\text{QCD}}}\frac{8\pi}{\alpha_{s}}\frac{M_{q}}{v}g_{s}^{2}C_{g}\,.\end{split}$
(2.4)

Instead, the coupling $\phi G\tilde{G}$ can be removed performing the
following ALP-dependent quark redefinition:

$q\rightarrow\exp\left[i\frac{\phi}{\Lambda}\left(Q_{V}+\lambda_{g}Q_{A}\gamma_{5}\right)\right]q\,,$
(2.5)

where $Q_{V}$ and $Q_{A}$ are arbitrary $3\times 3$ hermitian matrices. Such a
field redefinition is anomalous under both $SU(3)_{c}$ and $U(1)_{\text{em}}$,
therefore inducing the following shifts in the Lagrangian:

$\begin{split}\delta\mathcal{L}_{g}&=-g_{s}^{2}\,\frac{\phi}{\Lambda}G\tilde{G}\frac{\lambda_{g}}{16\pi^{2}}\,\text{tr}(Q_{A})\,,\\\
\delta\mathcal{L}_{\gamma}&=-e^{2}\,\frac{\phi}{\Lambda}F\tilde{F}\frac{N_{c}\lambda_{g}}{8\pi^{2}}\,\text{tr}(Q_{A}Q_{q}^{2})\,,\end{split}$
(2.6)

where $Q_{q}=\text{diag }(2/3,-1/3,-1/3)$ is the charge matrix for light
quarks. Requiring that $\lambda_{g}=32\pi^{2}\tilde{C}_{g}$ and
$\text{tr}(Q_{A})=\frac{1}{2}$, the term $\phi G\tilde{G}$ simply drops out of
the Lagrangian while the coupling $\tilde{C}_{\gamma}$ is shifted into

$\tilde{C}_{\gamma}\rightarrow\tilde{C}_{\gamma}-4N_{c}\tilde{C}_{g}^{\prime}\,\text{tr}(Q_{A}Q_{q}^{2})\,.$
(2.7)

The field redefinition (2.5) has yet two other consequences. Via the kinetic
term of quarks, it modifies the couplings to fermions according to

$\begin{split}Y_{S}&\rightarrow Y_{S}-Q_{V}\,,\\\ Y_{P}&\rightarrow
Y_{P}-\lambda_{g}Q_{A}=Y_{P}-32\pi^{2}\,\tilde{C}_{g}Q_{A}\,,\end{split}$
(2.8)

and it also affects the quark mass term, which gets dressed by the ALP field:

$M_{q}\quad\rightarrow\quad
e^{i\frac{\phi}{\Lambda}\lambda_{g}Q_{A}}M_{q}e^{i\frac{\phi}{\Lambda}\lambda_{g}Q_{A}}\equiv
M_{q}^{\phi}\,.$ (2.9)

Clearly, also the scalar coupling $\bar{q}\mathcal{Z}q$ and the energy-
momentum tensor are affected by such a rotation. However, it is simple to
convince oneself that these modification would only generate couplings of two
ALPs with quarks that are suppressed by at least two powers of $\Lambda$.

As a result, we end up with the following quark Lagrangian

$\begin{split}\mathcal{L}_{\text{ALP}}^{\text{QCD}}&=e^{2}\frac{c_{\gamma}}{\Lambda}\,\phi\,F^{\mu\nu}F_{\mu\nu}+e^{2}\frac{\tilde{c}_{\gamma}}{\Lambda}\,\phi\,F^{\mu\nu}\tilde{F}_{\mu\nu}+\frac{\partial_{\mu}\phi}{\Lambda}\,\bar{q}\gamma^{\mu}\left(Y_{V}+Y_{A}\gamma_{5}\right)q-\kappa\,\frac{\phi}{\Lambda}\,\theta^{\mu}_{\mu}+\frac{v}{\Lambda}\,\phi\,\bar{q}\mathcal{Z}q\,,\end{split}$
(2.10)

where we have defined the following parameters:

$\displaystyle c_{\gamma}$ $\displaystyle=$
$\displaystyle\,C_{\gamma}-\frac{\beta^{0}_{\text{QED}}}{\beta^{0}_{\text{QCD}}}\frac{g_{s}^{2}}{e^{2}}C_{g}\,,$
(2.11) $\displaystyle\tilde{c}_{\gamma}$ $\displaystyle=$
$\displaystyle\,\tilde{C}_{\gamma}-4N_{c}\text{ tr
}(Q_{A}Q_{q}^{2})\tilde{C}_{g}\,,$ (2.12) $\displaystyle Y^{ij}_{V}$
$\displaystyle=$ $\displaystyle Y^{ij}_{S}-Q^{ij}_{V}\,,$ (2.13)
$\displaystyle Y^{ij}_{A}$ $\displaystyle=$ $\displaystyle
Y_{P}^{ij}-32\pi^{2}\,Q_{A}^{ij}\tilde{C}_{g}\,,$ (2.14)
$\displaystyle\mathcal{Z}$ $\displaystyle=$ $\displaystyle
y_{S}+\frac{g_{s}^{2}C_{g}}{\beta^{0}_{\text{QCD}}}\frac{8\pi}{\alpha_{s}}\frac{M_{q}}{v}=y_{S}+\kappa\,\frac{M_{q}}{v}\,.$
(2.15)

Finally, making use of the completeness relation
$2\,T^{a}_{ab}T^{a}_{cd}=\delta_{ad}\delta_{bc}-1/N\delta_{ab}\delta_{cd}$
valid for the $SU(N)$ group, the quark currents can be written as

$\begin{split}&\bar{q}\,\gamma^{\mu}Y_{V}\,q=2\,\text{Tr}(Y_{V}T_{a})\,\bar{q}\,T^{a}\gamma^{\mu}q+\frac{1}{N}\text{Tr}(Y_{V})\,\bar{q}\,\gamma^{\mu}q\\\
&\bar{q}\,\gamma^{\mu}\gamma_{5}Y_{A}\,q=2\,\text{Tr}(Y_{A}T_{a})\,\bar{q}\,T^{a}\gamma^{\mu}\gamma_{5}q+\frac{1}{N}\text{Tr}(Y_{A})\,\bar{q}\,\gamma^{\mu}\gamma_{5}q\end{split}$
(2.16)

which in turn modifies (2.10) into

$\begin{split}\mathcal{L}_{\text{ALP}}^{\text{QCD}}&=\frac{\partial_{\mu}\phi}{\Lambda}\,\left[2\,\text{Tr}(Y_{V}T_{a})\,j_{V,a}^{\mu}+\frac{1}{N}\,\text{Tr}(Y_{V})\,j_{V}^{\mu}+2\,\text{Tr}(Y_{A}T_{a})\,j_{A,a}^{\mu}+\frac{1}{N}\,\text{Tr}(Y_{A})\,j_{A}^{\mu}\right]\\\
&\quad+\frac{v}{\Lambda}\,\phi\,\bar{q}\mathcal{Z}q-\kappa\,\frac{\phi}{\Lambda}\,\theta^{\mu}_{\mu}+e^{2}\frac{c_{\gamma}}{\Lambda}\,\phi\,F^{\mu\nu}F_{\mu\nu}+e^{2}\frac{\tilde{c}_{\gamma}}{\Lambda}\,\phi\,F^{\mu\nu}\tilde{F}_{\mu\nu}\,,\end{split}$
(2.17)

where we have defined the following currents:

$\begin{split}&j_{V,a}^{\mu}=\bar{q}\,T^{a}\gamma^{\mu}q\,,\qquad\qquad\qquad
j_{V}^{\mu}=\bar{q}\gamma^{\mu}q\,,\\\
&j_{A,a}^{\mu}=\bar{q}\,T^{a}\gamma^{\mu}\gamma_{5}\,q\,,\qquad\qquad\quad
j_{A}^{\mu}=\bar{q}\,\gamma^{\mu}\gamma_{5}\,q\,.\end{split}$ (2.18)

In the following, we will trade quarks for hadrons (mesons and baryons)
exploiting the hadron-quark duality.

###  2.1 ALP interactions with Mesons

At energies above few GeV, ALP interactions with quarks are correctly
described by the Lagrangian of Eq. (2.1). Instead, for the description of
strong interactions at energies below the GeV scale, we can exploit the
$\chi\text{pt}$ formalism [59, 60]. Let us consider first the massless QCD
Lagrangian based on the chiral symmetry group $G=SU(3)_{L}\times SU(3)_{R}$:

$\mathcal{L}_{\text{QCD}}^{0}=-\frac{1}{4}G^{a}_{\mu\nu}G_{a}^{\mu\nu}+i\bar{q}_{L}\gamma^{\mu}D_{\mu}q_{L}+i\bar{q}_{R}\gamma^{\mu}D_{\mu}q_{R}\,,$
(2.19)

where $q=(u,d,s)^{T}$. Chiral symmetry-breaking terms (like mass terms or
interactions with external gauge fields other than gluons) can be implemented
in $\mathcal{L}_{\text{QCD}}^{0}$ by introducing appropriate spurions
($a_{\mu},v_{\mu},s,p$) as external source fields:

$\begin{split}\mathcal{L}_{\text{QCD}}^{\text{ext}}&=\mathcal{L}_{\text{QCD}}^{0}+\bar{q}\gamma^{\mu}(v_{\mu}+a_{\mu}\gamma_{5})q+\bar{q}(s-ip\gamma_{5})q\\\
&=\mathcal{L}_{\text{QCD}}^{0}+\bar{q}\gamma^{\mu}(2r_{\mu}P_{R}+2\ell_{\mu}P_{L})q+\bar{q}(s-ip\gamma_{5})q\,.\end{split}$
(2.20)

Its chiral counterpart is then found to be

$\mathcal{L}_{\chi\text{pt}}^{\text{ext}}=\frac{f_{\pi}^{2}}{4}\text{Tr}\left[D_{\mu}\Sigma^{\dagger}D^{\mu}\Sigma+\Sigma^{\dagger}\chi+\chi^{\dagger}\Sigma\right]+\mathcal{O}(p^{4})\,,$
(2.21)

where $\Sigma(x)=\exp\left[i\lambda_{a}\pi_{a}(x)/f_{\pi}\right]$ is the
mesonic matrix transforming as $\Sigma(x)\rightarrow L\Sigma(x)R^{\dagger}$
under $SU(3)_{L}\times SU(3)_{R}$, $\pi_{a}(x)$ are the Goldstone boson fields
of $SU(3)_{L}\times SU(3)_{R}\to SU(3)_{V}$ spontaneous symmetry breaking and
$f_{\pi}=92.4\pm 0.3$ MeV is the pion decay constant. Moreover, we defined

$D_{\mu}\Sigma=\partial_{\mu}\Sigma-
ir_{\mu}\,\Sigma+i\Sigma\,\ell_{\mu}\qquad\text{and}\qquad\chi=2B_{0}\,(s+ip)\,.$
(2.22)

The duality existing between the Lagrangians (2.20) and (2.21) can be
established via the path-integral description of the generating functional
$Z\left[v_{\mu},a_{\mu},s,p\right]$:

$\exp(iZ)=\int\mathcal{D}q\,\mathcal{D}\bar{q}\,\mathcal{D}G_{\mu}\,\exp\left(i\int
d^{4}x\,\mathcal{L}_{\text{QCD}}^{\text{ext}}\right)=\int\mathcal{D}\Sigma\exp\left(i\int
d^{4}x\,\mathcal{L}_{\chi\text{pt}}^{\text{ext}}\right)\,,$ (2.23)

which can be used to construct the chiral counterparts of fermionic bilinears
by taking the functional derivatives of the QCD and the $\chi$pt actions with
respect to the appropriate external sources.

For instance, the scalar Dirac bilinear $\bar{q}_{i}\mathcal{Z}_{ij}q_{j}$ can
be obtained by taking a derivative of $\mathcal{L}^{\text{ext}}_{\text{QCD}}$
with respect to the external source $\chi$ specialized to $s=\mathcal{Z}$. Its
chiral counterpart can be constructed by simply taking the same functional
derivatives but acting on $\mathcal{L}^{\text{ext}}_{\chi\text{pt}}$:

$\bar{q}_{i}\mathcal{Z}_{ij}q_{j}=-\mathcal{Z}_{ij}\frac{\partial\mathcal{L}^{\text{ext}}_{\text{QCD}}}{\partial\mathcal{Z}_{ij}}\equiv-\mathcal{Z}_{ij}\frac{\partial\mathcal{L}^{\text{ext}}_{\chi\text{pt}}}{\partial\mathcal{Z}_{ij}}=-\frac{f_{\pi}^{2}}{2}B_{0}\text{Tr}\left[\mathcal{Z}(\Sigma+\Sigma^{\dagger})\right]\,.$
(2.24)

Quantities that are obtained by applying Noether’s theorem to the QCD
Lagrangian have their chiral counterpart in quantities that are constructed by
applying the same procedure to $\mathcal{L}_{\chi\text{pt}}$ instead.

In particular, we will need the $SU(3)_{L/R}$ currents and the energy-momentum
tensor, for which we find the following expressions:

$\begin{split}j^{\mu,a}_{L/R}&=\frac{\partial\mathcal{L}_{\chi\text{pt}}}{\partial(\partial_{\mu}\Sigma)}\delta_{L/R}\Sigma+\frac{\partial\mathcal{L}_{\chi\text{pt}}}{\partial(\partial_{\mu}\Sigma^{\dagger})}\delta_{L/R}\Sigma^{\dagger}\,,\\\
\theta^{\mu\nu}&=\frac{\partial\mathcal{L}_{\chi\text{pt}}}{\partial(\partial_{\mu}\Sigma)}\partial^{\nu}\Sigma+\frac{\partial\mathcal{L}_{\chi\text{pt}}}{\partial(\partial_{\mu}\Sigma^{\dagger})}\partial^{\nu}\Sigma^{\dagger}-\eta^{\mu\nu}\mathcal{L}_{\chi\text{pt}}\,,\end{split}$
(2.25)

where $\delta_{L}\Sigma=-i\alpha^{a}_{L}T_{a}\Sigma$,
$\delta_{R}\Sigma=i\alpha^{a}_{R}\Sigma T_{a}$,
$\delta_{L}\Sigma^{\dagger}=i\alpha^{a}_{L}\Sigma^{\dagger}T_{a}$,
$\delta_{R}\Sigma^{\dagger}=-i\alpha^{a}_{R}T_{a}\Sigma^{\dagger}$, and
$T_{a}=\lambda_{a}/2$ are the generators of the group $SU(3)_{L/R}$. As a
result, the currents of (2.2) and (2.18) and the energy-momentum tensor have
the following chiral representation:

$\begin{split}&j_{V,a}^{\mu}=j_{R,a}^{\mu}+j_{L,a}^{\mu}=i\frac{f_{\pi}^{2}}{2}\text{Tr}\left[T_{a}(D^{\mu}\Sigma^{\dagger})\Sigma+T_{a}(D^{\mu}\Sigma)\Sigma^{\dagger}\right]\,,\\\
&j_{A,a}^{\mu}=j_{R,a}^{\mu}-j_{L,a}^{\mu}=i\frac{f_{\pi}^{2}}{2}\text{Tr}\left[T_{a}(D^{\mu}\Sigma^{\dagger})\Sigma-
T_{a}(D^{\mu}\Sigma)\Sigma^{\dagger}\right]\,,\\\
&j_{V}^{\mu}=j_{R}^{\mu}+j_{L}^{\mu}=i\frac{f_{\pi}^{2}}{2}\text{Tr}\left[(D^{\mu}\Sigma^{\dagger})\Sigma+(D^{\mu}\Sigma)\Sigma^{\dagger}\right]\,,\\\
&j_{A}^{\mu}=j_{R}^{\mu}-j_{L}^{\mu}=i\frac{f_{\pi}^{2}}{2}\text{Tr}\left[(D^{\mu}\Sigma^{\dagger})\Sigma-(D^{\mu}\Sigma)\Sigma^{\dagger}\right]\,,\\\
&\theta^{\mu\nu}=\frac{f_{\pi}^{2}}{4}\text{Tr}\left[\partial^{\mu}\Sigma^{\dagger}\partial^{\nu}\Sigma+\partial^{\mu}\Sigma\partial^{\nu}\Sigma^{\dagger}\right]-\eta^{\mu\nu}\left[\frac{f_{\pi}^{2}}{4}\text{Tr}\left(\partial^{\mu}\Sigma^{\dagger}\partial_{\mu}\Sigma\right)+\frac{f_{\pi}^{2}}{2}B_{0}\text{Tr}\left(M_{q}(\Sigma+\Sigma^{\dagger})\right)\right]\,.\end{split}$
(2.26)

The bosonization procedure outlined before leads to the following chiral
Lagrangian:

$\begin{split}\mathcal{L}_{\text{ALP}}^{\chi\text{pt}}&=\frac{\partial_{\mu}\phi}{\Lambda}\,\left[2\,\text{Tr}(Y_{V}T_{a})\,j^{\mu,a}_{V}+2\,\text{Tr}(Y_{A}T_{a})\,j^{\mu,a}_{A}+\frac{1}{N}\text{Tr}(Y_{A})\,j^{\mu,a}_{A}\right]\\\
&+\kappa\,\frac{f_{\pi}^{2}}{2}\,\frac{\phi}{\Lambda}\,\left[\text{Tr}(\partial^{\mu}\Sigma\partial_{\mu}\Sigma^{\dagger})+4B_{0}\text{Tr}\left[M_{q}(\Sigma+\Sigma^{\dagger})\right]\right]-\frac{f_{\pi}^{2}}{2}\frac{v}{\Lambda}\,B_{0}\,\phi\,\text{Tr}\left[\mathcal{Z}(\Sigma+\Sigma^{\dagger})\right]\\\
&+e^{2}\frac{c_{\gamma}}{\Lambda}\,\phi\,F^{\mu\nu}F_{\mu\nu}+e^{2}\frac{\tilde{c}_{\gamma}}{\Lambda}\,\phi\,F^{\mu\nu}\tilde{F}_{\mu\nu}+\mathcal{O}\left(\frac{1}{\Lambda^{2}}\right)\,,\end{split}$
(2.27)

where $j_{V,A}^{\mu,a}$ are given in (2.26) and where we have neglected the
ALP interactions with the diagonal vectorial quark current, being the latter a
conserved current. Hereafter, we also neglect the ALP interaction with the
anomalous singlet axial current, that is related to the $\eta^{\prime}$ meson,
as in this case the convergence of the standard $\chi\text{pt}$ approach is
questionable. For a recent account of ALP-$\eta^{\prime}$ interactions see
however Ref. [61].

Moreover, the Lagrangian density in (2.27) has to be supplemented with the
ALP-dependent mass term of the QCD Lagrangian:

$\begin{split}\mathcal{L}_{\text{Mass}}^{\chi\text{pt}}&=\frac{f_{\pi}^{2}}{2}B_{0}\text{Tr}\left[M^{\phi}_{q}\Sigma^{\dagger}+\Sigma
M^{\phi\dagger}_{q}\right]\\\
&=\frac{f_{\pi}^{2}}{2}B_{0}\text{Tr}\left[M_{q}(\Sigma^{\dagger}+\Sigma)\right]+i\frac{\phi}{\Lambda}\lambda_{g}\frac{f_{\pi}^{2}}{2}B_{0}\text{Tr}\left[\left\\{M_{q},Q_{A}\right\\}(\Sigma^{\dagger}-\Sigma)\right]\\\
&-\frac{1}{2}\frac{\phi^{2}}{\Lambda^{2}}\lambda_{g}^{2}\frac{f_{\pi}^{2}}{2}B_{0}\text{Tr}\left[\left\\{\left\\{M_{q},Q_{A}\right\\},Q_{A}\right\\}(\Sigma^{\dagger}+\Sigma)\right]+\mathcal{O}\left(\frac{1}{\Lambda^{2}}\right)\,.\end{split}$
(2.28)

###  2.2 ALP interactions with Baryons

In this Section, we will concisely deal with the construction of an EFT
describing the interaction of baryons with mesons (see, e.g. [62, 63]) as well
as the inclusion of ALP interactions. The baryon octet $\mathcal{B}(x)$ is
conveniently described by a $3\times 3$ matrix transforming in the adjoint
representation of $SU(3)_{V}$

$\mathcal{B}=\begin{bmatrix}\frac{\Sigma_{0}}{\sqrt{2}}+\frac{\Lambda_{0}}{\sqrt{6}}&\Sigma^{+}&p\\\
\Sigma^{-}&-\frac{\Sigma_{0}}{\sqrt{2}}+\frac{\Lambda_{0}}{\sqrt{6}}&n\\\
\Xi^{-}&\Xi^{0}&-\frac{2}{\sqrt{6}}\Lambda_{0}\end{bmatrix}\,.$ (2.29)

Differently from the case of mesons, we can no longer consistently organize
the chiral Lagrangian for relativistic baryons in a series expansion in the
small parameter $p/\Lambda_{\chi}$, with $\Lambda_{\chi}=4\pi f_{\pi}\approx
1$ GeV. Indeed, higher dimension operators for relativistic baryons would
necessarily be of order $m_{B}/\Lambda_{\chi}\sim\mathcal{O}(1)$.

A consistent derivative expansion for baryons in a chiral effective theory can
be nonetheless developed by treating the baryon fields as heavy fields [64].
The fundamental concept of this approach is that in the chiral limit the
momentum transferred from one baryon to another via pion exchange is small if
compared to the average baryon mass, so that the velocity of baryons can be
safely considered to be conserved by such processes (it can however change as
a consequence of external currents or interactions, other than strong ones).

With this idea in mind one can then parametrize the baryon momentum as
$p^{\mu}=m_{B}v^{\mu}+k^{\mu}$, where $v^{\mu}$ is the velocity of the baryon,
$m_{B}$ its mass and $k^{\mu}$ the residual momentum measuring the amount by
which the baryon is off-shell because of its interactions. In the heavy baryon
limit, the Dirac propagator simplifies to

$i\frac{\not{p}+m_{B}}{p^{2}-m^{2}_{B}+i\epsilon}=i\frac{m_{B}\not{v}+m_{B}+\not{k}}{2m_{B}v\cdot
k+k^{2}+i\epsilon}\rightarrow\leavevmode\nobreak\ i\frac{1+\not{v}}{2v\cdot
k+i\epsilon}\,,$ (2.30)

where in the last step we have expanded for $k\ll m_{B}$. Hence, the
propagator of the heavy baryon contains a velocity-dependent projection
operator, $(1+\not{v})/2$, and it is convenient to decompose the original
quark field $\mathcal{B}(x)$ as follows

$\mathcal{B}(x)=e^{-im_{B}v\cdot x}[B_{v}(x)+H_{v}(x)]\,,$ (2.31)

where

$B_{v}(x)=e^{im_{B}v\cdot x}\frac{1+\not{v}}{2}\mathcal{B}(x)\,,\qquad
H_{v}(x)=e^{im_{B}v\cdot x}\frac{1-\not{v}}{2}\mathcal{B}(x)\,,$ (2.32)

and $B_{v}(x)$ can be identified as the definite-velocity (light-component)
baryon field. The exponential prefactor subtracts $m_{B}v^{\mu}$ from the
heavy quark momentum. As a result, the $B_{v}(x)$ field induces effects at
leading order while the effects of $H_{v}(x)$ are suppressed by powers of
$1/m_{B}$. Finally, neglecting $H_{v}(x)$ and substituting Eq. (2.31) into the
Lagrangian for the heavy baryon field,
$\bar{\mathcal{B}}(i\not{D}-m_{B})\mathcal{B}$, gives
$\bar{B}_{v}i\not{D}B_{v}$.

The interactions of baryons with mesons are accounted for by introducing the
quantities

$\begin{split}\xi(x)&=\exp\left[i\frac{\pi(x)}{2f_{\pi}}\right]\,,\\\
\mathcal{A}^{\mu}&=\frac{i}{2}(\xi\partial^{\mu}\xi^{\dagger}-\xi^{\dagger}\partial^{\mu}\xi)=\frac{\partial^{\mu}\pi}{2f_{\pi}}+\dots\,,\\\
\mathcal{V}^{\mu}&=\frac{1}{2}(\xi\partial^{\mu}\xi^{\dagger}+\xi^{\dagger}\partial^{\mu}\xi)=\frac{1}{8}\frac{\left[\pi,\partial^{\mu}\pi\right]}{f_{\pi}^{2}}+\dots\,,\end{split}$
(2.33)

which allow one to build the leading-order heavy baryon Lagrangian [65, 66]:

$\mathcal{L}_{\text{HB}}=i\text{Tr}(\bar{B}_{v}\gamma^{\mu}D_{\mu}B_{v})-D\,\text{Tr}(\bar{B}_{v}\gamma^{\mu}\gamma_{5}\left\\{\mathcal{A}_{\mu},B_{v}\right\\})-F\,\text{Tr}(\bar{B}_{v}\gamma^{\mu}\gamma_{5}\left[\mathcal{A}_{\mu},B_{v}\right])\,,$
(2.34)

where the covariant derivative is defined as
$D_{\mu}X=\partial_{\mu}X+\left[\mathcal{V}_{\mu},X\right]$. Noether’s
currents for baryons can be constructed by following the usual procedure,
yielding:

$\begin{split}(j^{a,\mu}_{A})_{\text{B}}&=\frac{1}{2}\text{Tr}\left(\bar{B}_{v}\gamma^{\mu}\left[\xi
T^{a}\xi^{\dagger}-\xi^{\dagger}T^{a}\xi,B_{v}\right]\right)+\frac{D}{2}\text{Tr}\left(\bar{B}_{v}\gamma^{\mu}\gamma_{5}\left\\{\xi
T^{a}\xi^{\dagger}+\xi^{\dagger}T^{a}\xi,B_{v}\right\\}\right)\\\
&\qquad+\frac{F}{2}\text{Tr}\left(\bar{B}_{v}\gamma^{\mu}\gamma_{5}\left[\xi
T^{a}\xi^{\dagger}+\xi^{\dagger}T^{a}\xi,B_{v}\right]\right)\,,\\\
(j^{a,\mu}_{V})_{\text{B}}&=-\frac{1}{2}\text{Tr}\left(\bar{B}_{v}\gamma^{\mu}\left[\xi
T^{a}\xi^{\dagger}+\xi^{\dagger}T^{a}\xi,B_{v}\right]\right)-\frac{D}{2}\text{Tr}\left(\bar{B}_{v}\gamma^{\mu}\gamma_{5}\left\\{\xi
T^{a}\xi^{\dagger}-\xi^{\dagger}T^{a}\xi,B_{v}\right\\}\right)\\\
&\qquad-\frac{F}{2}\text{Tr}\left(\bar{B}_{v}\gamma^{\mu}\gamma_{5}\left[\xi
T^{a}\xi^{\dagger}-\xi^{\dagger}T^{a}\xi,B_{v}\right]\right)\,,\\\
(j^{\mu}_{A})_{\text{B}}&=\frac{1}{2}\text{Tr}\left(\bar{B}_{v}\gamma^{\mu}\left[\xi\xi^{\dagger}-\xi^{\dagger}\xi,B_{v}\right]\right)+\frac{D}{2}\text{Tr}\left(\bar{B}_{v}\gamma^{\mu}\gamma_{5}\left\\{\xi\xi^{\dagger}+\xi^{\dagger}\xi,B_{v}\right\\}\right)\\\
&\qquad+\frac{F}{2}\text{Tr}\left(\bar{B}_{v}\gamma^{\mu}\gamma_{5}\left[\xi\xi^{\dagger}+\xi^{\dagger}\xi,B_{v}\right]\right)\,,\\\
(j^{\mu}_{V})_{\text{B}}&=-\frac{1}{2}\text{Tr}\left(\bar{B}_{v}\gamma^{\mu}\left[\xi\xi^{\dagger}+\xi^{\dagger}\xi,B_{v}\right]\right)-\frac{D}{2}\text{Tr}\left(\bar{B}_{v}\gamma^{\mu}\gamma_{5}\left\\{\xi\xi^{\dagger}-\xi^{\dagger}\xi,B_{v}\right\\}\right)\\\
&\qquad-\frac{F}{2}\text{Tr}\left(\bar{B}_{v}\gamma^{\mu}\gamma_{5}\left[\xi\xi^{\dagger}-\xi^{\dagger}\xi,B_{v}\right]\right)\,.\end{split}$
(2.35)

In a two-flavour setting the fundamental baryonic fields are the proton and
the neutron, combined in the nucleon doublet $\mathcal{N}=(p,n)^{T}$. Its
heavy-baryon counterpart is defined as before, $N_{v}(x)=\exp(im_{N}v\cdot
x)\frac{1+\not{v}}{2}\mathcal{N}$ and the leading-order Lagrangian is found to
be [65, 66]

$\mathcal{L}_{\text{HN}}=i\bar{N}_{v}\gamma^{\mu}D_{\mu}N_{v}-g_{A}\bar{N}_{v}\gamma^{\mu}\gamma_{5}\mathcal{A}_{\mu}N_{v}\,,$
(2.36)

where now $D_{\mu}=\partial_{\mu}+\mathcal{V}_{\mu}$. In this case the
Noether’s currents read:

$\begin{split}(j^{a,\mu}_{A})_{\text{N}}&=\frac{1}{2}\bar{N}_{v}\gamma^{\mu}\left[\xi
T^{a}\xi^{\dagger}-\xi^{\dagger}T^{a}\xi\right]N_{v}+\frac{g_{A}}{2}\bar{N}_{v}\gamma^{\mu}\gamma_{5}\left[\xi
T^{a}\xi^{\dagger}+\xi^{\dagger}T^{a}\xi\right]N_{v}\,,\\\
(j^{a,\mu}_{V})_{\text{N}}&=-\frac{1}{2}\bar{N}_{v}\gamma^{\mu}\left[\xi
T^{a}\xi^{\dagger}+\xi^{\dagger}T^{a}\xi\right]N_{v}-\frac{g_{A}}{2}\bar{N}_{v}\gamma^{\mu}\gamma_{5}\left[\xi
T^{a}\xi^{\dagger}-\xi^{\dagger}T^{a}\xi\right]N_{v}\,,\\\
(j^{\mu}_{A})_{\text{N}}&=\frac{1}{2}\bar{N}_{v}\gamma^{\mu}\left[\xi\xi^{\dagger}-\xi^{\dagger}\xi\right]N_{v}+\frac{g_{A}}{2}\bar{N}_{v}\gamma^{\mu}\gamma_{5}\left[\xi\xi^{\dagger}+\xi^{\dagger}\xi\right]N_{v}\,,\\\
(j^{\mu}_{V})_{\text{N}}&=-\frac{1}{2}\bar{N}_{v}\gamma^{\mu}\left[\xi\xi^{\dagger}+\xi^{\dagger}\xi\right]N_{v}-\frac{g_{A}}{2}\bar{N}_{v}\gamma^{\mu}\gamma_{5}\left[\xi\xi^{\dagger}-\xi^{\dagger}\xi\right]N_{v}\,.\end{split}$
(2.37)

The ALP interactions with baryons can be obtained starting from (2.1) and
exploiting the hadron-quark duality as well as the quark rotation (2.5) to get
rid of the term $\phi G\tilde{G}$. As a result, we obtain

$\begin{split}\mathcal{L}_{\text{ALP}}^{\text{QCD}}&=\frac{\partial_{\mu}\phi}{\Lambda}\,\left[2\,\text{Tr}(Y_{V}T_{a})\,(j_{V,a}^{\mu})_{\text{B}}+\frac{1}{N}\,\text{Tr}(Y_{V})\,(j_{V}^{\mu})_{\text{B}}+2\,\text{Tr}(Y_{A}T_{a})\,(j_{A,a}^{\mu})_{\text{B}}+\frac{1}{N}\,\text{Tr}(Y_{A})\,(j_{A}^{\mu})_{\text{B}}\right]\\\
&+\frac{v}{\Lambda}\,\phi\,\bar{q}y_{S}q+g^{2}_{s}\frac{C_{g}}{\Lambda}\,\phi\,G^{a}_{\mu\nu}G_{a}^{\mu\nu}+e^{2}\frac{c_{\gamma}}{\Lambda}\,\phi\,F^{\mu\nu}F_{\mu\nu}+e^{2}\frac{\tilde{c}_{\gamma}}{\Lambda}\,\phi\,F^{\mu\nu}\tilde{F}_{\mu\nu}\,.\end{split}$
(2.38)

Note that the trace anomaly condition (2.2) has not been employed so far to
remove the term $\phi GG$. The hadronization of such term in the framework of
heavy baryons will be discussed in the next Sections. Moreover, in contrast to
the case of the chiral Lagrangian for mesons, there are now no interaction
terms stemming from heavy baryons masses as they are suppressed by powers of
$1/m_{B}$.

##  3 ALP interactions for $N_{f}=2$

The general results presented in the previous Section can now be specialized
to the low-energy setting of interest. Hereafter, we will focus on a two-
flavours setting ($N_{f}=2$) where the strange quark and its phenomenological
impact can be safely neglected. Throughout this Section, we take
$Q_{V}=\text{diag}\,(q^{u}_{V},q^{d}_{V})$ and
$Q_{A}=\text{diag}\,(q^{u}_{A},q^{d}_{A})$.

###  3.1 ALP interactions with Mesons

We are now ready to expand Eq. (2.27) and (2.28) in terms of the constituent
meson fields. Keeping ALP interactions with no more than three mesons, and
employing the Gell-Mann Okubo relation $B_{0}=m_{\pi}^{2}/(m_{u}+m_{d})$, we
find

$\begin{split}\mathcal{L}_{\text{ALP}}^{\chi\text{pt}}&=\kappa\,\frac{\phi}{\Lambda}\,\left[\partial^{\mu}\pi_{0}\partial_{\mu}\pi_{0}+2\partial^{\mu}\pi^{+}\partial_{\mu}\pi^{-}-2m_{\pi}^{2}(\pi_{0}^{2}+2\pi^{+}\pi^{-})\right]+\frac{v}{2}\frac{\phi}{\Lambda}m_{\pi}^{2}\frac{\mathcal{Z}_{u}+\mathcal{Z}_{d}}{m_{u}+m_{d}}(\pi_{0}^{2}+2\pi^{+}\pi^{-})\\\
&+(Y_{A}^{u}-Y_{A}^{d})\frac{f_{\pi}}{\Lambda}\,\partial^{\mu}\phi\,\partial_{\mu}\pi_{0}+2\frac{(Y_{A}^{u}-Y_{A}^{d})}{3f_{\pi}\Lambda}\,\partial^{\mu}\phi\,(\pi_{0}\pi^{-}\partial_{\mu}\pi^{+}+\pi_{0}\pi^{+}\partial_{\mu}\pi^{-}-2\pi^{+}\pi^{-}\partial_{\mu}\pi^{0})\\\
&+i\frac{(q_{V}^{u}-q_{V}^{d})}{\Lambda}\,\partial^{\mu}\phi\,(\pi^{-}D_{\mu}\pi^{+}-\pi^{+}D_{\mu}\pi^{-})\,,\end{split}$
(3.1)

where $D_{\mu}\pi^{\pm}=(\partial_{\mu}\pm ieA_{\mu})\pi^{\pm}$. Moreover,
from the mass term we obtain the following interactions

$\mathcal{L}_{\text{m}}^{\chi\text{pt}}=2m_{\pi}^{2}\lambda_{g}\,\frac{m_{u}q_{u}-m_{d}q_{d}}{m_{u}+m_{d}}\,\frac{f_{\pi}}{\Lambda}\left[\phi\,\pi_{0}-\frac{1}{6f^{2}_{\pi}}\,\phi\,\pi_{0}(\pi^{2}_{0}+2\pi^{+}\pi^{-})\right]-2m_{\pi}^{2}\lambda_{g}^{2}\frac{f_{\pi}^{2}}{\Lambda^{2}}\frac{m_{u}q_{u}^{2}+m_{d}q_{d}^{2}}{m_{u}+m_{d}}\,\phi^{2}\,.$
(3.2)

Non-perturbative effects generate a mass term for the ALP even if it was
originally absent, in analogy to the case of the QCD axion. Eqs. (3.1) and
(3.2) contain both a kinetic mixing and a mass mixing term coupling the ALP to
the neutral pion. These can be schematically written as

$\mathcal{L}^{\text{ALP
mixing}}_{\chi\text{pt}}=\frac{1}{2}\partial^{\mu}\varphi^{T}\,\mathbf{Z}\,\partial_{\mu}\varphi-\frac{1}{2}\varphi^{T}\,\mathbf{M}\,\varphi\,,\qquad\text{with}\qquad\varphi=\begin{pmatrix}\phi\\\
\pi_{0}\end{pmatrix}\,,$ (3.3)

where

$\begin{split}&\mathbf{Z}=\begin{bmatrix}1&\epsilon\\\
\epsilon&1\end{bmatrix}\,,\qquad\qquad\qquad\mathbf{M}=\begin{bmatrix}m_{\phi}^{2}&-\epsilon\,\alpha\\\
-\epsilon\,\alpha&m_{\pi}^{2}\end{bmatrix}\,,\end{split}$ (3.4)

and where we have defined

$\epsilon=(Y_{A}^{u}-Y_{A}^{d})\frac{f_{\pi}}{\Lambda}\,,\quad\alpha=2\frac{m_{\pi}^{2}}{(Y_{A}^{u}-Y_{A}^{d})}\lambda_{g}\frac{m_{u}q^{u}_{A}-m_{d}q_{A}^{d}}{m_{u}+m_{d}}\,,\quad
m_{\phi}^{2}=M_{\phi}^{2}+4m_{\pi}^{2}(\lambda_{g})^{2}\frac{f_{\pi}^{2}}{\Lambda^{2}}\frac{m_{u}(q_{A}^{u})^{2}+m_{d}(q_{A}^{d})^{2}}{m_{u}+m_{d}}\,.$
(3.5)

Here, we have included a potential explicit mass term for the ALP, denoted as
$M_{\phi}$. After shifting the field doublet $\varphi$ to make the kinetic
terms canonical and diagonalising the mass matrix ${\mathbf{M}}$, we end up
with the following Lagrangian

$\begin{split}\mathcal{L}_{\text{ALP+
Mass}}^{\chi\text{pt}}&=-\frac{m^{2}_{\phi}}{2}\phi^{2}+\kappa\,\frac{\phi}{\Lambda}\,\left[\partial^{\mu}\pi_{0}\partial_{\mu}\pi_{0}+2\partial^{\mu}\pi^{+}\partial_{\mu}\pi^{-}-2m_{\pi}^{2}(\pi_{0}^{2}+2\pi^{+}\pi^{-})\right]\\\
&+\frac{v}{2}\frac{\phi}{\Lambda}m_{\pi}^{2}\frac{\mathcal{Z}_{u}+\mathcal{Z}_{d}}{m_{u}+m_{d}}(\pi_{0}^{2}+2\pi^{+}\pi^{-})-\alpha\frac{Y_{A}^{u}-Y_{A}^{d}}{6f_{\pi}\Lambda}\,\phi\,\pi_{0}(\pi_{0}^{2}+2\pi^{+}\pi^{-})\\\
&+2\frac{(Y_{A}^{u}-Y_{A}^{d})}{3f_{\pi}\Lambda}\,\partial^{\mu}\phi\,(\pi_{0}\pi^{-}\partial_{\mu}\pi^{+}+\pi_{0}\pi^{+}\partial_{\mu}\pi^{-}-2\pi^{+}\pi^{-}\partial_{\mu}\pi^{0})\\\
&+i\frac{(q_{V}^{u}-q_{V}^{d})}{\Lambda}\,\partial^{\mu}\phi\,(\pi^{-}D_{\mu}\pi^{+}-\pi^{+}D_{\mu}\pi^{-})+e^{2}\frac{c_{\gamma}}{\Lambda}\,\phi\,F^{\mu\nu}F_{\mu\nu}+e^{2}\frac{\tilde{c}_{\gamma}}{\Lambda}\,\phi\,F^{\mu\nu}\tilde{F}_{\mu\nu}\,,\end{split}$
(3.6)

where in the above expression the fields and masses are the physical ones,
defined as111Note that this formula is valid for
$|m_{\pi}^{2}-m_{\phi}^{2}|\gg m_{\pi}^{2}f_{\pi}/\Lambda$, that is satisfied
barring a fine-tuned region of parameter space.

$\begin{split}&\phi\rightarrow\phi+\epsilon\frac{m_{\pi}^{2}+\alpha}{m_{\pi}^{2}-m_{\phi}^{2}}\pi_{0}\,,\qquad\quad\leavevmode\nobreak\
m_{\phi}^{2}\rightarrow
m_{\phi}^{2}-\epsilon^{2}\frac{(m_{\phi}^{2}+\alpha)^{2}}{m_{\pi}^{2}-m_{\phi}^{2}}\,,\\\
&\pi_{0}\rightarrow\pi_{0}-\epsilon\frac{m_{\phi}^{2}+\alpha}{m_{\pi}^{2}-m_{\phi}^{2}}\phi\,,\qquad\quad
m_{\pi}^{2}\rightarrow
m_{\pi}^{2}+\epsilon^{2}\frac{(m_{\pi}^{2}+\alpha)^{2}}{m_{\pi}^{2}-m_{\phi}^{2}}\,.\end{split}$
(3.7)

Note that the kinetic and mass mixings between the neutral pion and the ALP
induce also a modification of the following four-pion interactions

$\displaystyle\mathcal{L}_{4\pi}$
$\displaystyle=\frac{m_{\pi}^{2}}{24f_{\pi}^{2}}(\pi_{0}^{4}+4\pi_{0}^{2}\pi^{+}\pi^{-}+4\pi^{+}\pi^{+}\pi^{-}\pi^{-})+\frac{1}{6f_{\pi}^{2}}\left[\pi^{-}\pi^{-}D^{\mu}\pi^{+}D_{\mu}\pi^{+}+\pi^{+}\pi^{+}D^{\mu}\pi^{-}D_{\mu}\pi^{-}-2\pi^{+}\pi^{-}\partial^{\mu}\pi_{0}\partial_{\mu}\pi^{0}\right.$
$\displaystyle\left.+2\pi_{0}\partial^{\mu}\pi_{0}(\pi^{-}D_{\mu}\pi^{+}+\pi^{+}D_{\mu}\pi^{-})-2D^{\mu}\pi^{+}D_{\mu}\pi^{-}(\pi_{0}^{2}+\pi^{+}\pi^{-})\right]+\frac{e^{2}}{16\pi^{2}}\frac{\pi_{0}}{f_{\pi}}F^{\mu\nu}\tilde{F}_{\mu\nu}\,,$
(3.8)

which leads to the additional ALP interaction terms

$\begin{split}\mathcal{L}^{\text{ALP}}_{4\pi}&=-\epsilon\frac{m_{\phi}^{2}+\alpha}{m_{\pi}^{2}-m_{\phi}^{2}}\left\\{\frac{e^{2}}{16\pi^{2}}\frac{\phi}{f_{\pi}}F^{\mu\nu}\tilde{F}_{\mu\nu}+\frac{m_{\pi}^{2}}{6f_{\pi}^{2}}(\phi\,\pi_{0}^{3}+2\phi\,\pi_{0}\pi^{+}\pi^{-})-\frac{1}{3f_{\pi}^{2}}\left[-2\pi^{+}\pi^{-}\partial^{\mu}\phi\partial_{\mu}\pi^{0}\right.\right.\\\
&\left.\left.+\,\phi\,\partial^{\mu}\pi_{0}(\pi^{-}D_{\mu}\pi^{+}+\pi^{+}D_{\mu}\pi^{-})+\pi_{0}\partial^{\mu}\phi(\pi^{-}D_{\mu}\pi^{+}+\pi^{+}D_{\mu}\pi^{-})-2\phi\,\pi_{0}\,D^{\mu}\pi^{+}D_{\mu}\pi^{-}\right]\right\\}\,.\end{split}$
(3.9)

A common choice in the literature for the parameters $q_{A}^{u}$ and
$q_{A}^{d}$ is [15]

$q_{A}^{u}=\frac{1}{2}\frac{m_{d}}{m_{u}+m_{d}}\qquad\text{and}\qquad
q_{A}^{d}=\frac{1}{2}\frac{m_{u}}{m_{u}+m_{d}}\,,$ (3.10)

which corresponds to $\alpha=0$. However, as noted in [51], a more convenient
choice corresponds to set $\alpha=-m_{\phi}^{2}$ as in this case the
interactions in $\mathcal{L}^{\text{ALP}}_{4\pi}$ vanish. This choice can be
implemented by setting

$q_{A}^{u}=\frac{1}{2}\frac{m_{d}}{m_{u}+m_{d}}-\frac{m_{\phi}^{2}}{m_{\pi}^{2}-m_{\phi}^{2}}\frac{\Delta^{A}_{ud}}{2\lambda_{g}}\qquad\text{and}\qquad
q_{A}^{d}=\frac{1}{2}\frac{m_{u}}{m_{u}+m_{d}}+\frac{m_{\phi}^{2}}{m_{\pi}^{2}-m_{\phi}^{2}}\frac{\Delta^{A}_{ud}}{2\lambda_{g}}\,,$
(3.11)

where we have defined the coupling constant

$\Delta^{A}_{ud}=Y_{P}^{u}-Y_{P}^{d}-\frac{\lambda_{g}}{2}\frac{m_{d}-m_{u}}{m_{d}+m_{u}}=\frac{m_{\pi}^{2}-m_{\phi}^{2}}{m_{\pi}^{2}}(Y_{A}^{u}-Y_{A}^{d})\,.$
(3.12)

In this case the ALP-Meson interaction Lagrangian simply reduces to

$\begin{split}\mathcal{L}_{\text{ALP+
Mass}}^{\chi\text{pt}}&=-\frac{m^{2}_{\phi}}{2}\phi^{2}+\kappa\,\frac{\phi}{\Lambda}\,\left[\partial^{\mu}\pi_{0}\partial_{\mu}\pi_{0}+2\partial^{\mu}\pi^{+}\partial_{\mu}\pi^{-}\right]\\\
&+\frac{\phi}{\Lambda}\,(\pi_{0}^{2}+2\pi^{+}\pi^{-})m_{\pi}^{2}\left[\frac{v}{2}\frac{\mathcal{Z}_{u}+\mathcal{Z}_{d}}{m_{u}+m_{d}}-2\,\kappa+\pi_{0}\frac{m_{\phi}^{2}}{m_{\pi}^{2}-m_{\phi}^{2}}\frac{\Delta^{A}_{ud}}{6f_{\pi}}\right]\\\
&+\frac{2}{3}\frac{\Delta^{A}_{ud}}{f_{\pi}\Lambda}\frac{m_{\pi}^{2}}{m_{\pi}^{2}-m_{\phi}^{2}}\,\partial^{\mu}\phi\,(\pi_{0}\pi^{-}\partial_{\mu}\pi^{+}+\pi_{0}\pi^{+}\partial_{\mu}\pi^{-}-2\pi^{+}\pi^{-}\partial_{\mu}\pi^{0})\\\
&+i\frac{(q_{V}^{u}-q_{V}^{d})}{\Lambda}\,\partial^{\mu}\phi\,(\pi^{-}D_{\mu}\pi^{+}-\pi^{+}D_{\mu}\pi^{-})+e^{2}\frac{c_{\gamma}}{\Lambda}\,\phi\,F^{\mu\nu}F_{\mu\nu}+e^{2}\frac{\tilde{c}_{\gamma}}{\Lambda}\,\phi\,F^{\mu\nu}\tilde{F}_{\mu\nu}\,,\end{split}$
(3.13)

where now

$\begin{split}m^{2}_{\phi}&=M_{\phi}^{2}+m_{\pi}^{2}(\lambda_{g})^{2}\frac{m_{u}m_{d}}{(m_{u}+m_{d})^{2}}\frac{f_{\pi}^{2}}{\Lambda^{2}}+m_{\pi}^{2}\frac{M_{\phi}^{4}}{(m_{\pi}^{2}-M_{\phi}^{2})^{2}}(\Delta^{A}_{ud})^{2}\frac{f_{\pi}^{2}}{\Lambda^{2}}+\mathcal{O}\left(\frac{1}{\Lambda^{4}}\right)\,.\end{split}$
(3.14)

###  3.2 ALP interactions with Nucleons

The ALP interactions with nucleons stem from the derivative couplings to
vectorial and axial currents, the direct couplings to the QCD energy-momentum
tensor, and the quark bilinear $\bar{q}\mathcal{Z}q$, see Eq. (2.38).

Focusing on the derivative interactions of an ALP with two nucleons, we find
the following Lagrangian at leading order:

$\begin{split}\mathcal{L}_{\text{ALP}}^{\partial
N}&=\frac{\partial_{\mu}\phi}{\Lambda}\left[\text{Tr}(Y_{A}\sigma_{3})j_{A,3}^{\mu}+\frac{1}{2}\text{Tr}(Y_{A})j_{A}^{\mu}\right]\\\
&=\frac{\partial_{\mu}\phi}{\Lambda}\left[\frac{g_{A}}{2}(Y_{A}^{u}-Y_{A}^{d})(\bar{p}_{v}\gamma^{\mu}\gamma_{5}p_{v}-\bar{n}_{v}\gamma^{\mu}\gamma_{5}n_{v})+\frac{g_{0}}{2}(Y_{A}^{u}+Y_{A}^{d})(\bar{p}_{v}\gamma^{\mu}\gamma_{5}p_{v}+\bar{n}_{v}\gamma^{\mu}\gamma_{5}n_{v})\right]\,.\end{split}$
(3.15)

In order to obtain the couplings between one ALP, two nucleons and any desired
number of pions it is then enough to expand up to next-to-leading order the
mesonic terms $\xi$ and $\xi^{\dagger}$ appearing in the definitions in Eq.
(2.37), having well in mind that in this case also the singlet currents play a
role.

Now, the coefficients $g_{0}$ and $g_{A}$ appearing in Eq. (3.15) can be
rewritten in terms of the spin component of a given quark type in the nucleon,
defined as $s^{\mu}\Delta_{q}=\bra{N}\bar{q}\gamma^{\mu}\gamma_{5}q\ket{N}$,
where $N=\\{p,n\\}$. In order to do so, we recall that
$2s^{\mu}=\bra{N}\bar{N}_{v}\gamma^{\mu}\gamma_{5}N_{v}\ket{N}$ and match the
single-nucleon matrix elements in the quark-axion Lagrangian (2.38) and in the
effective ALP-nucleon Lagrangian (3.15), that is
$\bra{N}\mathcal{L}_{\text{ALP}}^{\text{Nuc.,
der.}}\ket{N}=\bra{N}\mathcal{L}_{\text{ALP}}^{\text{QCD}}\ket{N}$ [53, 18].

The outcome of this procedure sets $g_{0}=\Delta_{u}+\Delta_{d}$ and
$g_{A}=\Delta_{u}-\Delta_{d}$, allowing us to write down the derivative
coupling of ALPs with nucleons in a compact, matrix form:

$\mathcal{L}_{\text{ALP}}^{\partial
N}=\frac{\partial_{\mu}\phi}{\Lambda}\bar{N}_{v}\tilde{C}_{\phi\text{N}}\gamma^{\mu}\gamma_{5}N_{v}\,.$
(3.16)

Here, defining
$\tilde{C}_{\phi\text{N}}=\text{diag}(\tilde{C}_{\phi\text{p}},\tilde{C}_{\phi\text{n}})$
we find

$\begin{split}\tilde{C}_{\phi\text{p}}&=Y_{A}^{u}\Delta_{u}+Y_{A}^{d}\Delta_{d}=Y_{P}^{u}\Delta_{u}+Y_{P}^{d}\Delta_{d}-\frac{\lambda_{g}}{2}\left[\frac{m_{u}\Delta_{d}}{m_{u}+m_{d}}+\frac{m_{d}\Delta_{u}}{m_{u}+m_{d}}-\frac{m_{\phi}^{2}}{m_{\pi}^{2}-m_{\phi}^{2}}\frac{\Delta_{u}-\Delta_{d}}{\lambda_{g}}\Delta^{A}_{ud}\right]\,,\\\
\tilde{C}_{\phi\text{n}}&=Y_{A}^{u}\Delta_{d}+Y_{A}^{d}\Delta_{u}=Y_{P}^{u}\Delta_{d}+Y_{P}^{d}\Delta_{u}-\frac{\lambda_{g}}{2}\left[\frac{m_{u}\Delta_{u}}{m_{u}+m_{d}}+\frac{m_{d}\Delta_{d}}{m_{u}+m_{d}}+\frac{m_{\phi}^{2}}{m_{\pi}^{2}-m_{\phi}^{2}}\frac{\Delta_{u}-\Delta_{d}}{\lambda_{g}}\Delta^{A}_{ud}\right]\,,\end{split}$
(3.17)

where $\Delta_{u}=0.858(22)$ and $\Delta_{d}=-0.418(22)$ [67].

There is also a non-derivative type of interaction of the ALP with nucleons,
given by

$\mathcal{L}_{\text{ALP}}^{N}=\frac{\phi}{\Lambda}C_{\phi\text{NN}}\bar{N}_{v}N_{v}\,,$
(3.18)

which originates from CP-even couplings of the ALP field with either quarks or
gluons [68, 69]. Since the ALP has no impact on the hadronic matrix element,
we can then simply determine the effective coupling between an ALP and
nucleons by requiring the equivalence

$\sum_{q}\zeta_{q}\bar{q}q\,\phi+g_{\phi
gg}\,\phi\,GG\qquad\longrightarrow\qquad\bra{N}\sum_{q}\zeta_{q}\bar{q}q+g_{\phi
gg}\,GG\ket{N}\,\phi\,\bar{N}N\,,$ (3.19)

and computing the hadronic matrix element of the scalar quark and gluon
densities. This can be achieved by applying the relations

$\begin{split}&\bra{p}\bar{u}u\ket{p}=\frac{\sigma_{u}}{m_{u}}\,,\qquad\bra{p}\bar{d}d\ket{p}=\frac{\sigma_{d}}{m_{d}}\,,\qquad\bra{p}\bar{s}s\ket{p}=\frac{\sigma_{s}}{m_{s}}\,,\\\
&\qquad\qquad\bra{p}GG\ket{p}=-\frac{8\pi}{9\alpha_{s}}(m_{p}-\sigma_{u}-\sigma_{d}-\sigma_{s})\,,\end{split}$
(3.20)

to the Lagrangian in Eq. (2.1), obtaining the effective scalar coupling of an
ALP to protons:

$C_{\phi\text{pp}}=\frac{v}{m_{u}}y_{q,S}^{u}\,\sigma_{u}+\frac{v}{m_{d}}y_{q,S}^{d}\,\sigma_{d}+\frac{v}{m_{s}}y_{q,S}^{s}\,\sigma_{s}-\frac{32\pi^{2}C_{g}}{9}\,(m_{p}-\sigma_{u}-\sigma_{d}-\sigma_{s})\,.$
(3.21)

Due to iso-spin symmetry the difference between $C_{\phi\text{pp}}$ and
$C_{\phi\text{nn}}$ is negligible and the scalar coupling to nucleons can be
approximated by a universal value $C_{\phi\text{NN}}\simeq C_{\phi\text{pp}}$.
Explicit values for the various quantities as well as the procedure employed
in order to obtain them can be found in [69] and references therein.

Clearly, due to the ALP-neutral pion mixing, one should take into account also
the ALP-nucleon interactions that originate from the pion-nucleon ones upon
rotation to the physical basis. More explicitly, from Eq. (2.36) one has the
following pion-nucleon interaction:

$\mathcal{L}_{N\pi}\supset-
g_{A}\bar{N}_{v}\gamma^{\mu}\gamma_{5}\frac{\partial_{\mu}\pi}{2f_{\pi}}N_{v}\supset-\frac{g_{A}}{2f_{\pi}}\partial_{\mu}\pi_{0}(\bar{p}_{v}\gamma^{\mu}\gamma_{5}p_{v}-\bar{n}_{v}\gamma^{\mu}\gamma_{5}n_{v})\,,$
(3.22)

where we stopped at order $\mathcal{O}(\pi\bar{N}N)$ in the expansion in Eq.
(2.33).

The mixing described in Eq. (3.7) then necessarily shifts the couplings of the
ALP to nucleons according to

$\delta\mathcal{L}_{N\pi}=-{2f_{\pi}}\varepsilon\frac{m_{\phi}^{2}+\alpha}{m_{\pi}^{2}-m_{\phi}^{2}}\,\partial_{\mu}\phi\,(\bar{p}_{v}\gamma^{\mu}\gamma_{5}p_{v}-\bar{n}_{v}\gamma^{\mu}\gamma_{5}n_{v})\,.$
(3.23)

Once again, however, the choice $\alpha=-m_{\phi}^{2}$ allows us to get rid of
this contribution.

###  3.3 Summary and Jarlskog invariants

Throughout this Section we have constructed the low-energy effective
Lagrangian describing the interaction of an ALP with light particles: photons,
pions and nucleons. We have found a Lagrangian that can be decomposed into two
pieces featuring opposite CP transformation properties, namely

$\begin{split}\left(\mathcal{L}_{\text{ALP}}^{\chi\text{pt}}\right)^{\rm CP-
even}&=e^{2}\frac{c_{\gamma}}{\Lambda}\,\phi\,F^{\mu\nu}F_{\mu\nu}+\frac{v}{\Lambda}y_{\ell,S}^{ij}\,\phi\,\bar{\ell}_{i}\ell_{j}+\kappa\,\frac{\phi}{\Lambda}\,\left[\partial^{\mu}\pi_{0}\partial_{\mu}\pi_{0}+2\partial^{\mu}\pi^{+}\partial_{\mu}\pi^{-}\right]\\\
&+\frac{\phi}{\Lambda}\,(\pi_{0}^{2}+2\pi^{+}\pi^{-})m_{\pi}^{2}\left[\frac{v}{2}\frac{\mathcal{Z}_{u}+\mathcal{Z}_{d}}{m_{u}+m_{d}}-2\,\kappa\right]+\frac{\phi}{\Lambda}C_{\phi\text{NN}}\bar{N}_{v}N_{v}\end{split}$
(3.24)

and

$\begin{split}\left(\mathcal{L}_{\text{ALP}}^{\chi\text{pt}}\right)^{\rm CP-
odd}&=e^{2}\frac{\tilde{c}_{\gamma}}{\Lambda}\,\phi\,F^{\mu\nu}\tilde{F}_{\mu\nu}+i\frac{v}{\Lambda}y_{\ell,P}^{ij}\,\phi\,\bar{\ell}_{i}\gamma_{5}\ell_{j}+\frac{\phi}{\Lambda}\,\pi_{0}(\pi_{0}^{2}+2\pi^{+}\pi^{-})m_{\pi}^{2}\left[\frac{m_{\phi}^{2}}{m_{\pi}^{2}-m_{\phi}^{2}}\frac{\Delta^{A}_{ud}}{6f_{\pi}\Lambda}\right]\\\
&+\frac{2}{3}\frac{\Delta^{A}_{ud}}{f_{\pi}\Lambda}\frac{m_{\pi}^{2}}{m_{\pi}^{2}-m_{\phi}^{2}}\,\partial^{\mu}\phi\,(\pi_{0}\pi^{-}\partial_{\mu}\pi^{+}+\pi_{0}\pi^{+}\partial_{\mu}\pi^{-}-2\pi^{+}\pi^{-}\partial_{\mu}\pi^{0})+\frac{\partial_{\mu}\phi}{\Lambda}\bar{N}_{v}\tilde{C}_{\phi\text{N}}\gamma^{\mu}\gamma_{5}N_{v}\,,\end{split}$
(3.25)

where explicit expressions for the parameters appearing in the above
Lagrangians can be found throughout the previous Sections. Note that,
regardless of the scalar or pseudoscalar nature of the ALP field, CP is
necessarily violated if at least one CP-even and CP-odd coupling is
simultaneously present.

A particularly convenient way to account for CP-violating effects is to employ
Jarlskog invariants. See Ref. [47] for a detailed analysis at the quark level.
In our case, they are reported in Table 1. To simplify their interpretation,
we also report below the compositions of all the coefficients appearing in
Table 1 in terms of the microscopic parameters of Eq. 2.1

$\displaystyle c_{\gamma}=$
$\displaystyle\,C_{\gamma}-\frac{\beta^{0}_{\text{QED}}}{\beta^{0}_{\text{QCD}}}\frac{g_{s}^{2}}{e^{2}}C_{g}\,,$
(3.26) $\displaystyle\tilde{c}_{\gamma}=$
$\displaystyle\,\tilde{C}_{\gamma}-4N_{c}\text{ tr
}(Q_{A}Q_{q}^{2})\tilde{C}_{g}\,,$ (3.27) $\displaystyle\kappa=$
$\displaystyle\frac{8\pi}{\alpha_{s}}\frac{g_{s}^{2}C_{g}}{\beta^{0}_{\text{QCD}}}\,,$
(3.28) $\displaystyle\mathcal{Z}=$ $\displaystyle
y_{S}+\frac{g_{s}^{2}C_{g}}{\beta^{0}_{\text{QCD}}}\frac{8\pi}{\alpha_{s}}\frac{M_{q}}{v}\,,$
(3.29) $\displaystyle\Delta^{A}_{ud}=$ $\displaystyle
Y_{P}^{u}-Y_{P}^{d}-16\pi^{2}\tilde{C}_{g}\frac{m_{d}-m_{u}}{m_{d}+m_{u}}\,,\qquad\Delta_{u}=0.858(22)\,,\qquad\Delta_{d}=-0.418(22)$
(3.30) $\displaystyle\tilde{C}_{\phi\text{p}}=$ $\displaystyle
Y_{P}^{u}\Delta_{u}+Y_{P}^{d}\Delta_{d}-16\pi^{2}\tilde{C}_{g}\left[\frac{m_{u}\Delta_{d}}{m_{u}+m_{d}}+\frac{m_{d}\Delta_{u}}{m_{u}+m_{d}}-\frac{m_{\phi}^{2}}{m_{\pi}^{2}-m_{\phi}^{2}}\frac{\Delta_{u}-\Delta_{d}}{32\pi^{2}\tilde{C}_{g}}\Delta^{A}_{ud}\right]\,,$
(3.31) $\displaystyle\tilde{C}_{\phi\text{n}}=$ $\displaystyle
Y_{P}^{u}\Delta_{d}-Y_{P}^{d}\Delta_{u}-16\pi^{2}\tilde{C}_{g}\left[\frac{m_{u}\Delta_{u}}{m_{u}+m_{d}}+\frac{m_{d}\Delta_{d}}{m_{u}+m_{d}}+\frac{m_{\phi}^{2}}{m_{\pi}^{2}-m_{\phi}^{2}}\frac{\Delta_{u}-\Delta_{d}}{32\pi^{2}\tilde{C}_{g}}\Delta^{A}_{ud}\right]\,,$
(3.32) $\displaystyle C_{\phi\text{pp}}=$
$\displaystyle\frac{v}{m_{u}}y_{q,S}^{u}\,\sigma_{u}+\frac{v}{m_{d}}y_{q,S}^{d}\,\sigma_{d}+\frac{v}{m_{s}}y_{q,S}^{s}\,\sigma_{s}-\frac{32\pi^{2}C_{g}}{9}\,(m_{p}-\sigma_{u}-\sigma_{d}-\sigma_{s})\,,$
(3.33) $\displaystyle C_{\phi\text{nn}}=$
$\displaystyle\frac{v}{m_{d}}y_{q,S}^{u}\,\sigma_{d}+\frac{v}{m_{u}}y_{q,S}^{d}\,\sigma_{u}+\frac{v}{m_{s}}y_{q,S}^{s}\,\sigma_{s}-\frac{32\pi^{2}C_{g}}{9}\,(m_{p}-\sigma_{u}-\sigma_{d}-\sigma_{s})\,.$
(3.34)

| $c_{\gamma}$ | $y_{\ell,S}$ | $\kappa$ | $\mathcal{Z}$ | $C_{\phi\text{NN}}$
---|---|---|---|---|---
$\tilde{c}_{\gamma}$ | $\tilde{c}_{\gamma}\,c_{\gamma}$ | $\tilde{c}_{\gamma}\,y_{\ell,S}$ | $\tilde{c}_{\gamma}\,\kappa$ | $\tilde{c}_{\gamma}\,\mathcal{Z}$ | $\tilde{c}_{\gamma}\,C_{\phi\text{NN}}$
$y_{\ell,P}$ | $y_{\ell,P}\,c_{\gamma}$ | $y_{\ell,P}\,y_{\ell,S}$ | $y_{\ell,P}\,\kappa$ | $y_{\ell,P}\,\mathcal{Z}$ | $y_{\ell,P}\,C_{\phi\text{NN}}$
$\Delta_{ud}^{A}$ | $\Delta_{ud}^{A}\,c_{\gamma}$ | $\Delta_{ud}^{A}\,y_{\ell,S}$ | $\Delta_{ud}^{A}\,\kappa$ | $\Delta_{ud}^{A}\,\mathcal{Z}$ | $\Delta_{ud}^{A}\,C_{\phi\text{NN}}$
$\tilde{C}_{\phi\text{N}}$ | $\tilde{C}_{\phi\text{N}}\,c_{\gamma}$ | $\tilde{C}_{\phi\text{N}}\,y_{\ell,S}$ | $\tilde{C}_{\phi\text{N}}\,\kappa$ | $\tilde{C}_{\phi\text{N}}\,\mathcal{Z}$ | $\tilde{C}_{\phi\text{N}}\,C_{\phi\text{NN}}$
Table 1: Jarlskog invariants emerging from the interactions in (3.24) and
(3.25) .

##  4 ALP interactions for $N_{f}=3$

In this Section, we are going to generalize the previous results to the case
of three flavours, $N_{f}=3$. In particular, as in the $N_{f}=2$ case, we will
first discuss the couplings of ALPs to mesons and then to baryons.

###  4.1 ALP interactions with Mesons

In a three-flavour setting we will allow for flavour-violating interactions in
the down-sector, i.e. the possibility of an ALP-induced mixing between the
down and strange quarks occurring via the matrices

$\begin{split}&Y_{A}=\begin{bmatrix}Y_{A}^{u}&0&0\\\ 0&Y_{A}^{d}&Y_{A}^{ds}\\\
0&Y_{A}^{ds*}&Y_{A}^{s}\end{bmatrix}\qquad\text{and}\qquad
Q_{A}=\begin{bmatrix}q_{u}&0&0\\\ 0&q_{d}&q_{ds}\\\
0&q_{ds}^{*}&q_{s}\end{bmatrix}\,,\\\ &Y_{V}=\begin{bmatrix}0&0&0\\\
0&0&Y_{V}^{ds}\\\ 0&Y_{V}^{ds*}&0\end{bmatrix}\quad\qquad\text{and}\qquad
Q_{V}=\begin{bmatrix}q_{V}^{u}&0&0\\\ 0&q_{V}^{d}&0\\\
0&0&q_{V}^{s}\end{bmatrix}\,.\end{split}$ (4.1)

We can then expand Eqs. (2.27) and (2.28) in terms of the constituent meson
fields. Up to order $\mathcal{O}(\Lambda^{-2})$ we find the following terms
describing the interaction of an ALP with no more than three mesons:

$\begin{split}\mathcal{L}_{\text{ALP}}^{\chi\text{pt}}&=\mathcal{L}_{\text{ALP,
}\kappa}^{\chi\text{pt}}+\mathcal{L}_{\text{ALP,
}\mathcal{Z}}^{\chi\text{pt}}+\mathcal{L}_{\text{ALP,
KM}}^{\chi\text{pt}}+\mathcal{L}_{\text{ALP,
ADI}}^{\chi\text{pt}}+\mathcal{L}_{\text{ALP, VDI}}^{\chi\text{pt}}\\\
&+\mathcal{L}_{\text{ALP, MM}}^{\chi\text{pt}}+\mathcal{L}_{\text{ALP,
MI}}^{\chi\text{pt}}+\mathcal{L}_{\text{ALP,
MT}}^{\chi\text{pt}}+\mathcal{O}\left(\frac{1}{\Lambda^{2}}\right)\,,\end{split}$
(4.2)

which contains both a kinetic mixing (KM) term

$\mathcal{L}_{\text{ALP,
KM}}^{\chi\text{pt}}=(Y_{A}^{u}-Y_{A}^{d})\,\frac{f_{\pi}}{\Lambda}\,\partial\phi\,\partial\pi_{0}+\frac{Y_{A}^{u}+Y_{A}^{d}-2Y_{A}^{s}}{\sqrt{3}}\,\frac{f_{\pi}}{\Lambda}\,\partial\eta\,\partial\phi+\sqrt{2}\,\frac{f_{\pi}}{\Lambda}(Y_{A}^{sd}\,\partial\bar{K_{0}}\partial\phi+Y_{A}^{sd*}\,\partial
K_{0}\partial\phi)$ (4.3)

and a mass mixing term (MM)

$\begin{split}\mathcal{L}_{\text{ALP,
MM}}^{\chi\text{pt}}&=2\lambda_{g}m_{\pi^{\pm}}^{2}\frac{m_{u}q_{u}-m_{d}q_{d}}{m_{u}+m_{d}}\frac{f_{\pi}}{\Lambda}\,\phi\,\pi_{0}+\frac{2}{\sqrt{3}}\lambda_{g}m_{\pi^{\pm}}^{2}\frac{m_{u}q_{u}+m_{d}q_{d}-2m_{s}q_{s}}{m_{u}+m_{d}}\frac{f_{\pi}}{\Lambda}\,\phi\,\eta_{0}\\\
&+\sqrt{2}\lambda_{g}m_{\pi^{\pm}}^{2}\frac{m_{d}+m_{s}}{m_{u}+m_{d}}\frac{f_{\pi}}{\Lambda}\,(q_{ds}\,\bar{K}_{0}\,\phi+q_{ds}^{*}\,K_{0}\,\phi)\,.\end{split}$
(4.4)

In addition to these, from the quark mass term we also extract a mass term for
the ALP field (MT)

$\mathcal{L}_{\text{ALP,
MT}}^{\chi\text{pt}}=-\frac{1}{2}\frac{4\,(\lambda_{g})^{2}m_{\pi^{\pm}}^{2}}{m_{u}+m_{d}}\frac{f_{\pi}^{2}}{\Lambda^{2}}\left[m_{u}q_{u}^{2}+m_{d}(q_{d}^{2}+|q_{ds}|^{2})+m_{s}(q_{s}^{2}+|q_{ds}|^{2})\right]\phi^{2}$
(4.5)

and a set of ALP-meson non-derivative interactions (MI):

$\begin{split}\mathcal{L}_{\text{ALP,
MI}}^{\chi\text{pt}}&=-\frac{2}{3f_{\pi}\Lambda}\frac{\lambda_{g}m_{\pi^{\pm}}^{2}}{m_{u}+m_{d}}\,\phi\,\left[\sqrt{2}\left(\bar{K}_{0}K^{+}\pi^{-}+K_{0}K^{-}\pi^{+}\right)(m_{u}q_{u}+m_{d}q_{d}+m_{s}q_{s})-2m_{d}q_{d}\pi_{0}\bar{K}_{0}K_{0}\right.\\\
&\left.+\sqrt{3}m_{s}q_{s}\bar{K}_{0}K_{0}\eta+m_{s}q_{s}\bar{K}_{0}K_{0}\pi_{0}-\frac{4}{3\sqrt{3}}m_{s}q_{s}\eta^{3}+\frac{m_{d}q_{d}}{18}\left(\sqrt{3}\eta^{3}-9\eta^{2}\pi_{0}+9\sqrt{3}\eta\pi_{0}^{2}-9\pi_{0}^{3}\right)\right.\\\
&\left.+\frac{m_{u}q_{u}}{18}\left(\sqrt{3}\eta^{3}+9\eta^{2}\pi_{0}+9\sqrt{3}\eta\pi_{0}^{2}+9\pi_{0}^{3}\right)-K^{-}K^{+}\left(\sqrt{3}m_{s}q_{s}\eta-(m_{s}q_{s}+2m_{u}q_{u})\pi^{0}\right)\right.\\\
&\left.+\pi^{+}\pi^{-}\left(\sqrt{3}(m_{u}q_{u}+m_{d}q_{d})\eta+(m_{u}q_{u}-m_{d}q_{d})\pi^{0}\right)\right]\\\
&-\frac{\lambda_{g}}{f_{\pi}\Lambda}\frac{m_{d}+m_{s}}{m_{u}+m_{d}}m_{\pi^{\pm}}^{2}\,\phi\,\left(q_{ds}\bar{K_{0}}+q_{ds}^{*}K_{0}\right)\left(2\bar{K}_{0}K_{0}+2K^{+}K^{-}+2\pi^{+}\pi^{-}+\eta^{2}+\pi_{0}^{2}\right)\,.\end{split}$
(4.6)

The other terms in the Lagrangian stem either from the term proportional to
the energy-momentum tensor (subscript $\kappa$),

$\begin{split}\mathcal{L}_{\text{ALP,
}\kappa}^{\chi\text{pt}}&=\kappa\,\frac{\phi}{\Lambda}\,\left[\partial\pi_{0}\partial\pi_{0}+\partial\eta\partial\eta+2\partial\pi^{+}\partial\pi^{-}+2\partial
K^{+}\partial K^{-}+2\partial K_{0}\partial\bar{K}_{0}\right.\\\
&\qquad\qquad\left.-\frac{4m_{\pi^{\pm}}^{2}}{m_{u}+m_{d}}\left((m_{d}+m_{s})\bar{K}_{0}K_{0}+(m_{u}+m_{s})\,K^{+}K^{-}+(m_{d}+m_{u})\,\pi^{+}\pi^{-}\right.\right.\\\
&\qquad\qquad\left.\left.+(m_{d}+m_{u}+4m_{s})\,\frac{\eta^{2}}{6}+(m_{u}-m_{d})\,\frac{\eta}{\sqrt{3}}\pi_{0}+(m_{u}+m_{d})\,\frac{\pi_{0}^{2}}{2}\right)\right]\,,\end{split}$
(4.7)

from the scalar coupling to the matrix $\mathcal{Z}$ (subscript $\mathcal{Z}$)

$\begin{split}\mathcal{L}_{\text{ALP,
}\mathcal{Z}}^{\chi\text{pt}}&=\frac{\phi}{\Lambda}\frac{m_{\pi^{\pm}}^{2}\,v}{m_{u}+m_{d}}\left[\mathcal{Z}_{s}\left(\bar{K}_{0}K_{0}+K^{+}K^{-}+\frac{2}{3}\eta^{2}\right)+\mathcal{Z}_{d}\left(\bar{K}_{0}K_{0}-+\pi^{+}\pi^{-}+\frac{\eta^{2}}{6}-\frac{\eta}{\sqrt{3}}\pi_{0}+\frac{\pi_{0}^{2}}{2}\right)\right.\\\
&\qquad\qquad\qquad\quad\left.+\mathcal{Z}_{u}\left(K^{+}K^{-}+\pi^{+}\pi^{-}+\frac{\eta^{2}}{6}+\frac{\eta}{\sqrt{3}}\pi_{0}+\frac{\pi_{0}^{2}}{2}\right)\right]\,,\end{split}$
(4.8)

or from derivative interactions. These can be further distinguished in axial
(ADI) and vectorial (VDI) derivative interactions; we find:

$\displaystyle\mathcal{L}_{\text{ALP, ADI}}^{\chi\text{pt}}$
$\displaystyle=+\frac{Y_{A}^{u}-Y_{A}^{d}}{6f_{\pi}\Lambda}\,\partial\phi\,\left[2\sqrt{3}\partial\eta\left(\bar{K}_{0}K_{0}-K^{+}K^{-}\right)-2\partial\pi_{0}\left(\bar{K}_{0}K_{0}+K^{+}K^{-}+4\pi^{+}\pi^{-}\right)\right.$
$\displaystyle\quad\left.-\sqrt{3}\eta\left(\partial\bar{K}_{0}K_{0}+\bar{K}_{0}\partial
K_{0}-DK^{+}K^{-}-DK^{-}K^{+}\right)\right.$
$\displaystyle\quad\left.+\pi_{0}\left(\partial\bar{K}_{0}K_{0}+\bar{K}_{0}\partial
K_{0}+DK^{+}K^{-}+DK^{-}K^{+}+4D\pi^{+}\pi^{-}+4D\pi^{-}\pi^{+}\right)\right.$
$\displaystyle\quad\left.+3\sqrt{2}\left(\partial\bar{K}_{0}K^{+}\pi^{-}+\partial
K_{0}K^{-}\pi^{+}-DK^{+}\pi^{-}\bar{K}_{0}-DK^{-}\pi^{+}K_{0}\right)\right]$
$\displaystyle\quad+\frac{Y_{A}^{u}+Y_{A}^{d}-2Y_{A}^{s}}{6f_{\pi}\Lambda}\,\partial\phi\,\left[-2\sqrt{3}\partial\eta\left(\bar{K}_{0}K_{0}+K^{+}K^{-}\right)+2\partial\pi_{0}\left(\bar{K}_{0}K_{0}-K^{+}K^{-}\right)\right.$
$\displaystyle\quad\left.+\sqrt{3}\eta\left(\partial\bar{K}_{0}K_{0}+\bar{K}_{0}\partial
K_{0}+DK^{+}K^{-}+DK^{-}K^{+}\right)\right.$ (4.9)
$\displaystyle\quad\left.-\pi_{0}\left(\partial\bar{K}_{0}K_{0}+\bar{K}_{0}\partial
K_{0}-DK^{+}K^{-}-DK^{-}K^{+}\right)\right.$
$\displaystyle\quad\left.+\sqrt{2}\left(DK^{+}\pi^{-}\bar{K}_{0}+DK^{-}\pi^{+}K_{0}+\partial\bar{K}_{0}K^{+}\pi^{-}+\partial
K_{0}K^{-}\pi^{+}-2D\pi^{+}K^{-}K_{0}-2D\pi^{-}K^{+}\bar{K}_{0}\right)\right]$
$\displaystyle\quad+\frac{1}{6f_{\pi}\Lambda}\partial\phi\left[Y_{A}^{ds}\left(4\sqrt{2}\partial
K_{0}\bar{K}_{0}\bar{K}_{0}+4\sqrt{2}DK^{+}K^{-}\bar{K}_{0}-2DK^{-}(\sqrt{2}K^{+}\bar{K}_{0}-2\sqrt{3}\pi^{+}\eta)\right.\right.$
$\displaystyle\quad\left.\left.+2D\pi^{+}(\sqrt{3}K^{-}\eta-3K^{-}\pi_{0}-\sqrt{2}\pi^{-}\bar{K}_{0})+4\sqrt{2}D\pi^{-}\pi^{+}\bar{K}_{0}\right.\right.$
$\displaystyle\quad\left.\left.+\partial\eta\,(3\sqrt{2}\bar{K}_{0}\eta-\sqrt{6}\bar{K}_{0}\pi_{0}+2\sqrt{3}K^{-}\pi^{+})+\partial\pi_{0}(\bar{K}_{0}\eta+\sqrt{2}\bar{K}_{0}\pi_{0}+6K^{-}\pi^{+})\right.\right.$
$\displaystyle\quad\left.\left.+\sqrt{2}\,\partial\bar{K}_{0}(-4\bar{K}_{0}K_{0}-2K^{+}K^{-}-2\pi^{+}\pi^{-}-3\eta^{2}+2\sqrt{3}\eta\pi_{0}-\pi_{0}^{2})\right)+\text{h.c.}\right]$

and

$\displaystyle\mathcal{L}_{\text{ALP, VDI}}^{\chi\text{pt}}$
$\displaystyle=+i\frac{q_{V}^{u}-q_{V}^{d}}{2\Lambda}\,\partial\phi\,\left[\partial\bar{K}_{0}K_{0}-\partial
K_{0}\bar{K}_{0}+DK^{+}K^{-}-DK^{-}K^{+}+2D\pi^{+}\pi^{-}-2D\pi^{-}\pi^{+}\right]$
$\displaystyle\quad-i\frac{q_{V}^{u}+q_{V}^{d}-2q_{V}^{s}}{2\Lambda}\,\partial\phi\,\left[\partial\bar{K}_{0}K_{0}-\partial
K_{0}\bar{K}_{0}+DK^{-}K^{+}-DK^{+}K^{-}\right]$ (4.10)
$\displaystyle\quad+i\left[\frac{Y_{V}^{ds}}{2\Lambda}\,\partial\phi\,\left(-\sqrt{6}\partial\eta\bar{K}_{0}+\sqrt{6}\partial\bar{K}_{0}\eta+\sqrt{2}\partial\pi_{0}\bar{K}_{0}-\sqrt{2}\partial\bar{K}_{0}\pi_{0}+2DK^{-}\pi^{+}-2D\pi^{+}K^{-}\right)-\text{h.c.}\right]\,.$

The two terms in Eqs. (4.3) and (4.5) are responsible for both kinetic and
mass mixing between the ALP field and the neutral mesons
$\pi_{0},\eta,\bar{K}_{0},K_{0}$. These fields can be put in a canonical form
by repeating the procedure outlined in Section 3 for the full $5\times 5$
mixing matrices.

However, this does not allow us to obtain analytical results for the mixing,
and a better strategy is to block-diagonalize the various mixing terms: in
this case, as we will see, the results are analytical and start differing from
the exact ones at order $\mathcal{O}(\Lambda^{-2})$ in the mixing terms and at
order $\mathcal{O}(\Lambda^{-4})$ in the mass terms.

First of all we define the small parameter $\xi=f_{\pi}/\Lambda$ and

$A=Y^{u}_{A}-Y^{d}_{A}\,,\qquad
B=\frac{Y_{A}^{u}+Y_{A}^{d}-2Y_{A}^{s}}{\sqrt{3}}\,,\qquad
C=2\,\text{Re}Y_{A}^{ds}\,,\qquad G=-2\,\text{Im}Y_{A}^{ds}\,,$ (4.11)

as well as

$\begin{split}&\alpha^{\prime}=2\,(m_{u}q_{u}-m_{d}q_{d})\lambda_{g}\frac{m_{\pi^{\pm}}^{2}}{m_{u}+m_{d}}\,,\qquad\beta^{\prime}=\frac{2}{\sqrt{3}}\,(m_{u}q_{u}+m_{d}q_{d}-2m_{s}q_{s})\lambda_{g}\frac{m_{\pi^{\pm}}^{2}}{m_{u}+m_{d}}\,,\qquad\\\
&\gamma^{\prime}=2\,\text{Re}(q_{ds})(m_{d}+m_{s})\lambda_{g}\frac{m_{\pi^{\pm}}^{2}}{m_{u}+m_{d}}\,,\qquad\delta^{\prime}=-2\,\text{Im}(q_{ds})(m_{d}+m_{s})\lambda_{g}\frac{m_{\pi^{\pm}}^{2}}{m_{u}+m_{d}}\,.\end{split}$
(4.12)

Then we can start diagonalizing the sub-sector responsible for the mixing
between the ALP and the neutral pion. According to our results in Eq. (3.7) we
obtain

$\begin{split}&\phi^{\prime}=\phi+\xi
A\frac{m_{\pi_{0}}^{2}+(\alpha^{\prime}/A)}{m_{\pi_{0}}^{2}-m_{\phi}^{2}}\pi_{0}\,,\\\
&\pi_{0,\text{ph}}=\pi_{0}-\xi
A\frac{m_{\phi}^{2}+(\alpha^{\prime}/A)}{m_{\pi_{0}}^{2}-m_{\phi}^{2}}\phi\,,\end{split}$
(4.13)

which can be inverted, yielding

$\begin{split}&\phi=\phi^{\prime}-\xi
A\frac{m_{\pi_{0}}^{2}+(\alpha^{\prime}/A)}{m_{\pi_{0}}^{2}-m_{\phi}^{2}}\pi_{0}\,,\\\
&\pi_{0}=\pi_{0,\text{ph}}+\xi
A\frac{m_{\phi}^{2}+(\alpha^{\prime}/A)}{m_{\pi_{0}}^{2}-m_{\phi}^{2}}\phi^{\prime}\,.\end{split}$
(4.14)

Now it is apparent what we anticipated previously: the only modifications to
the mixing of ALPs with the other mesons start appearing at order
$\mathcal{O}(\xi^{2})=\mathcal{O}(\Lambda^{-2})$ and can be safely neglected.
For instance, the mass mixing term between the ALP and the $\eta$ meson is
shifted according to

$\beta^{\prime}\xi\,\phi\,\eta=\beta^{\prime}\xi\,\left(\phi^{\prime}-\xi
A\frac{m_{\pi_{0}}^{2}+(\alpha^{\prime}/A)}{m_{\pi_{0}}^{2}-m_{\phi}^{2}}\pi_{0}\right)\,\eta=\beta^{\prime}\xi\,\phi^{\prime}\,\eta+\mathcal{O}(\xi^{2})\,.$
(4.15)

The ALP-induced mixing in the $\pi_{0}-\eta$ sector thus emerges only at order
$\xi^{2}$, yielding mass corrections of order $\xi^{4}$ and higher; therefore,
we can safely neglect this kind of second-order effects in our analysis. The
reasoning we have followed here is completely general and applies as well as
to the kinetic mixing term, besides being valid of course also for the other
possible mixing effects with the remaining neutral mesons.

We repeat this procedure also for the remaining fields, finding in the end
that, up to $\mathcal{O}(\Lambda^{-2})$

$\begin{split}&\phi_{\text{ph}}=\phi+\xi
A\frac{m_{\pi_{0}}^{2}+(\alpha^{\prime}/A)}{m_{\pi_{0}}^{2}-m_{\phi}^{2}}\pi_{0}+\xi
B\frac{m_{\eta}^{2}+(\beta^{\prime}/B)}{m_{\eta}^{2}-m_{\phi}^{2}}\eta+\xi
C\frac{m_{K_{0}}^{2}+(\gamma^{\prime}/C)}{m_{K_{0}}^{2}-m_{\phi}^{2}}\pi_{6}++\xi
G\frac{m_{\pi}^{2}+(\delta^{\prime}/G)}{m_{K_{0}}^{2}-m_{\phi}^{2}}\pi_{7}\,,\\\
&\pi_{0}=\pi_{0,\text{ph}}+\xi
A\frac{m_{\phi}^{2}+(\alpha^{\prime}/A)}{m_{\pi_{0}}^{2}-m_{\phi}^{2}}\phi_{\text{ph}}\,,\\\
&\eta=\eta_{\text{ph}}+\xi
B\frac{m_{\phi}^{2}+(\beta^{\prime}/B)}{m_{\eta}^{2}-m_{\phi}^{2}}\phi_{\text{ph}}\,,\\\
&\pi_{6}=\pi_{6,\text{ph}}+\xi
C\frac{m_{\phi}^{2}+(\gamma^{\prime}/C)}{m_{K_{0}}^{2}-m_{\phi}^{2}}\phi_{\text{ph}}\,,\\\
&\pi_{7}=\pi_{7,\text{ph}}+\xi
G\frac{m_{\phi}^{2}+(\delta^{\prime}/G)}{m_{K_{0}}^{2}-m_{\phi}^{2}}\phi_{\text{ph}}\,.\end{split}$
(4.16)

These mixing effects are then to be taken into account into the meson-meson
interaction terms stemming from both the kinetic and the mass one, as well as
into the anomalous coupling of the $\pi_{0}$ and the $\eta$ mesons with two
photons. Recalling that these interactions read

$\begin{split}\mathcal{L}_{\pi\text{KT}}&\supset\frac{1}{12f_{\pi}^{2}}\left[2\partial\bar{K}_{0}\partial\bar{K}_{0}K_{0}K_{0}+2\partial
K_{0}\partial
K_{0}\bar{K}_{0}\bar{K}_{0}-\partial\pi_{0}\partial\pi_{0}\left(\bar{K}_{0}K_{0}+K^{+}K^{-}+4\pi^{+}\pi^{-}\right)\right.\\\
&\quad\left.-\partial\bar{K}_{0}\partial
K_{0}\left(4\bar{K}_{0}K_{0}+2K^{+}K^{-}+3\eta^{2}-2\sqrt{3}\eta\pi_{0}+\pi_{0}^{2}+2\pi^{+}\pi^{-}\right)-3\partial\eta\partial\eta\left(\bar{K}_{0}K_{0}+K^{+}K^{-}\right)\right.\\\
&\quad\left.-\partial\bar{K}_{0}\left(-3\partial\eta\eta
K_{0}+\sqrt{3}\partial\pi_{0}\eta
K_{0}+\sqrt{3}\partial\eta\pi_{0}K_{0}-\partial\pi_{0}\pi_{0}K_{0}+\sqrt{6}\partial\eta
K^{+}K^{-}+3\partial\pi_{0}K^{+}K^{-}\right)\right.\\\ &\quad\left.-\partial
K_{0}\left(-3\partial\eta\eta\bar{K}_{0}+\sqrt{3}\partial\pi_{0}\eta\bar{K}_{0}+\sqrt{3}\partial\eta\pi_{0}\bar{K}_{0}-\partial\pi_{0}\pi_{0}\bar{K}_{0}+\sqrt{6}\partial\eta
K^{+}K^{-}+3\partial\pi_{0}K^{+}K^{-}\right)\right]\end{split}$ (4.17)

and

$\begin{split}\mathcal{L}_{\pi\text{MT}}&\supset\frac{m_{\pi^{\pm}}^{2}}{m_{u}+m_{d}}\frac{1}{f_{\pi}^{2}}\left[\frac{m_{u}+m_{d}+16m_{s}}{216}\eta^{4}+\frac{(m_{u}-m_{d})}{18\sqrt{3}}\eta^{3}\pi_{0}+\frac{\eta^{2}}{12}\left(\bar{K}_{0}K_{0}(m_{d}+3m_{s})+K^{+}K^{-}(m_{u}+3m_{s})\right.\right.\\\
&\quad\left.\left.+(\pi_{0}^{2}+2\pi^{+}\pi^{-})(m_{u}-m_{s})\right)+\frac{\eta}{12\sqrt{3}}\left(2(m_{u}-m_{d})\pi_{0}^{3}+\sqrt{2}(m_{u}+m_{d}-2m_{s})(\bar{K}_{0}K^{+}\pi^{-}+K_{0}K^{-}\pi^{+})\right.\right.\\\
&\quad\left.\left.+2\pi_{0}\left((m_{s}-m_{d})\bar{K}_{0}K_{0}+(m_{u}-m_{s})K^{+}K^{-}+2(m_{u}-m_{d})\pi^{+}\pi^{-}\right)\right)\right.\\\
&\left.\quad+\frac{m_{u}+m_{d}}{24}\pi_{0}^{4}+\frac{\pi_{0}^{2}}{12}\left(\bar{K}_{0}K_{0}(3m_{d}+m_{s})+K^{+}K^{-}(3m_{u}+m_{s})+2\pi^{+}\pi^{-}(m_{u}+m_{d})\right)\right.\\\
&\quad\left.+\frac{m_{u}-m_{d}}{6\sqrt{2}}\,\pi_{0}\,\left(\bar{K}_{0}K^{+}\pi^{-}+K_{0}K^{-}\pi^{+}\right)+\frac{\bar{K}_{0}K_{0}}{6}\left(K^{+}K^{-}(m_{u}+m_{d}+2m_{s})+\pi^{+}\pi^{-}(m_{u}+2m_{d}+m_{s})\right)\right]\,,\end{split}$
(4.18)

one can perform the needed shifts in the neutral meson fields. For instance,
one finds

$\begin{split}\mathcal{L}_{\pi\text{KT+MT}}^{A}&=\xi
A\frac{m_{\phi}^{2}+\alpha^{\prime}/A}{m_{\pi_{0}}^{2}-m_{\phi}^{2}}\frac{1}{12f_{\pi}^{2}}\left[-2\partial\phi\partial\pi_{0}\left(\bar{K}_{0}K_{0}+K^{+}K^{-}+4\pi^{+}\pi^{-}\right)-\partial\bar{K}_{0}\partial
K_{0}\left(-2\sqrt{3}\eta\phi+2\pi_{0}\phi\right)\right.\\\
&\quad\left.-\partial\bar{K}_{0}\left(\sqrt{3}\partial\phi\eta
K_{0}+\sqrt{3}\partial\eta\phi
K_{0}-\partial\phi\pi_{0}K_{0}-\partial\pi_{0}\phi K_{0}+3\partial\phi
K^{+}K^{-}\right)\right.\\\ &\quad\left.-\partial
K_{0}\left(\sqrt{3}\partial\phi\eta\bar{K}_{0}+\sqrt{3}\partial\eta\phi\bar{K}_{0}-\partial\phi\pi_{0}\bar{K}_{0}-\partial\pi_{0}\phi\bar{K}_{0}+3\partial\phi
K^{+}K^{-}\right)\right]+\\\ &\quad\xi
A\frac{m_{\phi}^{2}+\alpha^{\prime}/A}{m_{\pi_{0}}^{2}-m_{\phi}^{2}}\frac{m_{\pi^{\pm}}^{2}}{m_{u}+m_{d}}\frac{1}{f_{\pi}^{2}}\left[\frac{m_{u}-m_{d}}{18\sqrt{3}}\eta^{3}\phi+\frac{\eta^{2}}{6}\phi\pi_{0}(m_{u}-m_{s})+\frac{\eta}{2\sqrt{3}}\pi_{0}^{2}\phi(m_{u}-m_{d})\right.\\\
&\quad\left.+\frac{\eta}{6\sqrt{3}}\phi\left((m_{s}-m_{d})\bar{K}_{0}K_{0}+(m_{u}-m_{s})K^{+}K^{-}+2(m_{u}-m_{d})\pi^{+}\pi^{-}\right)\right.\\\
&\left.\quad+\frac{m_{u}+m_{d}}{6}\phi\pi_{0}^{3}+\frac{\pi_{0}}{6}\phi\left(\bar{K}_{0}K_{0}(3m_{d}+m_{s})+K^{+}K^{-}(3m_{u}+m_{s})+2\pi^{+}\pi^{-}(m_{u}+m_{d})\right)\right]\\\
&\quad+\xi
A\frac{m_{\phi}^{2}+\alpha^{\prime}/A}{m_{\pi_{0}}^{2}-m_{\phi}^{2}}\frac{e^{2}}{16\pi^{2}f_{\pi}}\phi
F^{\mu\nu}\tilde{F}_{\mu\nu}\,,\end{split}$ (4.19)
$\begin{split}\mathcal{L}_{\eta\text{KT+MT}}^{B}&=\xi
B\frac{m_{\phi}^{2}+\beta^{\prime}/B}{m_{\eta}^{2}-m_{\phi}^{2}}\frac{1}{12f_{\pi}^{2}}\left[-6\partial\phi\partial\eta(\bar{K}_{0}K_{0}+K^{+}K^{-})\right.\\\
&\quad\left.-\partial\bar{K}_{0}\left(-3\partial\phi\eta
K_{0}-3\partial\eta\phi K_{0}+\sqrt{3}\partial\pi_{0}\phi
K_{0}+\sqrt{3}\partial\phi\pi_{0}K_{0}+\sqrt{6}\partial\phi
K^{+}K^{-}\right)-\partial\bar{K}_{0}\partial
K_{0}\left(6\eta\phi-2\sqrt{3}\phi\pi_{0}\right)\right.\\\
&\quad\left.-\partial
K_{0}\left(-3\partial\phi\eta\bar{K}_{0}-3\partial\eta\phi\bar{K}_{0}+\sqrt{3}\partial\pi_{0}\phi\bar{K}_{0}+\sqrt{3}\partial\phi\pi_{0}\bar{K}_{0}+\sqrt{6}\partial\phi
K^{+}K^{-}\right)\right]\\\ &\quad\xi
B\frac{m_{\phi}^{2}+\beta^{\prime}/B}{m_{\eta}^{2}-m_{\phi}^{2}}\frac{m_{\pi^{\pm}}^{2}}{m_{u}+m_{d}}\frac{1}{f_{\pi}^{2}}\left[\frac{m_{u}+m_{d}+16m_{s}}{54}\eta^{3}\phi+\frac{(m_{u}-m_{d})}{6\sqrt{3}}\eta^{2}\phi\pi_{0}+\frac{\eta\phi}{6}\left(\bar{K}_{0}K_{0}(m_{d}+3m_{s})\right.\right.\\\
&\quad\left.\left.+K^{+}K^{-}(m_{u}+3m_{s})+(\pi_{0}^{2}+2\pi^{+}\pi^{-})(m_{u}-m_{s})\right)+\frac{\phi}{12\sqrt{3}}\left(2(m_{u}-m_{d})\pi_{0}^{3}\right.\right.\\\
&\quad\left.\left.+\sqrt{2}(m_{u}+m_{d}-2m_{s})(\bar{K}_{0}K^{+}\pi^{-}+K_{0}K^{-}\pi^{+})+2\pi_{0}\left((m_{s}-m_{d})\bar{K}_{0}K_{0}+(m_{u}-m_{s})K^{+}K^{-}\right.\right.\right.\\\
&\quad\left.\left.\left.+2(m_{u}-m_{d})\pi^{+}\pi^{-}\right)\right)\right]+\xi
B\frac{m_{\phi}^{2}+\beta^{\prime}/B}{m_{\eta}^{2}-m_{\phi}^{2}}\frac{e^{2}}{16\pi^{2}f_{\pi}}\phi
F^{\mu\nu}\tilde{F}_{\mu\nu}\,,\end{split}$ (4.20)

and similarly for the mixing between the ALP field and the neutral kaons. We
do not report here the explicit expressions for such a mixing effect because,
as it will be clear soon, the charges $q_{ds}$ can be always chosen in such a
way to eliminate it completely. Indeed, a convenient choice consists in
minimizing the possible mixing effects in the mesonic sector. This amounts,
for instance, to requiring that

$\frac{\alpha^{\prime}}{A}=-m_{\phi}^{2}\,,\qquad\frac{\gamma^{\prime}}{C}=-m_{\phi}^{2}\,,\qquad\frac{\delta^{\prime}}{G}=-m_{\phi}^{2}\,,$
(4.21)

which corresponds to the choice

$\begin{split}q_{u}&=\frac{1}{2}\frac{m_{d}m_{s}}{m_{u}m_{d}+m_{u}m_{s}+m_{d}m_{s}}-\frac{m_{\phi}^{2}}{m_{\pi_{\pm}}^{2}-m_{\phi}^{2}}\frac{\Delta_{A}}{2\lambda_{g}}\,,\\\
q_{d}&=\frac{1}{2}\frac{m_{u}m_{s}}{m_{u}m_{d}+m_{u}m_{s}+m_{d}m_{s}}+\frac{m_{\phi}^{2}}{m_{\pi^{\pm}}^{2}-m_{\phi}^{2}}\frac{\Delta_{A}}{2\lambda_{g}}\,,\\\
q_{s}&=\frac{1}{2}\frac{m_{u}m_{d}}{m_{u}m_{d}+m_{u}m_{s}+m_{d}m_{s}}\,,\\\
q_{ds}&=-\frac{Y_{P}^{ds}}{\lambda_{g}}\frac{m_{\phi}^{2}(m_{u}+m_{d})}{(m_{d}+m_{s})m_{\pi^{\pm}}-m_{\phi}^{2}(m_{u}+m_{d})}\,,\end{split}$
(4.22)

where, also for future reference, we define

$\begin{split}\Delta_{A}&=Y_{P}^{u}+Y_{P}^{d}-\frac{\lambda_{g}}{2}\frac{m_{s}(m_{d}-m_{u})}{m_{u}m_{d}+m_{u}m_{s}+m_{d}m_{s}}=\frac{m_{\pi_{\pm}}^{2}-m_{\phi}^{2}}{m_{\pi_{\pm}}^{2}}(Y_{A}^{u}-Y_{A}^{d})\,,\\\
\Omega_{A}&=Y_{P}^{u}+Y_{P}^{d}-2Y_{P}^{s}=(Y_{A}^{u}+Y_{A}^{d}-2Y_{A}^{s})\,,\\\
\Xi_{A}&=Y_{P}^{ds}\frac{m_{\pi^{\pm}}^{2}(m_{d}+m_{s})}{(m_{d}+m_{s})m_{\pi^{\pm}}-m_{\phi}^{2}(m_{u}+m_{d})}\,,\\\
\Gamma_{A}&=Y_{P}^{u}+Y_{P}^{d}+Y_{P}^{s}-\frac{1}{2}=(Y_{A}^{u}+Y_{A}^{d}+Y_{A}^{s})\,.\end{split}$
(4.23)

It should be noted though that this choice, nor any other, cannot remove the
mixing between the ALP and both the $\pi_{0}$ and $\eta$ fields, as it is not
possible to satisfy simultaneously the condition in Eq. (4.21) and
$\text{Tr}(Q_{A})=1/2$.

In any case, with our choice (4.22), the ALP acquires a mass term given by

$m_{\text{ph}}^{2}=m_{\phi}^{2}-\xi^{2}B^{2}\frac{(m_{\phi}^{2}+\beta^{\prime}/B)^{2}}{m_{\pi^{\pm}}^{2}-m_{\phi}^{2}}=m_{\phi}^{2}-\Omega_{A}^{2}\frac{f_{\pi}^{2}}{\Lambda^{2}}\frac{M_{\phi}^{2}}{m_{\eta}^{2}-M_{\phi}^{2}}\left[1+\frac{\Delta_{A}}{\Omega_{A}}\frac{m_{d}-m_{u}}{m_{u}+m_{d}}\frac{m_{\pi^{\pm}}^{2}}{m_{\pi^{\pm}}^{2}-M_{\phi}^{2}}\right]+\mathcal{O}\left(\frac{1}{\Lambda^{4}}\right)\,,$
(4.24)

where

$\begin{split}m_{\phi}^{2}=M_{\phi}^{2}+m_{\pi^{\pm}}^{2}\frac{f_{\pi}^{2}}{\Lambda^{2}}&\left[(\lambda_{g})^{2}\,\frac{m_{u}m_{s}m_{d}}{m_{u}m_{d}+m_{u}m_{s}+m_{d}m_{s}}+\frac{4}{m_{u}+m_{d}}\,|\Xi_{A}|^{2}\frac{M_{\phi}^{4}}{m_{\pi^{\pm}}^{4}}\left(\frac{m_{d}+m_{u}}{m_{s}+m_{d}}\right)^{2}\right.\\\
&\left.\quad+\Delta_{A}^{2}\frac{M_{\phi}^{4}}{(m_{\pi^{\pm}}^{2}-M_{\phi}^{2})^{2}}\right]+\mathcal{O}\left(\frac{1}{\Lambda^{4}}\right)\,,\end{split}$
(4.25)

while the interaction terms (MI and ADI) now read

$\displaystyle\mathcal{L}_{\text{ALP, MI}}^{\chi\text{pt}}$
$\displaystyle=-\frac{1}{3f_{\pi}\Lambda}\frac{\lambda_{g}m_{\pi^{\pm}}^{2}}{m_{u}+m_{d}}\,\phi\,\left[\sqrt{2}\left(\bar{K}_{0}K^{+}\pi^{-}+K_{0}K^{-}\pi^{+}\right)\left(3\tilde{m}+\tilde{\Delta}_{A}(m_{d}-m_{u})\right)-2\left(\tilde{m}+m_{d}\tilde{\Delta}_{A}\right)\pi_{0}\bar{K}_{0}K_{0}\right.$
$\displaystyle\quad\left.+\sqrt{3}\tilde{m}\bar{K}_{0}K_{0}\eta+\tilde{m}\bar{K}_{0}K_{0}\pi_{0}-\frac{4}{3\sqrt{3}}\tilde{m}\eta^{3}+\frac{1}{18}\left(\tilde{m}+m_{d}\tilde{\Delta}_{A}\right)\left(\sqrt{3}\eta^{3}-9\eta^{2}\pi_{0}+9\sqrt{3}\eta\pi_{0}^{2}-9\pi_{0}^{3}\right)\right.$
$\displaystyle\quad\left.+\frac{1}{18}\left(\tilde{m}-m_{u}\tilde{\Delta}_{A}\right)\left(\sqrt{3}\eta^{3}+9\eta^{2}\pi_{0}+9\sqrt{3}\eta\pi_{0}^{2}+9\pi_{0}^{3}\right)-K^{-}K^{+}\left(\sqrt{3}\tilde{m}\eta-(3\tilde{m}-2\tilde{\Delta}_{A}m_{u})\pi^{0}\right)\right.$
$\displaystyle\quad\left.+\pi^{+}\pi^{-}\left(\sqrt{3}(2\tilde{m}+(m_{d}-m_{u})\tilde{\Delta}_{A})\eta+(m_{u}+m_{d})\tilde{\Delta}_{A}\pi^{0}\right)\right]$
$\displaystyle\quad+\frac{M_{\phi}^{2}}{f_{\pi}\Lambda}\,\phi\,\left(\Xi_{A}\bar{K_{0}}+\Xi_{A}^{*}K_{0}\right)\left(2\bar{K}_{0}K_{0}+2K^{+}K^{-}+2\pi^{+}\pi^{-}+\eta^{2}+\pi_{0}^{2}\right)$
(4.26)

and

$\displaystyle\mathcal{L}_{\text{ALP, ADI}}^{\chi\text{pt}}$
$\displaystyle=+\frac{1}{6f_{\pi}\Lambda}\frac{m_{\pi^{\pm}}^{2}}{m_{\pi^{\pm}}^{2}-m_{\phi}^{2}}\Delta_{A}\,\partial\phi\,\left[2\sqrt{3}\partial\eta\left(\bar{K}_{0}K_{0}-K^{+}K^{-}\right)-2\partial\pi_{0}\left(\bar{K}_{0}K_{0}+K^{+}K^{-}+4\pi^{+}\pi^{-}\right)\right.$
$\displaystyle\quad\left.-\sqrt{3}\eta\left(\partial\bar{K}_{0}K_{0}+\bar{K}_{0}\partial
K_{0}-DK^{+}K^{-}-DK^{-}K^{+}\right)\right.$
$\displaystyle\quad\left.+\pi_{0}\left(\partial\bar{K}_{0}K_{0}+\bar{K}_{0}\partial
K_{0}+DK^{+}K^{-}+DK^{-}K^{+}+4D\pi^{+}\pi^{-}+4D\pi^{-}\pi^{+}\right)\right.$
$\displaystyle\quad\left.+3\sqrt{2}\left(\partial\bar{K}_{0}K^{+}\pi^{-}+\partial
K_{0}K^{-}\pi^{+}-DK^{+}\pi^{-}\bar{K}_{0}-DK^{-}\pi^{+}K_{0}\right)\right]$
$\displaystyle\quad+\frac{\Omega_{A}}{6f_{\pi}\Lambda}\,\partial\phi\left[-2\sqrt{3}\partial\eta\left(\bar{K}_{0}K_{0}+K^{+}K^{-}\right)+2\partial\pi_{0}\left(\bar{K}_{0}K_{0}-K^{+}K^{-}\right)\right.$
$\displaystyle\quad\left.+\sqrt{3}\eta\left(\partial\bar{K}_{0}K_{0}+\bar{K}_{0}\partial
K_{0}+DK^{+}K^{-}+DK^{-}K^{+}\right)\right.$ (4.27)
$\displaystyle\quad\left.-\pi_{0}\left(\partial\bar{K}_{0}K_{0}+\bar{K}_{0}\partial
K_{0}-DK^{+}K^{-}+DK^{-}K^{+}\right)\right.$
$\displaystyle\quad\left.+\sqrt{2}\left(DK^{+}\pi^{-}\bar{K}_{0}+DK^{-}\pi^{+}K_{0}+\partial\bar{K}_{0}K^{+}\pi^{-}+\partial
K_{0}K^{-}\pi^{+}-2D\pi^{+}K^{-}K_{0}-2D\pi^{-}K^{+}\bar{K}_{0}\right)\right]$
$\displaystyle\quad+\frac{1}{6f_{\pi}\Lambda}\partial\phi\left[\Xi_{A}\left(4\sqrt{2}\partial
K_{0}\bar{K}_{0}\bar{K}_{0}+4\sqrt{2}DK^{+}K^{-}\bar{K}_{0}-2DK^{-}(\sqrt{2}K^{+}\bar{K}_{0}-2\sqrt{3}\pi^{+}\eta)\right.\right.$
$\displaystyle\quad\left.\left.+2D\pi^{+}(\sqrt{3}K^{-}\eta-3K^{-}\pi_{0}-\sqrt{2}\pi^{-}\bar{K}_{0})+4\sqrt{2}D\pi^{-}\pi^{+}\bar{K}_{0}\right.\right.$
$\displaystyle\quad\left.\left.+\partial\eta\,(3\sqrt{2}\bar{K}_{0}\eta-\sqrt{6}\bar{K}_{0}\pi_{0}+2\sqrt{3}K^{-}\pi^{+})+\partial\pi_{0}(\bar{K}_{0}\eta+\sqrt{2}\bar{K}_{0}\pi_{0}+6K^{-}\pi^{+})\right.\right.$
$\displaystyle\quad\left.\left.+\sqrt{2}\,\partial\bar{K}_{0}(-4\bar{K}_{0}K_{0}-2K^{+}K^{-}-2\pi^{+}\pi^{-}-3\eta^{2}+2\sqrt{3}\eta\pi_{0}-\pi_{0}^{2})\right)+\text{h.c.}\right]\,,$

where we have conveniently defined

$\tilde{m}\equiv\frac{m_{u}m_{d}m_{s}}{m_{u}m_{d}+m_{u}m_{s}+m_{d}m_{s}}\qquad\text{and}\qquad\tilde{\Delta}_{A}=\frac{m_{\phi}^{2}}{m_{\pi^{\pm}}^{2}-m_{\phi}^{2}}\frac{\Delta_{A}}{\lambda_{g}}\,.$
(4.28)

With the charge assignment in Eq. (4.22) then one finds that all of the
interactions between one ALP and at most three pseudoscalar mesons at order
$\mathcal{O}(\Lambda^{-2})$ are given by the sum of Eqs. (4.7), (4.8), (4.1),
(4.1) and (4.1), with an ALP mass given by Eqs. (4.24) and (4.25).

These have then to be supplemented with the explicit mixing term (4.20), where
$B$ and $\beta^{\prime}$ are specialized to our explicit choice of charges in
Eq. (4.22):

$\displaystyle\mathcal{L}_{\eta\text{KT+MT}}^{B}$
$\displaystyle=\frac{2\Omega_{A}}{\sqrt{3}}\frac{f_{\pi}}{\Lambda}\frac{m_{\phi}^{2}}{m_{\eta}^{2}-m_{\phi}^{2}}\left(1+\frac{\Delta_{A}}{2\Omega_{A}}\frac{m_{d}-m_{u}}{m_{d}+m_{u}}\frac{m_{\pi^{\pm}}^{2}}{m_{\pi^{\pm}}^{2}-m_{\phi}^{2}}\right)\left\\{\frac{1}{12f_{\pi}^{2}}\left[-\partial\bar{K}_{0}\partial
K_{0}\left(6\eta\phi-2\sqrt{3}\phi\pi_{0}\right)\right.\right.$
$\displaystyle\quad\left.\left.-\partial\bar{K}_{0}\left(-3\partial\phi\eta
K_{0}-3\partial\eta\phi K_{0}+\sqrt{3}\partial\pi_{0}\phi
K_{0}+\sqrt{3}\partial\phi\pi_{0}K_{0}+\sqrt{6}\partial\phi
K^{+}K^{-}\right)\right.\right.$ $\displaystyle\quad\left.\left.-\partial
K_{0}\left(-3\partial\phi\eta\bar{K}_{0}-3\partial\eta\phi\bar{K}_{0}+\sqrt{3}\partial\pi_{0}\phi\bar{K}_{0}+\sqrt{3}\partial\phi\pi_{0}\bar{K}_{0}+\sqrt{6}\partial\phi
K^{+}K^{-}\right)\right.\right.$
$\displaystyle\left.\left.\quad-6\partial\phi\partial\eta(\bar{K}_{0}K_{0}+K^{+}K^{-})\right]\right.$
(4.29)
$\displaystyle\quad\left.+\frac{m_{\pi^{\pm}}^{2}}{m_{u}+m_{d}}\frac{1}{f_{\pi}^{2}}\left[\frac{m_{u}+m_{d}+16m_{s}}{54}\eta^{3}\phi+\frac{(m_{u}-m_{d})}{6\sqrt{3}}\eta^{2}\phi\pi_{0}+\frac{\eta\phi}{6}\left(\bar{K}_{0}K_{0}(m_{d}+3m_{s})\right.\right.\right.$
$\displaystyle\quad\left.\left.\left.+K^{+}K^{-}(m_{u}+3m_{s})+(\pi_{0}^{2}+2\pi^{+}\pi^{-})(m_{u}-m_{s})\right)+\frac{\phi}{12\sqrt{3}}\left(2(m_{u}-m_{d})\pi_{0}^{3}\right.\right.\right.$
$\displaystyle\quad\left.\left.\left.+\sqrt{2}(m_{u}+m_{d}-2m_{s})(\bar{K}_{0}K^{+}\pi^{-}+K_{0}K^{-}\pi^{+})+2\pi_{0}\left((m_{s}-m_{d})\bar{K}_{0}K_{0}+(m_{u}-m_{s})K^{+}K^{-}\right.\right.\right.\right.$
$\displaystyle\quad\left.\left.\left.\left.+2(m_{u}-m_{d})\pi^{+}\pi^{-}\right)\right)\right]+\frac{e^{2}}{16\pi^{2}f_{\pi}}\phi
F^{\mu\nu}\tilde{F}_{\mu\nu}\right\\}\,.$

An interesting feature of the final Lagrangian is that, as it was the case for
the results in a two-flavour setting, one is always free to choose the
diagonal vectorial charges in such a way to get rid of the corresponding term
from the Lagrangian. However, in a three-flavour setting the flavour-changing
vectorial interactions are physical and the corresponding effects are to be
considered in the last line of Eq. (4.1).

Of course, one could implement a rotation of quark fields via a non-diagonal
vectorial charge matrix $Q_{V}$ in Eq. (2.5). This could be used in order to
rotate away indeed the last line of (4.1) at the price of introducing an
additional source of mixing via a mass term of the kind

$\begin{split}\begin{split}i\frac{\phi}{\Lambda}\frac{f_{\pi}^{2}}{2}B_{0}\text{Tr}\left[\left[M_{q},Q_{V}\right](\Sigma^{\dagger}-\Sigma)\right]-\frac{1}{2}\frac{\phi^{2}}{\Lambda^{2}}\frac{f_{\pi}^{2}}{2}B_{0}\text{Tr}\left[\left[Q_{V},\left[Q_{V},M_{q}\right]\right](\Sigma^{\dagger}+\Sigma)\right]+\mathcal{O}\left(\frac{1}{\Lambda^{2}}\right)\,,\end{split}\end{split}$
(4.30)

which would no longer vanish, since a non-diagonal $Q_{V}$ does not commute
with $M_{q}$. Note, also, that introducing a non-diagonal $Q_{V}$ matrix would
modify the value of the ALP mass with an extra contribution, in a way that is
completely analogous to the $Q_{A}$ case (but with a different structure of
commutators and anticommutators). Since we do not see any reason to prefer
this second approach over the one we have so far followed, we stick to our
choice and choose $Q_{V}$ to be diagonal.

###  4.2 ALP interactions with Baryons

We have already discussed the three possible terms from which an interaction
between ALPs and nucleons arise. These very terms are of course responsible
for the couplings of an ALP to baryons in a generalized three-flavour setting.
Exploring such kinds of interactions will be the subject of this Section.
First of all, we stress once again the fact that the axial and vectorial
currents in (2.38) will in general cointain also a baryonic component of the
form discussed in (2.35). These ought to be supplemented with the $U(1)_{A}$
axial current, resulting in the following Lagrangian describing the
interactions of an ALP and at most two baryons at a time:

$\begin{split}\mathcal{L}_{\text{ALP}}^{\partial
B}&=\frac{\partial_{\mu}\phi}{\Lambda}\left[\text{Tr}(Y_{A}\lambda_{a})j_{A,a}^{\mu}+\frac{1}{3}\text{Tr}(Y_{A})j_{A}^{\mu}+\text{Tr}(Y_{V}\lambda_{a})j_{V,a}^{\mu}\right]\\\
&=\frac{\partial_{\mu}\phi}{\Lambda}\left[\text{Tr}(Y_{A}\lambda_{a})\left[D\,\text{Tr}(\bar{B}_{v}\gamma^{\mu}\gamma_{5}\\{T_{a},B_{v}\\})+F\,\text{Tr}(\bar{B}_{v}\gamma^{\mu}\gamma_{5}\left[T_{a},B_{v}\right])\right]+\frac{1}{3}\text{Tr}(Y_{A})\,S\,\text{Tr}(\bar{B}_{v}\gamma^{\mu}\gamma_{5}B_{v})\right.\\\
&\qquad\qquad\left.-\text{Tr}(Y_{V}\lambda_{a})\text{Tr}(\bar{B}_{v}\gamma^{\mu}\left[T_{a},B_{v}\right])\right]\,,\end{split}$
(4.31)

where $a={3,8,6,7}$. Explicitly, with the choice in Eq. (4.21) we obtain

$\displaystyle\mathcal{L}_{\text{ALP}}^{\partial B}$
$\displaystyle=\frac{\partial_{\mu}\phi}{3\Lambda}\Gamma_{A}\,S\,\left[\bar{n}\gamma^{\mu}\gamma_{5}n+\bar{p}_{v}\gamma^{\mu}\gamma_{5}p_{v}+\bar{\Lambda}^{0}_{v}\gamma^{\mu}\gamma_{5}\Lambda^{0}_{v}+\bar{\Xi}^{0}_{v}\gamma^{\mu}\gamma_{5}\Xi^{0}_{v}+\bar{\Xi}^{-}_{v}\gamma^{\mu}\gamma_{5}\Xi^{-}_{v}\right.$
$\displaystyle\qquad\qquad\quad\left.+\bar{\Sigma}^{0}_{v}\gamma^{\mu}\gamma_{5}\Sigma^{0}_{v}+\bar{\Sigma}^{+}_{v}\gamma^{\mu}\gamma_{5}\Sigma^{+}_{v}+\bar{\Sigma}^{-}_{v}\gamma^{\mu}\gamma_{5}\Sigma^{-}_{v}\right]+$
$\displaystyle\quad\frac{\partial_{\mu}\phi}{6\Lambda}\frac{m_{\pi^{\pm}}^{2}}{m_{\pi^{\pm}}^{2}-m_{\phi}^{2}}\Delta_{A}\left[3\,(D+F)\,(\bar{p}_{v}\gamma^{\mu}\gamma_{5}p_{v}-\bar{n}_{v}\gamma^{\mu}\gamma_{5}n_{v})+3\,(F-D)\,(\bar{\Xi}_{v}^{0}\gamma^{\mu}\gamma_{5}\Xi_{v}^{0}-\bar{\Xi}_{v}^{-}\gamma^{\mu}\gamma_{5}\Xi_{v}^{-})\right.$
$\displaystyle\left.\qquad\qquad\qquad\qquad\qquad+2\sqrt{3}\,D\,(\bar{\Lambda}_{v}\gamma^{\mu}\gamma_{5}\Sigma^{0}_{v}+\bar{\Sigma}_{0}\gamma^{\mu}\gamma_{5}\Lambda_{v}^{0})+6\,F\,(\bar{\Sigma}_{v}^{+}\gamma^{\mu}\gamma_{5}\Sigma_{v}^{+}-\bar{\Sigma}_{v}^{-}\gamma^{\mu}\gamma_{5}\Sigma_{v}^{-})\right]+$
$\displaystyle\quad\frac{\partial_{\mu}\phi}{6\Lambda}\,\Omega_{A}\left[(3F-D)\,(\bar{p}_{v}\gamma^{\mu}\gamma_{5}p_{v}+\bar{n}_{v}\gamma^{\mu}\gamma_{5}n_{v})-(D+3F)\,(\bar{\Xi}^{0}_{v}\gamma^{\mu}\gamma_{5}\Xi_{v}^{0}+\bar{\Xi}_{v}^{-}\gamma^{\mu}\gamma_{5}\Xi_{v}^{-})\right.$
$\displaystyle\qquad\qquad\quad\left.+2\,D\,(\bar{\Sigma}_{v}^{-}\gamma^{\mu}\gamma_{5}\Sigma_{v}^{-}+\bar{\Sigma}_{v}^{+}\gamma^{\mu}\gamma_{5}\Sigma_{v}^{+}+\bar{\Sigma}^{0}_{v}\gamma^{\mu}\gamma_{5}\Sigma^{0}_{v}-\bar{\Lambda}^{0}_{v}\gamma^{\mu}\gamma_{5}\Lambda^{0}_{v}))\right]+$
(4.32)
$\displaystyle\quad\frac{\partial_{\mu}\phi}{6\Lambda}\left\\{\Xi_{A}\left[6(D-F)\,\bar{p}_{v}\gamma^{\mu}\gamma_{5}\Sigma_{v}^{+}+6\,(D+F)\,\bar{\Sigma}_{v}^{-}\gamma^{\mu}\gamma_{5}\Xi_{v}^{-}-\sqrt{6}\,(D+3F)\,\bar{n}_{v}\gamma^{\mu}\gamma_{5}\Lambda_{v}^{0}\right.\right.$
$\displaystyle\qquad\qquad\quad\left.\left.-\sqrt{6}\,(D-3F)\,\bar{\Lambda}_{v}\gamma^{\mu}\gamma_{5}\Xi_{v}^{0}-3\sqrt{2}\,(D-F)\bar{n}_{v}\gamma^{\mu}\gamma_{5}\Sigma^{0}_{v}-3\sqrt{2}\,(D+F)\,\bar{\Sigma}_{v}^{0}\gamma^{\mu}\gamma_{5}\Xi_{v}^{0}\right]+\text{h.c.}\right\\}-$
$\displaystyle\quad\frac{\partial_{\mu}\phi}{2\Lambda}(q_{V}^{u}-q_{V}^{d})\left[\bar{p}_{v}\gamma^{\mu}p_{v}-\bar{n}_{v}\gamma^{\mu}n_{v}+\bar{\Xi}^{0}_{v}\gamma^{\mu}\Xi^{0}_{v}-\bar{\Xi}^{-}_{v}\gamma^{\mu}\Xi^{-}_{v}+2\bar{\Sigma}^{+}_{v}\gamma^{\mu}\Sigma^{+}_{v}-2\bar{\Sigma}^{-}_{v}\gamma^{\mu}\Sigma^{-}_{v}\right]-$
$\displaystyle\quad\frac{\partial_{\mu}\phi}{2\Lambda}(q_{V}^{u}+q_{V}^{d}-2q_{V}^{s})\left[\bar{p}_{v}\gamma^{\mu}p_{v}+\bar{n}_{v}\gamma^{\mu}n_{v}-\bar{\Xi}^{0}_{v}\gamma^{\mu}\Xi^{0}_{v}-\bar{\Xi}^{-}_{v}\gamma^{\mu}\Xi^{-}_{v}\right]+$
$\displaystyle\quad\frac{\partial_{\mu}\phi}{2\Lambda}\left\\{Y_{V}^{ds}\left[\sqrt{6}\bar{\Lambda}_{v}\gamma^{\mu}\Xi^{0}_{v}-\sqrt{6}\bar{n}_{v}\gamma^{\mu}\Lambda_{v}^{0}+\sqrt{2}\bar{n}_{v}\gamma^{\mu}\Sigma^{0}_{v}-\sqrt{2}\bar{\Sigma}^{0}_{v}\gamma^{\mu}\Xi^{0}_{v}+2\bar{\Xi}^{-}_{v}\gamma^{\mu}\Sigma^{-}_{v}-2\bar{p}_{v}\gamma^{\mu}\Sigma^{+}_{v}\right]+\text{h.c.}\right\\}\,,$

where $F=0.441(26)$, $D=0.813(43)$ and $S=0.405(62)$ (see [55] based on the
lattice QCD results in [70]).

This expression can be further simplified by appropriately choosing the
entries of the vectorial matrix $Q_{V}$; setting
$q_{V}^{u}=q_{V}^{d}=q_{V}^{s}=0$ eliminates completely the last but least
line and the previous one from the Lagrangian.

In addition to the contributions in Eq. (4.31) we should also take into
account those stemming from the mixing of neutral mesons into an ALP and
stemming from the baryon-meson Lagrangian piece

$\displaystyle\mathcal{L}_{B\pi}$
$\displaystyle\supset-\frac{D}{2f_{\pi}}\,\text{Tr}(\bar{B}_{v}\gamma^{\mu}\gamma_{5}\left\\{\partial_{\mu}\pi,B_{v}\right\\})-\frac{F}{2f_{\pi}}\,\text{Tr}(\bar{B}_{v}\gamma^{\mu}\gamma_{5}\left[\partial_{\mu}\pi,B_{v}\right])$
$\displaystyle\supset-\frac{\partial_{\mu}\pi_{0}}{6f_{\pi}}\left[D\left(3\bar{p}_{v}\gamma^{\mu}\gamma_{5}p_{v}-3\bar{n}_{v}\gamma^{\mu}\gamma_{5}n_{v}+3\bar{\Xi}^{-}_{v}\gamma^{\mu}\gamma_{5}\Xi^{-}_{v}-3\bar{\Xi}^{0}_{v}\gamma^{\mu}\gamma_{5}\Xi^{0}_{v}+2\sqrt{3}(\bar{\Lambda}^{0}_{v}\gamma^{\mu}\gamma_{5}\Sigma^{0}_{v}+\bar{\Sigma}^{0}_{v}\gamma^{\mu}\gamma_{5}\Lambda^{0}_{v})\right)\right.$
$\displaystyle\qquad\qquad+\left.3F\left(\bar{p}_{v}\gamma^{\mu}\gamma_{5}p_{v}-\bar{n}_{v}\gamma^{\mu}\gamma_{5}n_{v}+2\bar{\Xi}^{0}_{v}\gamma^{\mu}\gamma_{5}\Xi^{0}_{v}-\bar{\Xi}^{-}_{v}\gamma^{\mu}\gamma_{5}\Xi^{-}_{v}+2\bar{\Sigma}^{+}_{v}\gamma^{\mu}\gamma_{5}\Sigma^{+}_{v}-\bar{\Sigma}^{-}_{v}\gamma^{\mu}\gamma_{5}\Sigma^{-}_{v}\right)\right]$
(4.33)
$\displaystyle\quad-\frac{\partial_{\mu}\eta}{2\sqrt{3}f_{\pi}}\left[D\left(\bar{p}_{v}\gamma^{\mu}\gamma_{5}p_{v}+\bar{n}_{v}\gamma^{\mu}\gamma_{5}n_{v}+\bar{\Xi}^{-}_{v}\gamma^{\mu}\gamma_{5}\Xi^{-}_{v}+\bar{\Xi}^{0}_{v}\gamma^{\mu}\gamma_{5}\Xi^{0}_{v}+2\bar{\Lambda}^{0}_{v}\gamma^{\mu}\gamma_{5}\Lambda^{0}_{v}-2\bar{\Sigma}^{0}_{v}\gamma^{\mu}\gamma_{5}\Sigma^{0}_{v}\right.\right.$
$\displaystyle\qquad\qquad\left.-2\bar{\Sigma}^{+}_{v}\gamma^{\mu}\gamma_{5}\Sigma^{+}_{v}-2\bar{\Sigma}^{-}_{v}\gamma^{\mu}\gamma_{5}\Sigma^{-}_{v}\right)+\left.3F\left(-\bar{p}_{v}\gamma^{\mu}\gamma_{5}p_{v}-\bar{n}_{v}\gamma^{\mu}\gamma_{5}n_{v}+\bar{\Xi}^{0}_{v}\gamma^{\mu}\gamma_{5}\Xi^{0}_{v}+\bar{\Xi}^{-}_{v}\gamma^{\mu}\gamma_{5}\Xi^{-}_{v}\right)\right]\,,$

originating from the first-order expansion of the second and third terms in
Eq. (2.34). We neglected terms involving neutral kaons as we can choose the
entries of the matrix $Q_{A}$ in such a way to get rid of them. This is,
however, not simultaneously possible for the $\pi_{0}$ and the $\eta$ and one
will have the following ALP interaction:

$\displaystyle\delta\mathcal{L}_{B\pi}$ $\displaystyle\supset-\xi
A\frac{m_{\phi}^{2}+(\alpha^{\prime}/A)}{m_{\pi_{0}}^{2}-m_{\phi}^{2}}\frac{\partial_{\mu}\phi}{6f_{\pi}}\cdot$
$\displaystyle\qquad\cdot\left[D\left(3\bar{p}_{v}\gamma^{\mu}\gamma_{5}p_{v}-3\bar{n}_{v}\gamma^{\mu}\gamma_{5}n_{v}+3\bar{\Xi}^{-}_{v}\gamma^{\mu}\gamma_{5}\Xi^{-}_{v}-3\bar{\Xi}^{0}_{v}\gamma^{\mu}\gamma_{5}\Xi^{0}_{v}+2\sqrt{3}(\bar{\Lambda}^{0}_{v}\gamma^{\mu}\gamma_{5}\Sigma^{0}_{v}+\bar{\Sigma}^{0}_{v}\gamma^{\mu}\gamma_{5}\Lambda^{0}_{v})\right)\right.$
$\displaystyle\qquad\qquad+\left.3F\left(\bar{p}_{v}\gamma^{\mu}\gamma_{5}p_{v}-\bar{n}_{v}\gamma^{\mu}\gamma_{5}n_{v}+\bar{\Xi}^{0}_{v}\gamma^{\mu}\gamma_{5}\Xi^{0}_{v}-\bar{\Xi}^{-}_{v}\gamma^{\mu}\gamma_{5}\Xi^{-}_{v}+2\bar{\Sigma}^{+}_{v}\gamma^{\mu}\gamma_{5}\Sigma^{+}_{v}-2\bar{\Sigma}^{-}_{v}\gamma^{\mu}\gamma_{5}\Sigma^{-}_{v}\right)\right]$
$\displaystyle\quad-\xi
B\frac{m_{\phi}^{2}+(\beta^{\prime}/B)}{m_{\eta}^{2}-m_{\phi}^{2}}\frac{\partial_{\mu}\phi}{2\sqrt{3}f_{\pi}}\cdot$
(4.34)
$\displaystyle\qquad\cdot\left[D\left(\bar{p}_{v}\gamma^{\mu}\gamma_{5}p_{v}+\bar{n}_{v}\gamma^{\mu}\gamma_{5}n_{v}+\bar{\Xi}^{-}_{v}\gamma^{\mu}\gamma_{5}\Xi^{-}_{v}+\bar{\Xi}^{0}_{v}\gamma^{\mu}\gamma_{5}\Xi^{0}_{v}+2\bar{\Lambda}^{0}_{v}\gamma^{\mu}\gamma_{5}\Lambda^{0}_{v}-2\bar{\Sigma}^{0}_{v}\gamma^{\mu}\gamma_{5}\Sigma^{0}_{v}\right.\right.$
$\displaystyle\qquad\qquad\left.-2\bar{\Sigma}^{+}_{v}\gamma^{\mu}\gamma_{5}\Sigma^{+}_{v}-2\bar{\Sigma}^{-}_{v}\gamma^{\mu}\gamma_{5}\Sigma^{-}_{v}\right)+\left.3F\left(-\bar{p}_{v}\gamma^{\mu}\gamma_{5}p_{v}-\bar{n}_{v}\gamma^{\mu}\gamma_{5}n_{v}+\bar{\Xi}^{0}_{v}\gamma^{\mu}\gamma_{5}\Xi^{0}_{v}+\bar{\Xi}^{-}_{v}\gamma^{\mu}\gamma_{5}\Xi^{-}_{v}\right)\right]\,.$

With our choice in Eq. (4.21) this amounts to

$\displaystyle\delta\mathcal{L}_{B\pi}$
$\displaystyle=\frac{M_{\phi}^{2}}{m_{\eta}^{2}-M_{\phi}^{2}}\bigg{[}1+\frac{m_{\pi^{\pm}}^{2}}{m_{\pi^{\pm}}^{2}-M_{\phi}^{2}}\frac{m_{d}-m_{u}}{m_{d}+m_{u}}\frac{\Delta_{A}}{\Omega_{A}}\bigg{]}\Omega_{A}\frac{\partial_{\mu}\phi}{6\Lambda}\cdot[(3F-D)\,(\bar{p}_{v}\gamma^{\mu}\gamma_{5}p_{v}+\bar{n}_{v}\gamma^{\mu}\gamma_{5}n_{v})$
$\displaystyle-(D+3F)\,(\bar{\Xi}^{0}_{v}\gamma^{\mu}\gamma_{5}\Xi_{v}^{0}+\bar{\Xi}_{v}^{-}\gamma^{\mu}\gamma_{5}\Xi_{v}^{-})+2D(\bar{\Sigma}_{v}^{-}\gamma^{\mu}\gamma_{5}\Sigma_{v}^{-}+\bar{\Sigma}_{v}^{+}\gamma^{\mu}\gamma_{5}\Sigma_{v}^{+}+\bar{\Sigma}^{0}_{v}\gamma^{\mu}\gamma_{5}\Sigma^{0}_{v}-\bar{\Lambda}^{0}_{v}\gamma^{\mu}\gamma_{5}\Lambda^{0}_{v}))].$
(4.35)

Summing this piece with the ones in Eq. (4.2) we finally obtain

$\displaystyle\mathcal{L}_{\text{ALP}}^{\text{BD, OC}}$
$\displaystyle=\frac{\partial_{\mu}\phi}{3\Lambda}\Gamma_{A}\,S\,\left[\bar{n}\gamma^{\mu}\gamma_{5}n+\bar{p}_{v}\gamma^{\mu}\gamma_{5}p_{v}+\bar{\Lambda}^{0}_{v}\gamma^{\mu}\gamma_{5}\Lambda^{0}_{v}+\bar{\Xi}^{0}_{v}\gamma^{\mu}\gamma_{5}\Xi^{0}_{v}+\bar{\Xi}^{-}_{v}\gamma^{\mu}\gamma_{5}\Xi^{-}_{v}\right.$
$\displaystyle\qquad\qquad\quad\left.+\bar{\Sigma}^{0}_{v}\gamma^{\mu}\gamma_{5}\Sigma^{0}_{v}+\bar{\Sigma}^{+}_{v}\gamma^{\mu}\gamma_{5}\Sigma^{+}_{v}+\bar{\Sigma}^{-}_{v}\gamma^{\mu}\gamma_{5}\Sigma^{-}_{v}\right]+$
$\displaystyle\quad\frac{\partial_{\mu}\phi}{6\Lambda}\frac{m_{\pi^{\pm}}^{2}}{m_{\pi^{\pm}}^{2}-m_{\phi}^{2}}\Delta_{A}\left[3\,(D+F)\,(\bar{p}_{v}\gamma^{\mu}\gamma_{5}p_{v}-\bar{n}_{v}\gamma^{\mu}\gamma_{5}n_{v})+3\,(F-D)\,(\bar{\Xi}_{v}^{0}\gamma^{\mu}\gamma_{5}\Xi_{v}^{0}-\bar{\Xi}_{v}^{-}\gamma^{\mu}\gamma_{5}\Xi_{v}^{-})\right.$
$\displaystyle\left.\qquad\qquad\qquad\qquad\qquad+2\sqrt{3}\,D\,(\bar{\Lambda}_{v}\gamma^{\mu}\gamma_{5}\Sigma^{0}_{v}+\bar{\Sigma}_{0}\gamma^{\mu}\gamma_{5}\Lambda_{v}^{0})+6\,F\,(\bar{\Sigma}_{v}^{+}\gamma^{\mu}\gamma_{5}\Sigma_{v}^{+}-\bar{\Sigma}_{v}^{-}\gamma^{\mu}\gamma_{5}\Sigma_{v}^{-})\right]+$
$\displaystyle\quad\frac{\partial_{\mu}\phi}{6\Lambda}\,\Omega_{A}\left[\frac{m_{\eta}^{2}}{m_{\eta}^{2}-m_{\phi}^{2}}+\frac{m_{\pi_{\pm}^{2}}}{m_{\pi_{\pm}^{2}}-m_{\phi}^{2}}\frac{m_{\phi}^{2}}{m_{\eta}^{2}-m_{\phi}^{2}}\frac{m_{d}-m_{u}}{m_{d}+m_{u}}\frac{\Delta_{A}}{\Omega_{A}}\right]\cdot$
(4.36)
$\displaystyle\qquad\qquad\qquad\cdot\left[(3F-D)\,(\bar{p}_{v}\gamma^{\mu}\gamma_{5}p_{v}+\bar{n}_{v}\gamma^{\mu}\gamma_{5}n_{v})-(D+3F)\,(\bar{\Xi}^{0}_{v}\gamma^{\mu}\gamma_{5}\Xi_{v}^{0}+\bar{\Xi}_{v}^{-}\gamma^{\mu}\gamma_{5}\Xi_{v}^{-})\right.$
$\displaystyle\qquad\qquad\qquad\quad\left.+2\,D\,(\bar{\Sigma}_{v}^{-}\gamma^{\mu}\gamma_{5}\Sigma_{v}^{-}+\bar{\Sigma}_{v}^{+}\gamma^{\mu}\gamma_{5}\Sigma_{v}^{+}+\bar{\Sigma}^{0}_{v}\gamma^{\mu}\gamma_{5}\Sigma^{0}_{v}-\bar{\Lambda}^{0}_{v}\gamma^{\mu}\gamma_{5}\Lambda^{0}_{v}))\right]+$
$\displaystyle\quad\frac{\partial_{\mu}\phi}{6\Lambda}\left\\{\Xi_{A}\left[6(D-F)\,\bar{p}_{v}\gamma^{\mu}\gamma_{5}\Sigma_{v}^{+}+6\,(D+F)\,\bar{\Sigma}_{v}^{-}\gamma^{\mu}\gamma_{5}\Xi_{v}^{-}-\sqrt{6}\,(D+3F)\,\bar{n}_{v}\gamma^{\mu}\gamma_{5}\Lambda_{v}^{0}\right.\right.$
$\displaystyle\qquad\qquad\quad\left.\left.-\sqrt{6}\,(D-3F)\,\bar{\Lambda}_{v}\gamma^{\mu}\gamma_{5}\Xi_{v}^{0}-3\sqrt{2}\,(D-F)\bar{n}_{v}\gamma^{\mu}\gamma_{5}\Sigma^{0}_{v}-3\sqrt{2}\,(D+F)\,\bar{\Sigma}_{v}^{0}\gamma^{\mu}\gamma_{5}\Xi_{v}^{0}\right]+\text{h.c.}\right\\}+$
$\displaystyle\quad\frac{\partial_{\mu}\phi}{2\Lambda}\left\\{Y_{V}^{ds}\left[\sqrt{6}\bar{\Lambda}_{v}\gamma^{\mu}\Xi^{0}_{v}-\sqrt{6}\bar{n}_{v}\gamma^{\mu}\Lambda_{v}^{0}+\sqrt{2}\bar{n}_{v}\gamma^{\mu}\Sigma^{0}_{v}-\sqrt{2}\bar{\Sigma}^{0}_{v}\gamma^{\mu}\Xi^{0}_{v}+2\bar{\Xi}^{-}_{v}\gamma^{\mu}\Sigma^{-}-2\bar{p}_{v}\gamma^{\mu}\Sigma^{+}_{v}\right]+\text{h.c.}\right\\}\,,$

where we set $q_{V}^{u}=q_{V}^{d}=q_{V}^{s}=0$.

There is a second kind of contribution to the ALP-baryon couplings, stemming
from the non-derivative couplings to the quark and gluonic scalar densities.
It can be generically parametrized as follows

$\mathcal{L}_{\text{ALP}}^{\text{Bar.,
ND}}=\frac{\phi}{\Lambda}C_{\phi\text{BB}}\bar{B}_{v}B_{v}\,.$ (4.37)

Since, once again, the ALP has no impact on the hadronic matrix element, i.e.
$\bra{B}\phi\,\theta^{\mu}_{\mu}\ket{B}=\phi\bra{B}\theta^{\mu}_{\mu}\ket{B}$,
we can then simply compute the effective coupling between an ALP and baryons
by requiring the equivalence

$\sum_{q}\zeta_{q}\bar{q}q\,\phi+g_{\phi
gg}\,\phi\,GG\qquad\longrightarrow\qquad\bra{B}\sum_{q}\zeta_{q}\bar{q}q+g_{\phi
gg}\,GG\ket{B}\,\phi\,\bar{B}B\,,$ (4.38)

and computing the hadronic matrix element of the scalar quark and gluon
densities.

These can be related to the hadronic matrix elements of the proton by making
use of the $SU(3)_{V}$ symmetry underlying the construction of the baryonic
Lagrangian itself [71]. In particular, it emerges that the matrix element of
the scalar gluonic density is approximately universal, i.e.

$\bra{B}GG\ket{B}=\bra{p}GG\ket{p}=-\frac{8\pi}{9\alpha_{s}}(m_{p}-\sigma_{u}-\sigma_{d}-\sigma_{s})\,,\quad\text{with}\quad
B=\\{p,n,\Sigma^{+},\Sigma^{-},\Sigma^{0},\Xi^{-},\Xi^{0},\Lambda^{0}\\}\,,$
(4.39)

while one obtains

$\begin{split}&\bra{p}\bar{u}u\ket{p}=\frac{\sigma_{u}}{m_{u}}\,,\qquad\bra{p}\bar{d}d\ket{p}=\frac{\sigma_{d}}{m_{d}}\,,\qquad\bra{p}\bar{s}s\ket{p}=\frac{\sigma_{s}}{m_{s}}\,,\\\
&\bra{n}\bar{u}u\ket{n}=\frac{\sigma_{d}}{m_{d}}\,,\qquad\bra{n}\bar{d}d\ket{n}=\frac{\sigma_{u}}{m_{u}}\,,\qquad\bra{n}\bar{s}s\ket{n}=\frac{\sigma_{s}}{m_{s}}\,,\\\
\end{split}$ (4.40)

for the nucleonic doublet, as it was the case in a two-flavour setting. As far
as the remaining baryons are concerned one instead has

$\begin{split}&\bra{\Sigma^{0}}\bar{u}u\ket{\Sigma^{0}}=\frac{1}{2}\left(\frac{\sigma_{u}}{m_{u}}+\frac{\sigma_{s}}{m_{s}}\right)\qquad\bra{\Sigma^{0}}\bar{d}d\ket{\Sigma^{0}}=\frac{1}{2}\left(\frac{\sigma_{u}}{m_{u}}+\frac{\sigma_{s}}{m_{s}}\right)\qquad\bra{\Sigma^{0}}\bar{s}s\ket{\Sigma^{0}}=\frac{\sigma_{d}}{m_{d}}\\\
&\bra{\Sigma^{+}}\bar{u}u\ket{\Sigma^{+}}=\frac{\sigma_{u}}{m_{u}}\qquad\bra{\Sigma^{+}}\bar{d}d\ket{\Sigma^{+}}=\frac{\sigma_{s}}{m_{s}}\qquad\bra{\Sigma^{+}}\bar{s}s\ket{\Sigma^{+}}=\frac{\sigma_{d}}{m_{d}}\\\
&\bra{\Sigma^{-}}\bar{u}u\ket{\Sigma^{-}}=\frac{\sigma_{s}}{m_{s}}\qquad\bra{\Sigma^{-}}\bar{d}d\ket{\Sigma^{-}}=\frac{\sigma_{u}}{m_{u}}\qquad\bra{\Sigma^{-}}\bar{s}s\ket{\Sigma^{-}}=\frac{\sigma_{d}}{m_{d}}\end{split}$
(4.41)

$\begin{split}&\bra{\Xi^{0}}\bar{u}u\ket{\Xi^{0}}=\frac{\sigma_{d}}{m_{d}}\quad\quad\,\,\,\bra{\Xi^{0}}\bar{d}d\ket{\Xi^{0}}=\frac{\sigma_{s}}{m_{s}}\qquad\\!\quad\bra{\Xi^{0}}\bar{s}s\ket{\Xi^{0}}=\frac{\sigma_{u}}{m_{u}}\\\
&\bra{\Xi^{-}}\bar{u}u\ket{\Xi^{-}}=\frac{\sigma_{s}}{m_{s}}\qquad\,\bra{\Xi^{-}}\bar{d}d\ket{\Xi^{-}}=\frac{\sigma_{d}}{m_{d}}\qquad\,\bra{\Xi^{-}}\bar{s}s\ket{\Xi^{-}}=\frac{\sigma_{u}}{m_{u}}\\\
&\bra{\Lambda^{0}}\bar{u}u\ket{\Lambda^{0}}=\frac{1}{6}\left(\frac{\sigma_{u}}{m_{u}}+\frac{\sigma_{s}}{m_{s}}+\frac{4\sigma_{d}}{m_{d}}\right)\quad\bra{\Lambda^{0}}\bar{d}d\ket{\Lambda^{0}}=\frac{1}{6}\left(\frac{\sigma_{u}}{m_{u}}+\frac{\sigma_{s}}{m_{s}}+\frac{4\sigma_{d}}{m_{d}}\right)\\\
&\bra{\Lambda^{0}}\bar{s}s\ket{\Lambda^{0}}=\frac{1}{3}\left(\frac{2\sigma_{u}}{m_{u}}+\frac{2\sigma_{s}}{m_{s}}-\frac{\sigma_{d}}{m_{d}}\right)\,.\\\
\end{split}$ (4.42)

These results can be then directly applied to Eq. (2.1) to extract the
following ALP-Baryon scalar couplings:

$\begin{split}&C_{\phi\text{pp}}=\frac{v}{m_{u}}y_{q,S}^{u}\,\sigma_{u}+\frac{v}{m_{d}}y_{q,S}^{d}\,\sigma_{d}+\frac{v}{m_{s}}y_{q,S}^{s}\,\sigma_{s}-\frac{32\pi^{2}C_{g}}{9}\,(m_{p}-\sigma_{u}-\sigma_{d}-\sigma_{s})\,,\\\
&C_{\phi\text{nn}}=\frac{v}{m_{d}}y_{q,S}^{u}\,\sigma_{d}+\frac{v}{m_{u}}y_{q,S}^{d}\,\sigma_{u}+\frac{v}{m_{s}}y_{q,S}^{s}\,\sigma_{s}-\frac{32\pi^{2}C_{g}}{9}\,(m_{p}-\sigma_{u}-\sigma_{d}-\sigma_{s})\,,\\\
&C_{\phi\Sigma^{+}\Sigma^{+}}=\frac{v}{m_{u}}y_{q,S}^{u}\,\sigma_{u}+\frac{v}{m_{s}}y_{q,S}^{d}\,\sigma_{s}+\frac{v}{m_{d}}y_{q,S}^{s}\,\sigma_{d}-\frac{32\pi^{2}C_{g}}{9}\,(m_{p}-\sigma_{u}-\sigma_{d}-\sigma_{s})\,,\\\
&C_{\phi\Sigma^{0}\Sigma^{0}}=\frac{v}{2}y_{q,S}^{u}\,\left(\frac{\sigma_{u}}{m_{u}}+\frac{\sigma_{s}}{m_{s}}\right)+\frac{v}{2}y_{q,S}^{d}\,\left(\frac{\sigma_{u}}{m_{u}}+\frac{\sigma_{s}}{m_{s}}\right)+\frac{v}{m_{d}}y_{q,S}^{s}\,\sigma_{d}-\frac{32\pi^{2}C_{g}}{9}\,(m_{p}-\sigma_{u}-\sigma_{d}-\sigma_{s})\,,\\\
&C_{\phi\Sigma^{-}\Sigma^{-}}=\frac{v}{m_{s}}y_{q,S}^{u}\,\sigma_{s}+\frac{v}{m_{u}}y_{q,S}^{d}\,\sigma_{u}+\frac{v}{m_{d}}y_{q,S}^{s}\,\sigma_{d}-\frac{32\pi^{2}C_{g}}{9}\,(m_{p}-\sigma_{u}-\sigma_{d}-\sigma_{s})\,,\end{split}$
(4.43)
$\begin{split}&C_{\phi\Xi^{0}\Xi^{0}}=\frac{v}{m_{d}}y_{q,S}^{u}\,\sigma_{d}+\frac{v}{m_{s}}y_{q,S}^{d}\,\sigma_{s}+\frac{v}{m_{u}}y_{q,S}^{s}\,\sigma_{u}-\frac{32\pi^{2}C_{g}}{9}\,(m_{p}-\sigma_{u}-\sigma_{d}-\sigma_{s})\,,\\\
&C_{\phi\Xi^{-}\Xi^{-}}=\frac{v}{m_{s}}y_{q,S}^{u}\,\sigma_{s}+\frac{v}{m_{d}}y_{q,S}^{d}\,\sigma_{d}+\frac{v}{m_{u}}y_{q,S}^{s}\,\sigma_{u}-\frac{32\pi^{2}C_{g}}{9}\,(m_{p}-\sigma_{u}-\sigma_{d}-\sigma_{s})\,,\\\
&C_{\phi\Lambda^{0}\Lambda^{0}}=\frac{v}{6}y_{q,S}^{u}\,\left(\frac{\sigma_{u}}{m_{u}}+\frac{\sigma_{s}}{m_{s}}+4\frac{\sigma_{d}}{m_{d}}\right)+\frac{v}{6}y_{q,S}^{d}\,\left(\frac{\sigma_{u}}{m_{u}}+\frac{\sigma_{s}}{m_{s}}+4\frac{\sigma_{d}}{m_{d}}\right)+\frac{v}{3}y_{q,S}^{s}\,\left(2\frac{\sigma_{u}}{m_{u}}+2\frac{\sigma_{s}}{m_{s}}-\frac{\sigma_{d}}{m_{d}}\right)\\\
&\qquad\qquad-\frac{32\pi^{2}C_{g}}{9}\,(m_{p}-\sigma_{u}-\sigma_{d}-\sigma_{s})\,,\end{split}$
(4.44)

where the explicit values for the various quantities and the procedure to
obtain them can be found in [69].

###  4.3 Summary and Jarlskog invariants

To summarize, we built the low-energy effective Lagrangian describing ALP
interactions with mesons, baryons and photons, finding that it can be
decomposed into two pieces with opposite CP transformation properties

$\displaystyle(\mathcal{L}_{\text{ALP}}^{\chi\text{pt}})^{\rm CP-even}$
$\displaystyle=e^{2}\frac{c_{\gamma}}{\Lambda}\,\phi\,F^{\mu\nu}F_{\mu\nu}+\mathcal{L}_{\text{ALP}}^{\kappa}+\mathcal{L}_{\text{ALP}}^{\mathcal{Z}}+\mathcal{L}_{\text{ALP}}^{\text{Bar.
ND}}$ (4.45) $\displaystyle(\mathcal{L}_{\text{ALP}}^{\chi\text{pt}})^{\rm CP-
odd}$
$\displaystyle=e^{2}\frac{\tilde{c}_{\gamma}}{\Lambda}\,\phi\,F^{\mu\nu}\tilde{F}_{\mu\nu}+\mathcal{L}_{\text{ALP}}^{\text{MI}}+\mathcal{L}_{\text{ALP}}^{\text{ADI}}+\mathcal{L}_{\text{ALP}}^{\eta\text{KT}+\text{MT}}+\mathcal{L}_{\text{ALP}}^{\text{BD,
OC}}\,,$ (4.46)

where the explicit expressions for the terms in the above Lagrangians have
been provided previously. The associated Jarlskog invariants are reported in
Table 2. To simplify their interpretation, we also report below the
compositions of all the coefficients appearing in Table 2 in terms of the
microscopic parameters of Eq. 2.1

$\displaystyle c_{\gamma}$ $\displaystyle=$
$\displaystyle\,C_{\gamma}-\frac{\beta^{0}_{\text{QED}}}{\beta^{0}_{\text{QCD}}}\frac{g_{s}^{2}}{e^{2}}C_{g}\,,$
(4.47) $\displaystyle\tilde{c}_{\gamma}$ $\displaystyle=$
$\displaystyle\,\tilde{C}_{\gamma}-4N_{c}\text{ tr
}(Q_{A}Q_{q}^{2})\tilde{C}_{g}\,,$ (4.48) $\displaystyle\kappa$
$\displaystyle=$
$\displaystyle\frac{8\pi}{\alpha_{s}}\frac{g_{s}^{2}C_{g}}{\beta^{0}_{\text{QCD}}}\,,$
(4.49) $\displaystyle\mathcal{Z}$ $\displaystyle=$ $\displaystyle
y_{S}+\frac{g_{s}^{2}C_{g}}{\beta^{0}_{\text{QCD}}}\frac{8\pi}{\alpha_{s}}\frac{M_{q}}{v}\,,$
(4.50) $\displaystyle\Delta_{A}$ $\displaystyle=$ $\displaystyle
Y_{P}^{u}+Y_{P}^{d}-16\pi^{2}\tilde{C}_{g}\frac{m_{s}(m_{d}-m_{u})}{m_{u}m_{d}+m_{u}m_{s}+m_{d}m_{s}}\,,$
(4.51) $\displaystyle\Omega_{A}$ $\displaystyle=$ $\displaystyle
Y_{P}^{u}+Y_{P}^{d}-2Y_{P}^{s}\,,$ (4.52) $\displaystyle\Xi_{A}$
$\displaystyle=$ $\displaystyle
Y_{P}^{ds}\frac{m_{\pi^{\pm}}^{2}(m_{d}+m_{s})}{(m_{d}+m_{s})m_{\pi^{\pm}}-m_{\phi}^{2}(m_{u}+m_{d})}\,,$
(4.53) $\displaystyle\Gamma_{A}$ $\displaystyle=$ $\displaystyle
Y_{P}^{u}+Y_{P}^{d}+Y_{P}^{s}-\frac{1}{2}\,,$ (4.54) $\displaystyle
Y_{V}^{ds}$ $\displaystyle=$ $\displaystyle Y_{S}^{ds}\,,$ (4.55)
$\displaystyle C_{\phi\text{BB}}$ $\displaystyle=$ $\displaystyle\text{see
equations\leavevmode\nobreak\ \eqref{eq:cpp3f} and \leavevmode\nobreak\
\eqref{eq:cpp3f_bis}}\,.$ (4.56)

| $c_{\gamma}$ | $y_{\ell,S}$ | $\kappa$ | $\mathcal{Z}$ | $Y_{V}^{ds}$ | $C_{\phi\text{BB}}$
---|---|---|---|---|---|---
$\tilde{c}_{\gamma}$ | $\tilde{c}_{\gamma}\,c_{\gamma}$ | $\tilde{c}_{\gamma}\,y_{\ell,S}$ | $\tilde{c}_{\gamma}\,\kappa$ | $\tilde{c}_{\gamma}\,\mathcal{Z}$ | $\tilde{c}_{\gamma}\,Y_{V}^{ds}$ | $\tilde{c}_{\gamma}\,C_{\phi\text{BB}}$
$y_{\ell,P}$ | $y_{\ell,P}\,c_{\gamma}$ | $y_{\ell,P}\,y_{\ell,S}$ | $y_{\ell,P}\,\kappa$ | $y_{\ell,P}\,\mathcal{Z}$ | $y_{\ell,P}\,Y_{V}^{ds}$ | $y_{\ell,P}\,C_{\phi\text{BB}}$
$\Delta_{A}$ | $\Delta_{A}\,c_{\gamma}$ | $\Delta_{A}\,y_{\ell,S}$ | $\Delta_{A}\,\kappa$ | $\Delta_{A}\,\mathcal{Z}$ | $\Delta_{A}\,Y_{V}^{ds}$ | $\Delta_{A}\,C_{\phi\text{BB}}$
$\Omega_{A}$ | $\Omega_{A}\,c_{\gamma}$ | $\Omega_{A}\,y_{\ell,S}$ | $\Omega_{A}\,\kappa$ | $\Omega_{A}\,\mathcal{Z}$ | $\Omega_{A}\,Y_{V}^{ds}$ | $\Omega_{A}\,C_{\phi\text{BB}}$
$\Xi_{A}$ | $\Xi_{A}\,c_{\gamma}$ | $\Xi_{A}\,y_{\ell,S}$ | $\Xi_{A}\,\kappa$ | $\Xi_{A}\,\mathcal{Z}$ | $\Xi_{A}\,Y_{V}^{ds}$ | $\Xi_{A}\,C_{\phi\text{BB}}$
$\Gamma_{A}$ | $\Gamma_{A}\,c_{\gamma}$ | $\Gamma_{A}\,y_{\ell,S}$ | $\Gamma_{A}\,\kappa$ | $\Gamma_{A}\,\mathcal{Z}$ | $\Gamma_{A}\,Y_{V}^{ds}$ | $\Gamma_{A}\,C_{\phi\text{BB}}$
Table 2: Jarlskog invariants emerging from the interactions in (4.45) and
(4.46)

##  5 Phenomenological applications

Although a detailed phenomenological analysis is beyond the scope of the
present work, in the following we will briefly discuss a few interesting CP-
violating observables which can be analysed by means of the ALP interactions
derived above in the the $N_{f}=2$ and $N_{f}=3$ frameworks.222For future
phenomenological applications, fully customisable FeynRules [72] models both
for $N_{f}=2$ and $N_{f}=3$ settings are provided as ancillary files.

###  5.1 Proton and neutron EDM

The effective Lagrangian given in (3.13) immediately allows one to compute the
contributions of a light ALP to the proton EDM. Working in the basis without
ALP-pion mixing, the only relevant topologies are the ones in Fig. 1.

$p$$p$$\gamma$$p$$p$$\phi$$p$$p$$\gamma$$\phi$$\gamma$$p$

$p$$p$$\gamma$$p$$p$$\gamma$$\gamma$$\gamma$$\phi$

Figure 1: Feynman diagrams contributing to the ALP-mediated effects to the
proton EDM, $d_{p}$.

Defining the electric dipole moment of a fermion $\psi$ via
$\mathcal{L}_{\text{dip.}}=-i\,(d_{\psi}/2)\,\bar{\psi}\sigma^{\mu\nu}\gamma_{5}\psi
F_{\mu\nu}$, we find that

$d_{p}\simeq-\frac{e\,Q_{p}}{4\pi^{2}\Lambda^{2}}\left[C_{\phi\text{pp}}\tilde{C}_{\phi\text{p}}+e^{2}m_{p}c_{\gamma}\tilde{C}_{\phi\text{p}}\left(6+2\ln\frac{\Lambda_{\text{ren}}^{2}}{m_{p}^{2}}\right)+e^{2}\tilde{c}_{\gamma}C_{\phi\text{pp}}\left(2+\ln\frac{\Lambda_{\text{ren}}^{2}}{m_{p}^{2}}\right)+3\frac{Q_{p}^{2}}{\pi^{2}}m_{p}e^{6}c_{\gamma}\tilde{c}_{\gamma}\ln^{2}\frac{\Lambda_{\text{ren}}}{m_{\phi}}\right]\,,$
(5.1)

where we worked in the limit $m_{\phi}\ll m_{p}$ and the renormalization scale
is set to $\Lambda_{\rm ren}\simeq m_{p}$.

A natural question is whether a similar effect can be induced for the neutron
EDM, $d_{n}$. Actually, no such contribution can be generated at leading order
in $\chi$pt, due to the fact that neutrons have null electric charge and they
do not have any minimal coupling to the electromagnetic field-strength tensor.
Nonetheless, couplings of neutrons to photons are generated at next-to-leading
order in $\chi$pt [73, 74]. In the $N_{f}=2$ case, the relevant relativistic
Lagrangian reads

$\mathcal{L}_{\gamma\text{N}}^{\text{NLO}}=\frac{F_{\mu\nu}}{4m_{N}}\left[\,\mathcal{C}_{p}\,\bar{p}\sigma^{\mu\nu}p+\mathcal{C}_{n}\,\bar{n}\sigma^{\mu\nu}n\,\right]\,,$
(5.2)

where $\mathcal{C}_{p}\simeq 1.79$ [75] and $\mathcal{C}_{n}\simeq 1.91$ [75]
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. We find the following order of magnitude estimate:

$d_{n}\sim-\frac{e\,\mathcal{C}_{n}}{4\pi^{2}\Lambda^{2}}\left[-\frac{3}{8}C_{\phi\text{nn}}\tilde{C}_{\phi\text{n}}\frac{\Lambda_{\text{ren}}^{2}}{m_{n}^{2}}+e^{2}(2m_{n}c_{\gamma}\tilde{C}_{\phi\text{n}}+3\tilde{c}_{\gamma}C_{\phi\text{nn}})\ln\frac{\Lambda_{\text{ren}}^{2}}{m_{n}^{2}}\right]\,.$
(5.3)

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 [76] and
$d_{n}<1.8\times 10^{-26}\,e$ cm (90% C.L.) [77, 78]. By turning on just two
couplings at a time and imposing these bounds, we can constrain some of the
low-energy Jarlskog invariants of the theory, see 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. To make contact with the notation in Ref. [43] here
we have defined $y_{P}^{q}=-2\frac{m_{q}}{v}Y_{P}^{q}$.

###  5.2 CP-violating ALP decays

A significant observable probing CP-violating effects is given by the ratio

$r(i,j)\equiv\frac{\text{BR}\left(\phi\rightarrow\pi_{i}\pi_{j}\right)}{\text{BR}\left(\phi\rightarrow\pi_{i}\pi_{j}\pi_{0}\right)}\,,\qquad\qquad(i,j)=\\{(+-),(00)\\}\,,$
(5.4)

which could be relevant in phenomenological analyses of hadronically coupled
GeV-scale ALPs, see e.g. [27]. The associated scattering amplitudes can in
turn be computed by making use of the Feynman rules extracted in a two-flavour
setting from the Lagrangian in (3.13). Starting from

$\begin{split}i\mathcal{M}(\phi(Q)\rightarrow\pi^{+}(p_{1})\pi^{-}(p_{2}))&=-i\frac{m_{\pi}^{2}}{\Lambda}\left[4\kappa-v\frac{\mathcal{Z}_{d}+\mathcal{Z}_{u}}{m_{d}+m_{u}}\right]\,,\\\
i\mathcal{M}(\phi(Q)\rightarrow\pi^{0}(p_{1})\pi^{0}(p_{2}))&=-i\frac{1}{\Lambda}\left[\kappa(2m_{\pi}^{2}+p_{1}\cdot
p_{2})-vm_{\pi}^{2}\frac{\mathcal{Z}_{d}+\mathcal{Z}_{u}}{m_{d}+m_{u}}\right]\,,\\\
i\mathcal{M}(\phi(Q)\rightarrow\pi^{+}(p_{1})\pi^{-}(p_{2})\pi^{0}(p_{3}))&=-i\frac{\Delta_{ud}^{A}}{f_{\pi}\Lambda}\frac{m_{\pi}^{2}m_{\phi}^{2}}{(m_{\phi}^{2}-m_{\pi}^{2})}\,,\\\
i\mathcal{M}(\phi(Q)\rightarrow\pi^{0}(p_{1})\pi^{0}(p_{2})\pi^{0}(p_{3}))&=-i\frac{\Delta_{ud}^{A}}{f_{\pi}\Lambda}\frac{m_{\pi}^{2}}{(m_{\phi}^{2}-m_{\pi}^{2})}\left[m_{\phi}^{2}-2\,Q\cdot
p_{3}\right]\,,\end{split}$ (5.5)

the decay rates of two-body and three-body processes are then computed making
use of the formulas:

$\begin{split}\Gamma_{\phi\rightarrow\pi_{i}\pi_{j}}&=\frac{g_{ij}}{64\pi^{2}}\frac{1}{m_{\phi}}\sqrt{1-\frac{4m_{\pi}^{2}}{m_{\phi}^{2}}}\int_{0}^{2\pi}d\varphi\int_{0}^{1}d\cos\theta|\bar{\mathcal{M}}|^{2}\,,\\\
\Gamma_{\phi\rightarrow\pi_{i}\pi_{j}\pi_{0}}&=\frac{m_{\phi}}{256\pi^{2}}g_{ij0}\int_{x_{1}^{\text{max}}}^{x_{1}^{\text{min}}}dx_{1}\int_{x_{2}^{\text{max}}}^{x_{2}^{\text{min}}}dx_{2}|\bar{\mathcal{M}}|^{2}\,,\end{split}$
(5.6)

where $g_{+-}=g_{+-0}=1$, $g_{00}=1/2$ and $g_{000}=1/6$ are the symmetry
factors for the final-state particles. In the three-body final space the
variables $x_{i}$ are defined via $x_{i}=2Q\cdot p_{i}$, while the extrema of
integration are

$x_{2}^{\text{max,min}}=\frac{x_{1}^{2}-3x_{1}+2-r_{\pi}(x_{1}-2)\pm\sqrt{-3r_{\pi}^{2}+2r_{\pi}(x_{1}-1)+(x_{1}^{2}-1)^{2}}\sqrt{x_{1}^{2}-4r_{\pi}}}{2(r_{\pi}-x_{1}+1)}\,,$
(5.7)

and

$x_{1}^{\text{max}}=1-3r_{\pi}\,,\qquad\qquad
x_{1}^{\text{min}}=2\sqrt{r_{\pi}}\,,$ (5.8)

where we have defined $r_{\pi}=m_{\pi}^{2}/m_{\phi}^{2}$. As a consequence one
finds

$\begin{split}&r(0,0)=48\pi^{2}\frac{f_{\pi}^{2}}{m_{\phi}^{2}}\frac{(1-r_{\pi})^{5/2}}{r_{\pi}^{2}}\frac{\left|vr_{\pi}\frac{\mathcal{Z}_{u}+\mathcal{Z}_{d}}{m_{u}+m_{d}}-\kappa\,\left(\frac{1}{2}+r_{\pi}\right)\right|^{2}}{|\Delta_{ud}^{A}|^{2}}\frac{1}{\int
dx_{1}dx_{2}}\,,\\\
&r(+,-)=16\pi^{2}\frac{f_{\pi}^{2}}{m_{\phi}^{2}}(1-r_{\pi})^{5/2}\frac{\left|4\kappa-v\frac{\mathcal{Z}_{u}+\mathcal{Z}_{d}}{m_{u}+m_{d}}\right|^{2}}{|\Delta_{ud}^{A}|^{2}}\frac{1}{\int
dx_{1}dx_{2}(x_{1}+x_{2}-1)^{2}}\,,\end{split}$ (5.9)

where we have used the fact that $x_{1}+x_{2}+x_{3}=2$ by definition. As an
example, we evaluate the above ratios for $m_{\phi}=m_{K}$, obtaining the
following numerical estimates

$r(0,0)\simeq 6.3\times
10^{3}\left|\frac{C_{g}-176\,y_{S}^{d}-176\,y_{S}^{u}}{\tilde{C}_{g}-454\,y_{P}^{d}+981\,y_{P}^{u}}\right|^{2}\,,\qquad
r(+,-)\simeq 4.2\times
10^{3}\left|\frac{C_{g}-367\,y_{S}^{d}-367\,y_{S}^{u}}{\tilde{C}_{g}-454\,y_{P}^{d}+981\,y_{P}^{u}}\right|^{2}\,.$
(5.10)

###  5.3 Kaon decays

Kaon decays are observables of paramount interest when dealing with CP
violation in the mesonic sector. In our setup, the relevant Lagrangian
describing charged kaon interactions is the following

$\begin{split}\mathcal{L}_{\text{ALP,
KC}}^{\chi\text{pt}}&=\left[i\frac{Y_{V}^{ds}}{2\Lambda}\,\partial\phi\,\left(2DK^{-}\pi^{+}-2D\pi^{+}K^{-}\right)+\frac{\Xi_{A}}{f_{\pi}\Lambda}\partial\phi\left(\partial\pi_{0}K^{-}\pi^{+}-D\pi^{+}K^{-}\pi_{0}\right)+\text{h.c.}\right]\,,\end{split}$
(5.11)

while in the neutral sector, choosing the basis in (4.21) in order to minimize
mesonic mixing effects, we have

$\displaystyle\mathcal{L}_{\text{ALP, KN}}^{\chi\text{pt}}$
$\displaystyle=\frac{\partial\phi}{6f_{\pi}\Lambda}\left[\Xi_{A}\left(4\sqrt{2}D\pi^{-}\pi^{+}\bar{K}_{0}-2\sqrt{2}D\pi^{+}\pi^{-}\bar{K}_{0}+\sqrt{2}\partial\pi_{0}\bar{K}_{0}\pi_{0}+\sqrt{2}\,\partial\bar{K}_{0}(-2\pi^{+}\pi^{-}-\pi_{0}^{2})\right)+\text{h.c.}\right]$
$\displaystyle+\frac{M_{\phi}^{2}}{f_{\pi}\Lambda}\,\phi\,\left(\Xi_{A}\bar{K_{0}}+\Xi_{A}^{*}K_{0}\right)\left(2\pi^{+}\pi^{-}+\pi_{0}^{2}\right)+\left[i\frac{Y_{V}^{ds}}{2\Lambda}\,\partial\phi\,\left(+\sqrt{2}\partial\pi_{0}\bar{K}_{0}-\sqrt{2}\partial\bar{K}_{0}\pi_{0}\right)+\text{h.c.}\right]\,,$
(5.12)

Passing to the mass basis, (5.3) becomes

$\displaystyle\mathcal{L}_{\text{ALP, KN}}^{\chi\text{pt}}$
$\displaystyle=-2\sqrt{2}\frac{M_{\phi}^{2}}{f_{\pi}\Lambda}\,\phi\,\left(2\pi^{+}\pi^{-}+\pi_{0}^{2}\right)\,\left[\text{
Re }\Xi_{A}\,K_{L}+i\text{ Im
}\Xi_{A}\,K_{S}\right]+\frac{\partial\phi}{3f_{\pi}\Lambda}\,\partial\pi_{0}\pi_{0}\left[\text{
Re }\Xi_{A}\,K_{L}+i\text{ Im }\Xi_{A}\,K_{S}\right]$
$\displaystyle\quad-\frac{1}{\Lambda}\,\partial\phi\,\partial\pi_{0}\left[i\text{
Im }Y_{V}^{ds}\,K_{L}+\text{ Re
}Y_{V}^{ds}\,K_{S}\right]+\frac{1}{\Lambda}\,\partial\phi\,\pi_{0}\left[i\text{
Im }Y_{V}^{ds}\,\partial K_{L}+\text{ Re }Y_{V}^{ds}\,\partial K_{S}\right]$
$\displaystyle\quad+\frac{\sqrt{2}}{3f_{\pi}\Lambda}\partial\phi\left[\Xi_{A}(2D\pi^{-}\pi^{+}-D\pi^{+}\pi^{-})(K_{L}-K_{S})+\Xi_{A}^{*}(2D\pi^{+}\pi^{-}-D\pi^{-}\pi^{+})(K_{S}+K_{L})\right]\,,$
(5.13)

where the parameters $Y_{V}^{ds}$ and $\Xi_{A}$ are defined in terms of
$Y^{ds}_{S}$ and $Y^{ds}_{P}$ via Eq. (4.55) and (4.53), respectively.

We can then extract the decay rates for the two-body final state decays of
either charged or neutral kaons:

$\begin{split}\Gamma(K^{+}\to\pi^{+}\phi)&=\frac{|Y_{V}^{ds}|^{2}}{\Lambda^{2}}\frac{m_{K}^{3}}{16\pi}\left(1-\frac{m_{\pi}^{2}}{m_{K}^{2}}\right)^{2}\,\sqrt{1-2\frac{(m_{\pi}^{2}+m_{\phi}^{2})}{m_{K}^{2}}+\frac{(m_{\phi}^{2}-m_{\pi}^{2})^{2}}{m_{K}^{4}}}\,,\\\
\Gamma(K_{L}\to\pi_{0}\,\phi)&=\frac{|\text{Im
}Y_{V}^{ds}|^{2}}{\Lambda^{2}}\frac{m_{K}^{3}}{16\pi}\left(1-\frac{m_{\pi}^{2}}{m_{K}^{2}}\right)^{2}\,\sqrt{1-2\frac{(m_{\pi}^{2}+m_{\phi}^{2})}{m_{K}^{2}}+\frac{(m_{\phi}^{2}-m_{\pi}^{2})^{2}}{m_{K}^{4}}}+\mathcal{O}(\varepsilon)\,.\end{split}$
(5.14)

Assuming $m_{\phi}\ll m_{K}$, we can then employ the BNL result,
$\text{BR}(K^{+}\to\pi^{+}\phi)<7.3\times 10^{-11}$ [79] and the KOTO one,
$\text{BR}(K_{L}\to\pi_{0}\phi)<2\times 10^{-9}$ [80], to put bounds on the
coupling $Y_{V}^{ds}$ appearing in the previous expressions:

$\begin{split}|Y_{V}^{ds}|\lesssim 1.6\times
10^{-14}\frac{\Lambda}{\text{TeV}}\qquad\text{ and }\qquad|\text{Im
}Y_{V}^{ds}|\lesssim 1.7\times
10^{-13}\frac{\Lambda}{\text{TeV}}\,.\end{split}$ (5.15)

As far as future perspectives are concerned, the NA62 experiment is expected
to improve by one order of magnitude the bounds on
$\text{BR}(K^{+}\to\pi^{+}\phi)$, while the KOTO experiment foresees a
possible improvement to $\text{BR}(K_{L}\to\pi_{0}\phi)\lesssim 10^{-11}$, see
e.g. Ref. [33].

Similarly, one can compute the three-body decay rates of both charged and
neutral kaons. We find that

$\begin{split}\frac{d\Gamma(K^{+}\to\pi^{+}\pi_{0}\phi)}{dx_{+}dx_{0}}&=\frac{|\Xi_{A}|^{2}}{f_{\pi}^{2}\Lambda^{2}}\frac{m_{K}^{5}}{512\pi^{3}}(x_{+}-x_{0})^{2}\,,\\\
\frac{d\Gamma(K_{L}\to\pi_{0}\pi_{0}\phi)}{dx_{0,a}dx_{0,b}}&=\frac{|\text{Re
}\Xi_{A}|^{2}}{f_{\pi}^{2}\Lambda^{2}}\frac{m_{K}^{5}}{256\pi^{3}}\left[\frac{2-x_{0,a}-x_{0,b}}{6}+\frac{m_{\phi}^{2}}{m_{K}^{2}}\left(2\sqrt{2}-\frac{1}{3}\right)\right]^{2}\,,\\\
\end{split}$ (5.16)

where we have defined $x_{i}=2p_{K}\cdot k_{i}/m_{K}^{2}$ with $p_{K}$ being
the kaon momentum and $k_{i}$ is the momentum belonging to one of the three
final-state particles. Neglecting the mass difference between charged and
neutral pions, the decay rate for these processes can be found by integrating
the differential expressions between

$x_{j}^{\text{max,
min}}=\frac{2-2r-3x_{i}+rx_{i}+x_{i}^{2}+4\varepsilon-2x_{i}\varepsilon\pm\sqrt{x_{i}^{2}-4\varepsilon}\,\sqrt{1-2r+r^{2}-2x_{i}+2rx_{i}+x_{i}^{2}-4r\varepsilon}}{2(1-x_{i}+\varepsilon)}$
(5.17)

and

$x_{i}^{\text{max}}=1-r-2\sqrt{\varepsilon r}\,,\qquad\qquad
x_{i}^{\text{min}}=2\sqrt{\varepsilon}\,,$ (5.18)

where $r=m_{\phi}^{2}/m_{K}^{2}$ and $\varepsilon=m_{\pi}^{2}/m_{K}^{2}$ and
the indices $i,j$ refer to pionic final-states only. In the limit $m_{K}\gg
m_{\phi},m_{\pi}$, we find the following simplified expressions for the total
decay rates:

$\Gamma(K^{+}\to\pi^{+}\pi_{0}\phi)\simeq\frac{|\Xi_{A}|^{2}}{f_{\pi}^{2}\Lambda^{2}}\frac{m_{K}^{5}}{6144\pi^{3}}\,,\qquad\Gamma(K_{L}\to\pi_{0}\pi_{0}\phi)\simeq\frac{|\text{Re
}\Xi_{A}|^{2}}{f_{\pi}^{2}\Lambda^{2}}\frac{m_{K}^{5}}{6144\pi^{3}}\frac{1}{6}\,.$
(5.19)

We can then employ the E787 result
$\text{BR}(K^{+}\to\pi_{0}\pi^{+}\phi)<3.8\times 10^{-5}$ [81] and the E391
one, $\text{BR}(K_{L}\to\pi_{0}\pi_{0}\phi)<0.7\times 10^{-6}$ [82], to put
bounds on the coupling $\Xi_{A}$ appearing in the previous expressions:

$\begin{split}|\Xi_{A}|\lesssim 3.33\times
10^{-12}\frac{\Lambda}{\text{TeV}}\qquad\text{ and }\qquad|\text{Re
}\Xi_{A}|\lesssim 5.45\times 10^{-13}\frac{\Lambda}{\text{TeV}}\,.\end{split}$
(5.20)

##  6 Conclusions

ALP theories are among the best motivated extensions beyond the SM, as they
can naturally solve long-standing problems in particle physics such as the
strong CP problem as well as the flavour and hierarchy problems while
providing at the same time a viable dark matter candidate. ALP interactions
with SM fermions and gauge bosons can be tested through a variety of
cosmological and astrophysical observables as well as through direct searches
at collider. Moreover, low-energy FCNC processes represent other powerful
indirect probes of ALPs. In this work, we have extended a previous analysis
[43] \- where CP-violating ALP scenarios have been discussed for the first
time in the most general terms - assuming ALP masses below the GeV scale,
where QCD cannot be treated perturbatively.

Starting with a CP-violating effective Lagrangian containing light quarks and
gluons, we have carefully discussed its matching onto a chiral effective
Lagrangian, described in terms of mesons and baryons, identifying the
correspondence of the high-energy Jarlskog invariants with those emerging in
the chiral theory. This work provides the necessary tools for phenomenological
analyses related to a light CP-violating ALP, connecting low-energy
observables, such as EDMs and flavour processes entailing mesons and baryons,
with the couplings of the underlying UV complete theory. An extensive analysis
of the phenomenological implications of our work is worthwhile and it is
postponed to a future study.

## Acknowledgements

This work received funding from the European Union’s Horizon 2020 research and
innovation programme under the Marie Skłodowska-Curie grant agreement n.
101086085 – ASYMMETRY and by the INFN Iniziative Specifica APINE. The work of
LDL and PP is supported by the by the Italian Ministry of University and
Research (MUR) via the PRIN 2022 project n. 2022K4B58X – AxionOrigins. The
work of LDL is funded by the European Union – NextGenerationEU 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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