MITP-23-046

August 31, 2023

$K^{-}\to\pi^{-}a$ at Next-to-Leading Order
in Chiral Perturbation Theory

Claudia Cornellaa, Anne Mareike Galdaa, Matthias Neuberta,b, and Daniel Wylerc

a PRISMA+ Cluster of Excellence & MITP, Johannes Gutenberg University
55099 Mainz, Germany
b Department of Physics & LEPP, Cornell University, Ithaca, NY 14853, U.S.A.
c Physik-Institut, Universität Zürich, CH-8057 Zürich, Switzerland

Abstract

###### Abstract

The weak decay $K^{-}\to\pi^{-}a$ is a powerful probe of axion-like particles
(ALPs). In this work, we provide a comprehensive analysis of this process
within chiral perturbation theory, extending existing calculations by
including complete next-to-leading order (NLO) contributions and isospin-
breaking corrections at first order in $(m_{u}-m_{d})$. We show that the
consistent incorporation of ALPs in the QCD and weak chiral Lagrangians
requires a non-trivial extension of the corresponding operator bases, which we
describe in detail. Furthermore, we show that in the presence of an ALP the
so-called weak mass term, which is unobservable in the Standard Model, is non-
redundant already at leading order. We find that NLO corrections associated
with flavor-violating ALP couplings modify the leading-order result by a few
percent, with only small uncertainties. On the contrary, the NLO corrections
proportional to flavor-conserving ALP couplings lead to an $\mathcal{O}(20\%)$
reduction relative to the leading-order predictions. These corrections are
accompanied by a large uncertainties mainly originating from the QCD low-
energy constant $L_{4,r}$ as well as from the presence of various unknown weak
low-energy constants. We emphasize the importance of a precise determination
of these coupling parameters for the successful study of new physics in light
meson decays.

## 1 Introduction

Axions and axion-like particles (collectively referred to as ALPs in this
work) are light pseudoscalar bosons arising in a large class of well-motivated
extensions of the Standard Model (SM). They offer an elegant solution to the
strong CP problem through the Peccei–Quinn mechanism Peccei:1977hh ;
Weinberg:1977ma ; Wilczek:1977pj ; Bardeen:1977bd ; Kim:1979if ;
Shifman:1979if ; Dine:1981rt ; Zhitnitsky:1980tq  and may also provide
insights into the flavor structure of the SM Calibbi:2016hwq ; Ema:2016ops .
More generally, ALPs can emerge as pseudo Nambu–Goldstone bosons in models
featuring explicit global symmetry breaking. Their mass is protected by an
approximate shift symmetry, allowing them to be much lighter than the
symmetry-breaking scale. Due to this symmetry, ALP couplings to SM particles
arise at dimension-five order and higher, and are therefore naturally
suppressed by powers of the scale of global symmetry breaking. This makes the
ALP a prominent example of a weakly coupled new-physics particle. In light of
the absence of any discoveries of new heavy degrees of freedom at the LHC and
at flavor factories, there has been renewed theoretical interest in ALP
models. Search strategies for ALPs use a large variety of cosmological
observations Cadamuro:2011fd ; Millea:2015qra , astrophysical measurements
Payez:2014xsa ; Jaeckel:2017tud , and collider probes Mimasu:2014nea ;
Jaeckel:2015jla ; Knapen:2016moh ; Brivio:2017ije ; Bauer:2017ris ;
Bauer:2018uxu , as well as precision studies of flavor-violating transitions
in the quark and lepton sectors Batell:2009jf ; Freytsis:2009ct ;
Dolan:2014ska ; MartinCamalich:2020dfe ; Bauer:2019gfk ; Bauer:2021mvw .

Low-energy weak-interaction processes impose some of the most stringent bounds
on ALP couplings to SM particles. This was realized shortly after the axion
was originally proposed Bardeen:1978nq ; Antoniadis:1981zw ; Krauss:1986bq ;
Bardeen:1986yb  and has been further explored in recent years Freytsis:2009ct
; Batell:2009jf ; Dobrich:2018jyi ; Cornella:2019uxs ; Guerrera:2021yss . In a
seminal paper Georgi:1986df , Georgi, Kaplan and Randall derived the effective
chiral Lagrangian accounting for interactions between a light ALP (with mass
below the scale of chiral symmetry breaking, $\mu_{\chi}=4\pi F_{\pi}\approx
1.6$ GeV) and light pseudoscalar mesons, enabling a model-independent
description that does not rely on the details of Peccei–Quinn symmetry
breaking. Despite the long history of the subject, even several recent studies
on weak decays, such as $K^{-}\to\pi^{-}a$ and $\pi^{-}\to
e^{-}\bar{\nu}_{e}\,a$, have omitted the contributions of relevant Feynman
diagrams, leading to incomplete expressions for the decay amplitudes (see e.g.
Bjorkeroth:2018dzu ; Ertas:2020xcc ; Gori:2020xvq ). Often, the amplitudes
were derived starting from the amplitude for an analogous decay process
involving a $\pi^{0}$ or $\eta$ meson, and then accounting for the (kinetic)
mixing of the ALP with these neutral mesons by means of mixing angles
$\theta_{\pi^{0}a}$ and $\theta_{\eta_{8}a}$, neglecting the fact that these
mixing angles are unphysical and depend on the parameters of the chiral
rotation used to eliminate the ALP–gluon coupling in the effective chiral
Lagrangian.

Recently, this problem has been reanalyzed and an important omission in the
representation of the weak-interaction quark currents in the effective theory
has been corrected Bauer:2021wjo , with important consequences. The chiral
Lagrangian for weak-interaction processes established in Bauer:2021wjo
predicts a result for the $K^{-}\to\pi^{-}a$ decay rate approximately 37 times
larger than the result obtained in Georgi:1986df  and used in several later
works. Based on this finding, a detailed phenomenological study has shown that
the decays $K^{\pm}\to\pi^{\pm}a$ and $K_{L}\to\pi^{0}a$ impose the strongest
constraints on the ALP couplings to gluons and light quarks in the region
where $m_{a}$ is lighter than about 300 MeV Bauer:2021mvw . These calculations
were performed at leading order (LO) in chiral perturbation theory, making it
challenging to assign them a theoretical uncertainty. Given the significance
of the subject, it is important to push the calculations to next-to-leading
order (NLO), which is the primary objective of the present work.

This paper is organized as follows: In Section 2 we outline how to
consistently incorporate the ALP in the strong- and weak-interaction chiral
Lagrangians at $\mathcal{O}(p^{2})$ and $\mathcal{O}(p^{4})$. We extend the
existing operator bases for the $\mathcal{O}(p^{4})$ Lagrangians by including
new operators which exist only in the presence of an ALP. We show that the
“weak mass term”, which is unobservable in the SM, yields a non-vanishing
contribution to the $K^{-}\to\pi^{-}a$ decay amplitude starting at LO in the
chiral expansion. We also comment on how the $\eta^{\prime}$ meson could be
consistently included in the chiral Lagrangian in the presence of an ALP. In
Section 3 we use these results to compute the $K^{-}\to\pi^{-}a$ decay
amplitude at NLO in the chiral expansion, including first-order isospin-
breaking corrections in the mass difference $(m_{u}-m_{d})$ at LO. We present
detailed numerical results for the magnitude of the NLO corrections and their
theoretical uncertainties. Finally, in Section 4 we briefly discuss the
phenomenological implications of our findings, before summarizing our main
results in Section 5. Explicit expressions detailing the calculations in
Section 3 can be found in a Mathematica notebook attached to the arXiv version
of this paper.

## 2 Effective Lagrangians

### 2.1 Effective ALP interactions at low energies

Our starting point is the effective Lagrangian describing the interactions of
the ALP with the particles of the SM just above the scale of spontaneous
breaking of the QCD chiral symmetry $G_{\chi}=SU(3)_{L}\times SU(3)_{R}\to
SU(3)_{V}$, i.e. $\mu_{\chi}=4\pi F_{\pi}\approx 1.6$ GeV. At this scale, the
active quark flavors are the light quarks, $u,d,s$, which we collectively
denote by $q=(u,d,s)$. The relevant effective Lagrangian up to dimension-5
order is

$\displaystyle\mathcal{L}_{\mathrm{eff}}$
$\displaystyle=\mathcal{L}_{\mathrm{QCD}}+\frac{1}{2}(\partial_{\mu}a)(\partial^{\mu}a)-\frac{m_{a,0}^{2}}{2}\,a^{2}+c_{GG}\,\frac{\alpha_{s}}{4\pi}\,\frac{a}{f}\,G_{\mu\nu}^{a}\tilde{G}^{\mu\nu,a}+c_{\gamma\gamma}\,\frac{\alpha}{4\pi}\,\frac{a}{f}\,F_{\mu\nu}\tilde{F}^{\mu\nu}$
$\displaystyle\quad+\frac{\partial_{\mu}a}{f}\left(\bar{q}_{L}k_{Q}\gamma^{\mu}q_{L}+\bar{q}_{R}k_{q}\gamma^{\mu}q_{R}\right),$
(2.1)

where $F_{\mu\nu}$ and $G_{\mu\nu}^{a}$ are the field-strength tensors of the
photon and the gluon fields, respectively. The corresponding dual field-
strength tensors are defined by
$\tilde{X}^{\mu\nu}=\frac{1}{2}\epsilon^{\mu\nu\alpha\beta}X_{\alpha\beta}$
for $X=F,G$ (with $\epsilon^{0123}=+1$). The parameter $f$ is the ALP decay
constant. The quantities $k_{Q}$ and $k_{q}$ are hermitian $3\times 3$
matrices parametrizing the ALP couplings to left-handed and right-handed
quarks, respectively. We omit ALP interactions with leptons as they are
irrelevant for this study. The QCD Lagrangian is given by

$\displaystyle\begin{aligned}
\mathcal{L}_{\mathrm{QCD}}=-\frac{1}{4}G_{\mu\nu}^{a}G^{\mu\nu,a}+\bar{q}i\not{D}q-\left(\bar{q}_{L}\,m_{q}\,q_{R}+\mathrm{h.c.}\right),\end{aligned}$
(2.2)

and we choose to work in the mass basis, where
$m_{q}=\mathrm{diag}(m_{u},m_{d},m_{s})$.

In terms of the ALP–quark couplings as defined in Bauer:2020jbp , our fermion
couplings are given by

$k_{Q}=\begin{pmatrix}[k_{U}]_{11}&0&0\\\ 0&[k_{D}]_{11}&[k_{D}]_{12}\\\
0&[k_{D}]_{21}&[k_{D}]_{22}\end{pmatrix}\,,\qquad
k_{q}=\begin{pmatrix}[k_{u}]_{11}&0&0\\\ 0&[k_{d}]_{11}&[k_{d}]_{12}\\\
0&[k_{d}]_{21}&[k_{d}]_{22}\end{pmatrix}\,.$ (2.3)

The matrices $k_{U}$ and $k_{D}$ are connected via the CKM matrix $V$ via
$k_{D}=V^{\dagger}k_{U}V$. Disregarding two-loop QED effects, the scale
dependence of the fermion couplings is dictated by the evolution equations
Bauer:2020jbp

$\displaystyle\frac{d}{d\ln\mu}\,k_{q}(\mu)=-\frac{d}{d\ln\mu}\,k_{Q}(\mu)=\frac{\alpha_{s}^{2}}{\pi^{2}}\,\tilde{c}_{GG}(\mu)\,\mathbbm{1}\,,$
(2.4)

with

$\displaystyle\tilde{c}_{GG}(\mu)=c_{GG}+\frac{1}{2}\,\big{[}\braket{k_{q}(\mu)}-\braket{k_{Q}(\mu)}\big{]}\,,$
(2.5)

where $\braket{\dots}$ denotes the trace over flavor indices. The flavor off-
diagonal couplings are scale independent. Solving the evolution equations, one
can relate the couplings defined at the low scale $\mu_{\chi}$ to ALP
couplings at the electroweak scale. Note the important fact that at the
electroweak scale flavor off-diagonal couplings
$\left[k_{D}\right]_{12}=\left[k_{D}\right]_{21}^{*}$ are generated from one-
loop diagrams involving $W$ bosons, even in models in which the ALP couplings
at very high energies are flavor diagonal Bauer:2020jbp ; Izaguirre:2016dfi ;
Gavela:2019wzg .

The quark bilinears in the effective Lagrangian (2.1) and the ALP–gluon
coupling can be written in the general form

$\displaystyle\mathcal{L}_{\mathrm{eff}}\ni\bar{q}(x)\big{[}l_{\mu}(x)\,\gamma^{\mu}P_{L}+r_{\mu}(x)\,\gamma^{\mu}P_{R}-s+i\gamma_{5}\,p\big{]}q(x)-\frac{\alpha_{s}}{8\pi}\,\theta(x)\,G_{\mu\nu}^{a}(x)\,\tilde{G}^{\mu\nu,a}(x)\,,$
(2.6)

where $s=m_{q}$, $p=0$, and we have defined the local ALP sources

$\displaystyle\begin{aligned}
l_{\mu}(x)=k_{Q}\,\frac{\partial_{\mu}a(x)}{f}\,,\qquad
r_{\mu}(x)=k_{q}\,\frac{\partial_{\mu}a(x)}{f}\,,\qquad\theta(x)=-2c_{GG}\,\frac{a(x)}{f}\,.\end{aligned}$
(2.7)

The left-handed and right-handed sources can be combined into a vector and an
axial-vector source according to

$\displaystyle
v_{\mu}=\frac{l_{\mu}+r_{\mu}}{2}=c^{v}\,\frac{\partial_{\mu}a}{2f}\,,\qquad
a_{\mu}=\frac{r_{\mu}-l_{\mu}}{2}=c^{a}\,\frac{\partial_{\mu}a}{2f}\,,$ (2.8)

where we have defined111For simplicity we denote the diagonal entries of these
matrices by $c^{v,a}_{uu}$, $c^{v,a}_{dd}$, and $c^{v,a}_{ss}$. Our parameters
$c_{qq}^{a}$ are identical to the couplings $c_{qq}$ used in Bauer:2021mvw .

$\displaystyle c^{v}=k_{q}+k_{Q}\,,\qquad c^{a}=k_{q}-k_{Q}\,.$ (2.9)

It is well established that performing the local chiral transformation
Bardeen:1986yb ; Georgi:1986df ; Srednicki:1985xd

$q(x)\to e^{\frac{i}{2}\theta(x)\,\kappa_{q}\gamma_{5}}\,q(x)$ (2.10)

removes the ALP coupling to gluons in (2.6) term as long as

$\braket{{\kappa}_{q}}=\kappa_{u}+\kappa_{d}+\kappa_{s}=1\,.$ (2.11)

Once this constraint is met, the parameters $\kappa_{q}$ must not affect
predictions for physical quantities Bauer:2020jbp .

### 2.2 QCD chiral Lagrangian at $\bm{\mathcal{O}(p^{2})}$

Below the scale $\mu_{\chi}$, the light pseudoscalar mesons $\pi$, $K$ and
$\eta$ take the place of light quarks as dynamical degrees of freedom. The ALP
interactions with these mesons are best described in terms of an effective
chiral Lagrangian involving the meson octet and the external sources in (2.7)
Georgi:1986df ; Bauer:2020jbp . The meson octet ${\Sigma}_{0}$ is defined as

$\Sigma_{0}(x)=\mathrm{exp}\left[\frac{i\sqrt{2}}{F}\,{\Phi}(x)\right],\qquad\Phi(x)=\lambda^{a}\,\pi^{a}(x)\,.$
(2.12)

Here, $F$ is the meson decay constant in the chiral limit (we use the
normalization $F\approx F_{\pi}=130.2$ MeV), and ${\lambda}^{a}$ (with
$a=1,\dots,8$) are the Gell-Mann matrices. With this definition, $\Sigma_{0}$
satisfies

$\displaystyle\mathrm{det}\,\Sigma_{0}=1\,,\qquad\braket{\Sigma_{0}\,i(D_{\mu}\Sigma_{0})^{\dagger}}=0\,,$
(2.13)

where the second equality follows from the first one by taking a derivative of
$\mathrm{det}\,\Sigma_{0}=\exp[\braket{\ln\Sigma_{0}}]$. An important point to
notice is that in presence of a non-zero $\theta$ the relevant group becomes
$G_{\chi}^{\prime}=U(3)_{L}\times U(3)_{R}$, and $\Sigma$ needs to be promoted
to a $U(3)$ matrix with the non-trivial determinant

$\mathrm{det}\,\Sigma(x)=e^{-i\,\theta(x)}\,.$ (2.14)

This condition ensures the consistency with the chiral transformation.
Gasser:1984gg . While there is no unique way of implementing this constraint,
we choose the option that reflects exactly the rotation in (2.10), namely

$\Sigma(x)=e^{-\frac{i}{2}\theta(x)\,\kappa_{q}}\,\Sigma_{0}(x)\,e^{-\frac{i}{2}\theta(x)\,{\kappa}_{q}}\,,$
(2.15)

which implies (2.14) due to the condition (2.11). The matrix $\Sigma$
transforms linearly under $G_{\chi}^{\prime}$, and we assign it to the
$(3_{L},\bar{3}_{R})$ representation, i.e. ${\Sigma}\to
g_{L}\Sigma\,g_{R}^{\dagger}$. This convention differs from the one adopted in
the seminal paper of Gasser and Leutwyler Gasser:1984gg , where the meson
octet, denoted by $U$, is assumed to transform in the $({3}_{R},\bar{3}_{L})$
representation of the chiral group. Given that $\theta$ transforms non-
linearly under $G_{\chi}^{\prime}$, i.e.,
$\theta\to\theta+i\ln\mathrm{det}(g_{L}g_{R}^{\dagger})$, it is useful to
introduce a corresponding covariant derivative

$\displaystyle
D_{\mu}\theta=\partial_{\mu}\theta-2\braket{a_{\mu}}=-\left(2c_{GG}+\braket{c^{a}}\right)\frac{\partial_{\mu}a}{f}\,,$
(2.16)

which is invariant under $G_{\chi}^{\prime}$ once the transformation of the
axial current is taken into account Gasser:1984gg .

We now have all the necessary ingredients to construct the most general
Lorentz-invariant Lagrangian at $\mathcal{O}(p^{2})$ which is (formally, when
$\chi$ and the other external sources are treated as a spurions) invariant
under $G_{\chi}^{\prime}$ in the presence of the sources in (2.7). Including
the ALP mass term explicitly, we obtain

$\displaystyle\mathcal{L}^{(p^{2})}_{\rm
QCD}=\frac{F^{2}}{8}\braket{(D_{\mu}\Sigma)\,(D^{\mu}\Sigma)^{\dagger}+\chi\Sigma^{\dagger}+\Sigma\chi^{\dagger}}+\frac{H_{0}}{2}\,(D_{\mu}\theta)(D^{\mu}\theta)+\frac{1}{2}m_{a,0}^{2}\,a^{2}\,,$
(2.17)

where

$\displaystyle\begin{aligned}
\chi&=2B_{0}\,(s+ip)=2B_{0}\,m_{q}\,,\\\\[8.53581pt]
iD_{\mu}\Sigma&=i\partial_{\mu}\Sigma+(QeA_{\mu}+l_{\mu})\,\Sigma-\Sigma\,(QeA_{\mu}+r_{\mu})\,,\\\
H_{0}&=\frac{f^{2}}{\left(2c_{GG}+\braket{c^{a}}\right)^{2}}\,,\end{aligned}$
(2.18)

where $Q=\text{diag}(\frac{2}{3},-\frac{1}{3},-\frac{1}{3})$ contains the
electric charges of the quarks. In the present work we ignore QED corrections,
so the photon field can be dropped for our purposes. The choice of $H_{0}$
ensures a canonical normalization for the ALP kinetic term. This Lagrangian is
equivalent to the one used in Bauer:2021wjo : substituting (2.15) into (2.17)
and using the properties in (2.13) yields equation (7) in Bauer:2021wjo .

As expected, the electrically neutral states in the theory, $a$, $\pi^{0}$ and
$\eta_{8}$, undergo both kinetic and mass mixing. In the following we
replicate the analysis performed in Bauer:2020jbp  for the $\pi^{0}$ – $a$
mixing, but also include the $\eta_{8}$. Expanding the QCD chiral Lagrangian
in the presence of the ALP up to quadratic order in the fields, we obtain

$\mathcal{L}^{\mathrm{ALP}}_{\chi}\supset\frac{1}{2}(\partial_{\mu}\varphi)^{T}Z\,(\partial_{\mu}\varphi)-\frac{1}{2}\,\varphi^{T}M^{2}\,\varphi\,,$
(2.19)

where $\varphi^{T}=(\pi^{0},\eta_{8},a)$, and

$\displaystyle Z$
$\displaystyle=\begin{pmatrix}1&0&\frac{F}{f}\,\frac{\hat{c}_{uu}^{a}-\hat{c}_{dd}^{a}}{2\sqrt{2}}\\\\[5.69054pt]
0&1&\frac{F}{f}\,\frac{\hat{c}_{uu}^{a}+\hat{c}_{dd}^{a}-2\hat{c}_{ss}^{a}}{2\sqrt{6}}\\\\[5.69054pt]
\frac{F}{f}\,\frac{\hat{c}_{uu}^{a}-\hat{c}_{dd}^{a}}{2\sqrt{2}}&\frac{F}{f}\,\frac{\hat{c}_{uu}^{a}+\hat{c}_{dd}^{a}-2\hat{c}_{ss}^{a}}{2\sqrt{6}}&1+\frac{F^{2}}{f^{2}}\,y_{\kappa}\end{pmatrix},$
(2.20) $\displaystyle M^{2}$
$\displaystyle=\begin{pmatrix}B_{0}(m_{u}+m_{d})&\frac{B_{0}(m_{u}-m_{d})}{\sqrt{3}}&\frac{\sqrt{2}B_{0}c_{GG}F\,g_{\kappa}(m_{u},-m_{d},0)}{f}\\\\[5.69054pt]
\frac{B_{0}(m_{u}-m_{d})}{\sqrt{3}}&\frac{B_{0}(4m_{s}+m_{u}+m_{d})}{3}&\frac{2\sqrt{2}c_{GG}B_{0}F\,g_{\kappa}(m_{u},m_{d},-2m_{s})}{\sqrt{3}f}\\\\[5.69054pt]
\frac{\sqrt{2}c_{GG}B_{0}F\,g_{\kappa}(m_{u},-m_{d},0)}{f}&\frac{2\sqrt{2}c_{GG}B_{0}F\,g_{\kappa}(m_{u},m_{d},-2m_{s})}{\sqrt{3}f}&m_{a,0}^{2}+\frac{2c_{GG}^{2}B_{0}F^{2}}{f^{2}}\,g_{\kappa^{2}}\end{pmatrix},$

with

$\displaystyle\begin{aligned}
g_{\kappa}(a,b,c)&=a\,\kappa_{u}+b\,\kappa_{d}+c\,\kappa_{s}\,,\\\
g_{\kappa^{2}}&=m_{u}\,\kappa^{2}_{u}+m_{d}\,\kappa^{2}_{d}+m_{s}\,\kappa^{2}_{s}\,,\\\\[1.42262pt]
\hat{c}_{qq}^{a}&=c_{qq}^{a}+2c_{GG}\,\kappa_{q}\,,\end{aligned}$ (2.21)

and we have defined

$\displaystyle
y_{\kappa}=\frac{1}{4}\braket{c^{a}c^{a}}+c_{GG}\,g_{\kappa}(c_{uu}^{a},c_{dd}^{a},c_{ss}^{a})+c_{GG}^{2}\,g_{\kappa^{2}}(1,1,1)\,.$
(2.22)

It is important to note that in the isospin-conserving limit, which
corresponds to setting $m_{u}=m_{d}\equiv\hat{m}$, the states $\pi^{0}$ and
$\eta_{8}$ do not mix, since $M^{2}_{12}=M^{2}_{21}=0$ in this limit.

To identify the physical mass eigenstates it is necessary to diagonalize both
the kinetic and the mass term. As outlined in Bauer:2020jbp , we first find an
orthogonal matrix $U_{Z}$ such that $U_{Z}^{T}\,Z\,U_{Z}=Z_{\rm{diag}}$. By
performing the rotation $\varphi\to U_{Z}\,\varphi$, the bilinear terms in the
Lagrangian become

$\mathcal{L}_{\chi}^{\mathrm{ALP}}\supset\frac{1}{2}(\partial_{\mu}\varphi)^{T}Z_{\mathrm{diag}}\,(\partial_{\mu}\varphi)-\frac{1}{2}\,\varphi^{T}(U_{Z}^{T}M^{2}\,U_{Z})\,\varphi\,.$
(2.23)

At this stage, kinetic terms are not yet canonically normalized. To correct
this, we rescale $\varphi\to Z_{\mathrm{diag}}^{-1/2}\,\varphi$, which
transforms the mass matrix into

$\displaystyle\hat{M}^{2}=Z_{\mathrm{diag}}^{-1/2}\,U_{Z}^{T}\,M^{2}\,U_{Z}\,Z_{\mathrm{diag}}^{-1/2}\,.$
(2.24)

The last step is to diagonalize $\hat{M}^{2}$, i.e., to find an orthogonal
matrix $U_{M}$ such that
$U_{M}^{T}\hat{M}^{2}U_{M}=M_{\mathrm{diag}}^{2}\equiv M_{\mathrm{phys}}^{2}$.
With the last rotation $\varphi\to U_{M}\,\varphi$, we finally obtain

$\displaystyle\mathcal{L}_{\chi}^{\mathrm{ALP}}\supset\frac{1}{2}(\partial_{\mu}\varphi_{\mathrm{phys}})^{T}(\partial_{\mu}\varphi_{\mathrm{phys}})-\frac{1}{2}\,\varphi^{T}_{\mathrm{phys}}\,M^{2}_{\mathrm{phys}}\,\varphi_{\mathrm{phys}}\,,$
(2.25)

where we have added the pedix “phys” to highlight that these are the physical
fields. They are related to those appearing at the beginning via

$\displaystyle\varphi=R\,\varphi_{\mathrm{phys}}\,,\quad\text{with}\quad
R=U_{Z}\,Z_{\mathrm{diag}}^{-1/2}\,U_{M}\,.$ (2.26)

Before finding the explicit rotations, we want to find the physical masses,
i.e. the eigenvalues of $\hat{M}^{2}$. Since $\hat{M}^{2}$ and $M^{2}\,Z^{-1}$
are similar matrices they have the same eigenvalues, and hence it is
sufficient to solve Bauer:2020jbp

$\mathrm{det}\left(M^{2}Z^{-1}-m^{2}\,\mathbbm{1}\right)=0\,.$ (2.27)

Keeping terms up to second order in $F/f$, and working in the isospin limit
where $m_{u}=m_{d}=\hat{m}$, we find

$\displaystyle\begin{aligned}
m_{\pi^{0}}^{2}&=\tilde{m}_{\pi^{0}}^{2}\left[1+\frac{F^{2}}{8f^{2}}\,\frac{\tilde{m}_{\pi^{0}}^{2}}{\tilde{m}_{\pi^{0}}^{2}-m_{a,0}^{2}}\left(c_{uu}^{a}-c_{dd}^{a}\right)^{2}\right],\\\
m_{\eta_{8}}^{2}&=\tilde{m}_{\eta_{8}}^{2}\left[1+\frac{F^{2}}{24f^{2}}\frac{\tilde{m}_{\eta_{8}}^{2}}{\tilde{m}_{\eta_{8}}^{2}-m_{a,0}^{2}}\left(4c_{GG}\,\frac{m_{s}-\hat{m}}{2m_{s}+\hat{m}}+c_{uu}^{a}+c_{dd}^{a}-2c^{a}_{ss}\right)^{2}\right],\\\
m_{a}^{2}&=m_{a,0}^{2}\left\\{1-\frac{F^{2}}{f^{2}}\left[\bar{y}+\frac{c_{GG}}{2}\,(c_{GG}+c_{uu}^{a}+c_{dd}^{a})+\frac{\tilde{m}_{\pi^{0}}^{2}}{8\,(\tilde{m}_{\pi^{0}}^{2}-m_{a,0}^{2})}\left(c_{uu}^{a}-c_{dd}^{a}\right)^{2}\right]\right\\}\\\
&\quad+\frac{F^{2}}{2f^{2}}\,c_{GG}^{2}\,\tilde{m}_{\pi^{0}}^{2}+\mathcal{O}\left(m_{a,0}^{2}\,\frac{\hat{m}}{m_{s}}\right),\end{aligned}$
(2.28)

where the quantity $\bar{y}$ in (2.28) is the $\kappa_{q}$-independent piece
of $y_{\kappa}$, obtained e.g. by setting $\kappa_{s}=1$ and
$\kappa_{u}=\kappa_{d}=0$ in (2.22). Note in particular that having set
$\braket{\kappa_{q}}=1$ the result for the physical masses is independent of
the individual $\kappa_{q}$ parameters.222In order to see the cancellation of
the $\kappa_{q}$ parameters it is crucial to take into account the
$y_{\kappa}$ term in the expression for $Z$ in (2.20), even though it is of
second order in $F/f$. This contribution was omitted in Bauer:2020jbp . The
parameters

$\displaystyle\begin{aligned}
\tilde{m}_{\pi^{0}}^{2}&=\tilde{m}_{\pi^{-}}^{2}=B_{0}\,(m_{u}+m_{d})\,,\\\
\tilde{m}_{\eta_{8}}^{2}&=\frac{B_{0}}{3}\,(m_{u}+m_{d}+4m_{s})\end{aligned}$
(2.29)

denote the zeroth-order contributions (in $F^{2}/f^{2}$) to the pion and
$\eta_{8}$ masses. The masses of the kaons do not receive corrections in $F/f$
and are given by

$\displaystyle\begin{aligned} m_{K^{-}}^{2}&=B_{0}\,(m_{u}+m_{s})\,,\\\
m_{\bar{K}^{0}}^{2}&=B_{0}\,(m_{d}+m_{s})\,.\end{aligned}$ (2.30)

In our analysis of the $K^{-}\to\pi^{-}a$ decay amplitude we work consistently
to first order in $F/f$, so the above mass corrections can be ignored. We will
also treat isospin-breaking effects in the quark-mass difference
$(m_{d}-m_{u})$ as a perturbation around the average value
$\hat{m}\equiv\frac{1}{2}(m_{u}+m_{d})$. In the isospin limit one has
$m_{K^{-}}^{2}=m_{\bar{K}^{0}}^{2}\equiv m_{K}^{2}$ and

$\displaystyle\tilde{m}_{\eta_{8}}^{2}=\frac{4m_{K}^{2}-m_{\pi}^{2}}{3}\,.$
(2.31)

The physical fields are obtained from the initial ones via the transformation

$\displaystyle R=\begin{pmatrix}1&0&\theta_{\pi^{0}a}\\\
0&1&\theta_{\eta_{8}a}\\\
\theta_{a\pi^{0}}&\theta_{a\eta_{8}}&1\end{pmatrix}+\mathcal{O}\left(\frac{F^{2}}{f^{2}}\right),$
(2.32)

where in the isospin limit the rotation angles are given by

$\displaystyle\begin{aligned}
\theta_{\pi^{0}a}&=-\frac{M^{2}_{13}-m_{a,0}^{2}\,Z_{13}}{\tilde{m}_{\pi^{0}}^{2}-m_{a,0}^{2}}=\frac{F}{f}\,\frac{(\hat{c}_{uu}^{a}-\hat{c}_{dd}^{a})\,m_{a,0}^{2}-2c_{GG}\,(\kappa_{u}-\kappa_{d})\,\tilde{m}_{\pi^{0}}^{2}}{2\sqrt{2}\,(\tilde{m}_{\pi^{0}}^{2}-m_{a,0}^{2})}\,,\\\
\theta_{\eta_{8}a}&=-\frac{M^{2}_{23}-m_{a,0}^{2}\,Z_{23}}{\tilde{m}_{\eta_{8}}^{2}-m_{a,0}^{2}}=\frac{F}{f}\,\frac{(\hat{c}_{uu}^{a}+\hat{c}_{dd}^{d}-2\hat{c}_{ss}^{a})\,m_{a,0}^{2}-2c_{GG}\,\tilde{m}_{\pi^{0}}^{2}+6c_{GG}\,\kappa_{s}\,\tilde{m}_{\eta_{8}}^{2}}{2\sqrt{6}\,(\tilde{m}_{\eta_{8}}^{2}-m_{a,0}^{2})}\,,\\\
\theta_{a\pi^{0}}&=\frac{M^{2}_{13}-\tilde{m}_{\pi^{0}}^{2}\,Z_{13}}{\tilde{m}_{\pi^{0}}^{2}-m_{a,0}^{2}}=-\frac{F}{f}\,\frac{(c^{a}_{uu}-c^{a}_{dd})\,\tilde{m}_{\pi^{0}}^{2}}{2\sqrt{2}\,(\tilde{m}_{\pi^{0}}^{2}-m_{a,0}^{2})}\,,\\\
\theta_{a\eta_{8}}&=\frac{M^{2}_{23}-\tilde{m}_{\eta_{8}}^{2}Z_{23}}{\tilde{m}_{\eta_{8}}^{2}-m_{a,0}^{2}}=-\frac{F}{f}\,\frac{(\hat{c}_{uu}^{a}+\hat{c}_{dd}^{a}-\hat{c}_{ss}^{a}+6c_{GG}\,\kappa_{s})\,\tilde{m}_{\eta_{8}}^{2}-2c_{GG}\,\tilde{m}_{\pi^{0}}^{2}}{2\sqrt{6}\,(\tilde{m}_{\eta_{8}}^{2}-m_{a,0}^{2})}\,.\end{aligned}$
(2.33)

In these expressions the quark masses have been written in terms of the meson
masses by using the relations (2.29). Note the important fact that these
mixing angles are not independent of the $\kappa_{q}$ parameters, and hence
they are not physical quantities. In the following we will apply the rotation
in (2.32) in order to remove mass and kinetic mixing of $a$, $\pi^{0}$ and
$\eta_{8}$ from the $\mathcal{O}(p^{2})$ Lagrangian. Mixing effects still
arise at $\mathcal{O}(p^{4})$, and we treat them as interaction vertices.
Isospin-breaking corrections will be included in our LO calculation in Section
3.1 at linear order in $(m_{u}-m_{d})$ and treated as small perturbations.
They are not included explicitly in (2.33).

### 2.3 A comment on the $\eta^{\prime}$ meson

Because of its large mass, the $\eta^{\prime}$ meson appears only indirectly
through the values of the low-energy constants and is usually not included as
a light degree of freedom in the chiral effective theory. However, its mixing
with the $\eta$ meson is known to be numerically important and it is needed to
consistently perform the large-$N_{c}$ limit. In the present context, it might
appear to be possibly even more important, since the $\eta^{\prime}$ is
(largely) an $SU(3)$ singlet, much like the ALP. In this section we sketch
briefly how the $\eta^{\prime}$ can be included in the chiral Lagrangian,
leaving a quantitative investigation of its contributions for future work.

The extension from eight to nine pseudoscalar mesons has already been
described in Gasser:1984gg  and, more recently, in Herrera-Siklody:1996tqr .
In a nutshell, the matrix $\Sigma$ is extended with a singlet field $\phi_{0}$
as

$\Sigma^{\prime}(x)=\Sigma_{0}(x)\,e^{i\frac{\phi_{0}(x)}{3}}\,,$ (2.34)

with $\det\Sigma_{0}=1$ as in (2.13). The new field $\phi_{0}$ is not
invariant under $U(1$) axial transformations, however its transformation is
precisely compensated by the transformation properties of the $\theta$
parameter, such that the combination $(\phi_{0}+\theta)$ is invariant under
$U(3)_{L}\times U(3)_{R}$. This fact allows one to replace all low-energy
coefficients in the chiral Lagrangian by arbitrary functions
$V_{i}=V_{i}(\phi_{0}+\theta)$, i.e. at $\mathcal{O}(p^{2})$ Gasser:1984gg ;
Herrera-Siklody:1996tqr

$\displaystyle\begin{aligned} \mathcal{L}^{(p^{2})}_{\rm
QCD}=V_{0}(\phi_{0}+\theta)+V_{1}(\phi_{0}+\theta)\braket{(D_{\mu}\Sigma^{\prime})(D^{\mu}\Sigma^{\prime})^{\dagger}}+V_{2}(\phi_{0}+\theta)\braket{\chi\Sigma^{\prime\dagger}+\Sigma^{\prime}\chi^{\dagger}},\end{aligned}$
(2.35)

where $V_{1}(0)=V_{2}(0)=F^{2}/8$. The potential $V_{0}$ can generate a mass
term for the field $(\phi_{0}+\theta)$, which is not protected by chiral
symmetry and hence can be arbitrarily heavy. We thus identify

$\displaystyle\eta_{1}\equiv\frac{F}{2\sqrt{3}}\left(\phi_{0}+\theta\right)$
(2.36)

with the heavy singlet meson $\eta_{1}$,333The “heavy” state $\eta_{1}$ mixes
with the “light” state $\eta_{8}$ to produce the physical mesons
$\eta^{\prime}$ and $\eta$. where the prefactor on the right-hand side has
been chosen such that $\eta_{1}$ has a canonically normalized kinetic term.
From (2.34), we then obtain

$\displaystyle\Sigma^{\prime}(x)=\Sigma_{0}(x)\,e^{i\,\frac{2}{\sqrt{3}F}\,\eta_{1}(x)}\,e^{-i\,\frac{\theta(x)}{3}}\,.$
(2.37)

Upon integrating out the heavy field $\eta_{1}$, this leaves us with

$\displaystyle\Sigma^{\prime}(x)=\Sigma_{0}(x)\,e^{-i\,\frac{\theta(x)}{3}}\,.$
(2.38)

This way of implementing the constraint
$\det\Sigma^{\prime}(x)=e^{-i\,\theta(x)}$ is a particular case of (2.15),
corresponding to the special choice $\kappa_{q}=\frac{1}{3}\mathbbm{1}$.

Given that $m_{\eta^{\prime}}\gg F$, we believe that not including the
$\eta^{\prime}$ as a propagating degree of freedom in the low-energy chiral
Lagrangian is a reasonable approximation. Nevertheless, it would be
interesting in the future to study the alternative approach, where $\eta_{1}$
is included as the ninth pseudoscalar meson.

### 2.4 Weak Lagrangian at $\bm{\mathcal{O}(G_{F}\,p^{2})}$

The decay $K^{-}\to\pi^{-}a$ involves an interplay of strong and non-leptonic
weak interactions. Starting from the current-current form of the effective
weak Lagrangian at energies well below the electroweak scale, it is possible
to classify the relevant four-quark operators according to their
transformation properties under the chiral group Pich:1985st ; Pich:1990mw .
Neglecting electroweak penguin operators,444This is a very good approximation
for our purposes. Including electroweak penguins, one would also obtain
operators transforming as $(8_{L},8_{R})$ Ecker:2000zr ; Cirigliano:2003gt .
one finds operators transforming as $(8_{L},1_{R})$ and $(27_{L},1_{R})$. In
kaon physics, the octet operators are dynamically enhanced and give the
leading contribution to the decay amplitudes. At $\mathcal{O}(p^{2})$, the
weak effective chiral Lagrangian can be written as

$\displaystyle\begin{aligned} \mathcal{L}_{\rm
weak}^{(p^{2})}&=\frac{F^{4}}{4}\,\Big{[}G_{8}\braket{\lambda_{+}L_{\mu}L^{\mu}}+G_{8}^{\prime}\braket{\lambda_{+}S}+G_{8}^{\theta}\,(D_{\mu}\theta)\braket{\lambda_{+}L^{\mu}}\\\
&\hskip
39.83368pt+G_{27}^{1/2}\,\mathcal{O}_{27}^{1/2}+G_{27}^{3/2}\,\mathcal{O}_{27}^{3/2}+\text{h.c.}\Big{]}\,,\end{aligned}$
(2.39)

where $\lambda_{+}=\frac{1}{2}(\lambda_{6}+i\lambda_{7})$, and the objects
$L_{\mu}$ and $S$ are the chiral representations of the left-handed current
$\bar{q}_{L}\gamma_{\mu}q_{L}$ and of the scalar current
$m_{q}(\bar{q}_{L}q_{R}+\bar{q}_{R}q_{L})$, respectively. We collect them in
Table 2.1, together with other building blocks we will need in the rest of the
paper. The terms shown explicitly in (2.39) mediate weak $s\to d$ transitions,
while their hermitian conjugate describe $d\to s$ transitions. The first octet
operator was written down as early as in 1967 by Cronin Cronin:1967jq . The
27-plet operators with isospin change $\Delta I=\frac{1}{2}$ and $\frac{3}{2}$
have the form Bernard:1985wf

$\displaystyle\begin{aligned}
\mathcal{O}_{27}^{1/2}&=L_{\mu\,32}L_{11}^{\mu}+L_{\mu\,31}L_{12}^{\mu}+2L_{\mu\,32}L_{22}^{\mu}-3L_{\mu\,32}L_{33}^{\mu}\,,\\\
\mathcal{O}_{27}^{3/2}&=L_{\mu\,32}L_{11}^{\mu}+L_{\mu\,31}L_{12}^{\mu}-L_{\mu\,32}L_{22}^{\mu}\,.\end{aligned}$
(2.40)

The second octet operator in (2.39), $\braket{\lambda_{+}S}$, is known as the
“weak mass term”. In absence of external fields, this operator can be removed
through field redefinitions. However, its presence leads to non-zero effects
when the ALP is present, as discussed in the next section. On the other hand,
the third octet operator is a novel object, which, to the best of our
knowledge, has not been previously discussed in the literature. It explicitly
involves the ALP source $\theta$.

The constants $G_{i}$ are defined as

$\displaystyle
G_{i}=-\frac{G_{F}}{\sqrt{2}}\,V_{ud}^{\ast}V_{us}\left(g_{i}-\frac{V_{td}^{\ast}\,V_{ts}}{V_{ud}^{\ast}V_{us}}\,g_{i}^{t}\right),$
(2.41)

with dimensionless constants $g_{i}$ and $g_{i}^{t}$. In the standard
definition of the CKM matrix the combination $V_{ud}^{\ast}V_{us}$ is real,
while the ratio of CKM elements (expressed here using the Wolfenstein
approximation)

$\displaystyle-\frac{V_{td}^{\ast}\,V_{ts}}{V_{ud}^{\ast}V_{us}}\approx
A^{2}\lambda^{4}\left(1-\rho+i\eta\right)$ (2.42)

carries a non-trivial CP-odd phase. This ratio is numerically very small, of
order $10^{-3}$, and we will neglect it for simplicity. Consequently, the
couplings $G_{i}$ become real, and adding the hermitian conjugate has the
effect of replacing $\lambda_{+}\to\lambda_{6}$ in the three octet operators,
while in the 27-plet operators one must add the corresponding operators with
the flavor indices on each current interchanged.

object | definition
---|---
$S$ | $\chi\Sigma^{\dagger}+\Sigma\chi^{\dagger}$
$P$ | $i(\chi\Sigma^{\dagger}-\Sigma\chi^{\dagger})$
$L_{\mu}$ | $\Sigma\,i(D_{\mu}\Sigma)^{\dagger}$
$R_{\mu}$ | $\Sigma^{\dagger}\,iD_{\mu}\Sigma$
$W_{\mu\nu}$ | $2(D_{\mu}L_{\nu}+D_{\nu}L_{\mu})$
$D_{\mu}\theta$ | $\partial_{\mu}\theta-2\braket{a_{\mu}}$
Table 2.1: Building blocks of the $\mathcal{O}(p^{2})$ and
$\mathcal{O}(p^{4})$ chiral Lagrangians. The definition of $L_{\mu}$ used here
coincides with the one in Bauer:2021wjo  up to an overall factor. Our
definitions differ from those used in Gasser:1984gg ; Kambor:1989tz  because
of the transformation properties assumed for $\Sigma$ (i.e. $\Sigma\to
g_{L}\Sigma\,g_{R}^{\dagger}$ here vs. $\Sigma\to
g_{R}\Sigma\,g_{L}^{\dagger}$ in Gasser:1984gg ; Kambor:1989tz ).

To make contact with the notation used elsewhere (e.g. in Kambor:1989tz ;
Cirigliano:2003gt ; Cirigliano:2011ny ; Pich:2021yll ), we consider the limit
of exact $SU(3)$ symmetry. In this case the relation
$g_{27}^{3/2}=5g_{27}^{1/2}$ holds and the combination of 27-plet operators
simplifies to

$G_{27}^{1/2}\,\mathcal{O}_{27}^{1/2}+G_{27}^{3/2}\,\mathcal{O}_{27}^{3/2}=9\,G_{27}^{1/2}\left(L_{\mu\,32}L^{\mu}_{11}+\frac{2}{3}L_{\mu\,31}L^{\mu}_{12}-\frac{1}{3}L_{\mu\,32}\braket{L^{\mu}}\right).$
(2.43)

This coincides with the $27$-plet as given in the cited references in the
absence of a $\theta$ source, since in this case $\braket{L^{\mu}}=0$, see
(2.13). In the presence of the ALP, we have instead

$\displaystyle\braket{L^{\mu}}=-D^{\mu}\theta\,,$ (2.44)

and the extra term gives a contribution to $G_{8}^{\theta}$.

The low-energy couplings are determined at the scale
$\nu_{\chi}=m_{\rho}\approx 770$ MeV through a phenomenological analysis of
the $\bar{K}\to\pi\pi$ decay rates at NLO in chiral perturbation theory.
Including isospin-breaking corrections, their values are Cirigliano:2011ny

$\displaystyle g_{8}=3.61\pm 0.28\,,\qquad g_{27}\equiv
9\,g_{27}^{1/2}=0.297\pm 0.028\,,$ (2.45)

where the errors are completely theory-dominated, and the $SU(3)$ limit has
been assumed for $g_{27}$. For the individual 27-plet couplings we then obtain
$g_{27}^{1/2}\approx 0.033\pm 0.003$ and $g_{27}^{3/2}\approx 0.165\pm 0.016$.
The fact that $g_{8}$ is about a factor 22 larger than $g_{27}^{3/2}$ is
referred to as the $\Delta I=\frac{1}{2}$ selection rule in non-leptonic kaon
decays. Because of this pronounced hierarchy, in this work we focus solely on
calculating octet contributions to the $K^{-}\to\pi^{-}a$ decay amplitude at
NLO. Meanwhile, for the 27-plet contributions, we will limit ourselves to
improving the LO predictions from Bauer:2021mvw  by adding isospin-breaking
corrections.

Contrary to $g_{8}$ and $g_{27}$, the low-energy couplings $g_{8}^{\prime}$
and $g_{8}^{\theta}$, which enter (2.39) via $G_{8}^{\prime}$ and
$G_{8}^{\theta}$, are currently unknown, since they do not intervene in SM
processes. If an ALP was to be found in the future, the decays
$K^{-}\to\pi^{-}a$ and $K_{L}\to\pi^{0}a$ would give access to these two
couplings, which would then be promoted to physical parameters. In principle,
it should be possible to calculate $g_{8}^{\prime}$ and $g_{8}^{\theta}$ using
lattice QCD, by finding an appropriate representation of the external sources
in terms of quark and gluon fields.

Figure 2.1: Schematic representation of the effective ALP–gluon coupling
$\tilde{c}_{GG}$ (top row) and of the effective coupling $g_{8}^{\theta}$
(bottom row). The crossed circle represents an effective weak vertex obtain
after integrating out the electroweak gauge bosons and heavy quarks of the SM.

Before turning to a detailed discussion of the weak mass term multiplying
$G_{8}^{\prime}$, we briefly comment about the new octet operator including
the $\theta$ source, which is the chiral representation of the quark operator

$\displaystyle\frac{F^{2}}{4}\,(D_{\mu}\theta)\braket{\lambda_{+}L^{\mu}}\longleftrightarrow\left(2c_{GG}+c_{uu}^{a}+c_{dd}^{a}+c_{ss}^{a}\right)\frac{(\partial_{\mu}a)}{f}\,\bar{d}_{L}\gamma^{\mu}s_{L}\,.$
(2.46)

As shown in Bauer:2020jbp , the combination of ALP couplings entering this
expression equates the effective ALP–gluon coupling $\tilde{c}_{GG}$ below the
scale of chiral symmetry breaking. This effective coupling results from the
diagrams shown in the upper row of Figure 2.1. The coupling $g_{8}^{\theta}$
thus accounts for the gluon-induced interactions of the ALP with the loop-
induced flavor-changing neutral current interaction present in the SM,
mediated by diagrams such as those shown in the lower row of Figure 2.1. One
might expect that this interaction is a very small effect, since it starts at
two-loop order in perturbation theory. However, in the low-energy regime
relevant here, the gluon interactions are non-perturbative and loop
suppression becomes meaningless. For example, it is well known that gluon-
induced interactions at low-energy yield a sizable contribution to the
effective ALP–photon coupling in the chiral Lagrangian, such that
$c_{\gamma\gamma}^{\rm eff}\approx c_{\gamma\gamma}-(1.92\pm 0.04)\,c_{GG}$
for $m_{a}^{2}\ll m_{\pi}^{2}$ GrillidiCortona:2015jxo ; Bauer:2020jbp , even
though also in this case the interaction starts at two-loop order in
perturbation theory.

### 2.5 Weak mass term in the presence of external sources

In (2.39) we have introduced the weak mass term as an octet operator of the
form $\braket{\lambda_{+}S}$ or, equivalently, $\braket{\lambda_{6}S}$. When
external sources are absent, there are several ways to show that this operator
has no physical significance at lowest order in the chiral expansion (see e.g.
Crewther:1985zt ; Bernard:1985wf ; Leurer:1987ih ). Moreover, at the
$\mathcal{O}(p^{4})$ level, the effects of this operator can be absorbed into
a redefinition of the couplings of the octet operators $W_{i}^{(8)}$ in the
weak chiral Lagrangian Kambor:1989tz .555Note that the weak mass term is not
strictly unobservable at $\mathcal{O}(p^{4})$, since it affects the running of
the low-energy couplings $E_{i,r}(\mu)$. Originally suggested by Crewther
Crewther:1985zt , the reparametrization invariance of the low-energy
Lagrangian under chiral transformations plays a key role in eliminating the
weak mass term from the LO chiral Lagrangian in the absence of external
sources (up to linear order in $G_{F}$). We discuss the argument here in some
detail, because the presence of an ALP introduces some significant changes.

In presence of the weak mass term, the $\mathcal{O}(p^{2})$ chiral Lagrangian
can be expressed as

$\displaystyle\mathcal{L}^{(p^{2})}=\frac{F^{2}}{8}\braket{(D_{\mu}\Sigma)(D^{\mu}\Sigma)^{\dagger}+\chi^{D}\Sigma^{\dagger}+\Sigma\,\chi^{D\dagger}}\,,$
(2.47)

where

$\displaystyle\chi^{D}=\left(1+2F^{2}G_{8}^{\prime}\,\lambda_{6}\right)\chi\,,$
(2.48)

contains linear (tadpole) terms in the meson fields, indicating that
$\Sigma=1$ is no longer the correct ground-state of the theory. The linear
terms can be removed, and the scalar potential be minimized, by means of an
infinitesimal chiral transformation
$\Sigma\to\Sigma^{\prime}=g_{L}\Sigma\,g_{R}^{\dagger}$ with

$\displaystyle g_{L}=1+i\alpha_{L}\,,\qquad g_{R}=1+i\alpha_{R}\,,$ (2.49)

and $\alpha_{L,R}=\mathcal{O}(G_{8}^{\prime})$. Under this transformation,
$\Sigma$ transforms as $\Sigma\to\Sigma+\delta\Sigma$, with

$\displaystyle\delta\Sigma=i\alpha_{L}\Sigma-i\Sigma\,\alpha_{R}\,.$ (2.50)

In the absence of external sources the kinetic term is unaffected by this
transformation, and the condition

$g_{L}\chi^{D}g_{R}^{\dagger}=\chi+\mathcal{O}(G_{F}^{2})$ (2.51)

ensures that after the transformation the ground-state is indeed at
$\Sigma=1$. This condition is met by choosing the parameters $\alpha_{L}$ and
$\alpha_{R}$ as

$\displaystyle\begin{aligned}
\alpha_{L}&=G_{8}^{\prime}F^{2}\left(\frac{m_{s}+m_{d}}{m_{s}-m_{d}}+\frac{m_{s}-m_{d}}{m_{s}+m_{d}}\right)\lambda_{7}\,,\\\
\alpha_{R}&=G_{8}^{\prime}F^{2}\left(\frac{m_{s}+m_{d}}{m_{s}-m_{d}}-\frac{m_{s}-m_{d}}{m_{s}+m_{d}}\right)\lambda_{7}\,.\end{aligned}$
(2.52)

In the absence of external sources, this choice removes the weak mass term
from the LO chiral Lagrangian, indicating that it is a redundant operator at
this order. Indeed, when computing the $K\to\pi\pi$ matrix elements of the
weak mass term, one finds that there are two tree-level diagrams which exactly
cancel each other. Beyond the LO the situation becomes more subtle. Since the
operator is redundant, one would find that also its one-loop matrix elements
does vanish,666To the best of our knowledge, this calculation has not yet been
done. but tree-level diagrams involving the weak mass term along with an
insertion of one of the operators $O_{i}$ in the $\mathcal{O}(p^{4})$ QCD
chiral Lagrangian (2.58) do not vanish. The reason is that some of the
operators in the $\mathcal{O}(p^{4})$ QCD chiral Lagrangian are not invariant
under the field redefinition in (2.50) Kambor:1989tz . Hence the weak mass
term is not a redundant operator at $\mathcal{O}(p^{4})$ and beyond.

If external sources are present, the weak mass term is in general not
redundant even at LO. In our case, the chiral Lagrangian contains the spurion
fields $\chi=2B_{0}\,m$ along with $l_{\mu}$ and $r_{\mu}$, and it is formally
invariant under chiral transformations only if we impose the hypothetical
transformation rules

$\displaystyle\chi\to g_{L}\,\chi\,g_{R}^{\dagger}\,,\qquad l_{\mu}\to
g_{L}\,l_{\mu}\,g_{L}^{\dagger}\,,\qquad r_{\mu}\to
g_{R}\,r_{\mu}\,g_{R}^{\dagger}\,.$ (2.53)

The fact that in reality the spurion $\chi$ is not invariant under a chiral
transformation was the key to eliminating the weak mass term in the first
place, alongside with the invariance of the kinetic term. However, in the
presence of the ALP currents, this invariance no longer holds. In particular,
the covariant derivative transforms as

$\displaystyle D_{\mu}\Sigma\to D_{\mu}\Sigma+i\left(\alpha_{L}D_{\mu}\Sigma-
D_{\mu}\Sigma\,\alpha_{R}\right)-[\alpha_{L},l_{\mu}]\,\Sigma+\Sigma\,[\alpha_{R},r_{\mu}]\,.$
(2.54)

The variation of the kinetic term in (2.47) then yields an extra contribution,
so that after the field redefinition we obtain

$\displaystyle\mathcal{L}^{(p^{2})}=\frac{F^{2}}{8}\braket{(D_{\mu}\Sigma)(D^{\mu}\Sigma)^{\dagger}+\chi\Sigma^{\dagger}+\Sigma\,\chi^{\dagger}}+\frac{iF^{2}}{4}\braket{[\alpha_{L},l_{\mu}]L^{\mu}+[\alpha_{R},r_{\mu}]R^{\mu}}\,,$
(2.55)

with $\alpha_{L,R}$ as given in (2.52). This term is non-zero if there are
sources $l_{\mu}$ and/or $r_{\mu}$ that do not commute with $\lambda_{7}$. In
the SM, the only possible sources $l_{\mu}$ and $r_{\mu}$ at low energies
arise from the photon, whose interactions are flavor-universal and thus
commute with $\lambda_{7}$. In the ALP model, however, we have

$\displaystyle\begin{aligned}
&=\frac{\partial_{\mu}a}{f}\left[\big{(}[k_{Q}]_{33}-[k_{Q}]_{22}\big{)}\,\lambda_{6}+\text{Re}\,[k_{Q}]_{23}\,(\sqrt{3}\,\lambda_{8}-\lambda_{3})\right],\\\
[i\lambda_{7},r_{\mu}]&=\frac{\partial_{\mu}a}{f}\left[\big{(}[k_{q}]_{33}-[k_{q}]_{22}\big{)}\,\lambda_{6}+\text{Re}\,[k_{q}]_{23}\,(\sqrt{3}\,\lambda_{8}-\lambda_{3})\right],\end{aligned}$
(2.56)

which in general can be non-zero. In our calculation of the $K^{-}\to\pi^{-}a$
decay amplitude in Section 3 we are going to use the weak mass term in the
transformed form (2.55), which has the advantage of not introducing tadpole
vertices at $\mathcal{O}(p^{2})$. We will compute its contribution only at LO.
In fact, phenomenological bounds impose very tight constraints on flavor off-
diagonal ALP couplings in the down-quark sector, especially for the quarks of
the first two generations Bauer:2021mvw . Since the ALP couplings in the mass
basis are obtained from the couplings in the interaction basis via a CKM-like
rotation, the most plausible way to assure this is to impose the conditions

$\displaystyle[k_{Q}]_{33}=[k_{Q}]_{22}\,,\qquad[k_{q}]_{33}=[k_{q}]_{22}\,,\qquad[k_{Q}]_{23}=[k_{q}]_{23}=0$
(2.57)

in the UV theory far above the electroweak scale, i.e., on the couplings in
(2.3). Corrections to these relations are generated by one-loop matching
contributions at the electroweak scale through the weak interactions of the SM
Bauer:2021mvw ; Izaguirre:2016dfi ; Gavela:2019wzg . Since the weak mass term
itself is of first order in $G_{F}$, these effects are then of second order in
weak interactions and can safely be neglected.

### 2.6 QCD Lagrangian at $\bm{\mathcal{O}(p^{4})}$

At NLO in the chiral expansion, the operators entering the QCD Lagrangian have
been constructed in the seminal paper Gasser:1984gg  under some specific
assumptions about the external sources. As long as the conditions $\theta=0$,
$\braket{a_{\mu}}=0$ and $\braket{v_{\mu}}=0$ hold, the basis contains 12
operators $O_{i}$, whose coefficients are called $L_{i}$ for $i=1,\dots,10$
and $H_{1,2}$. In Table 2.2 we show the definitions of the operators $O_{i}$
for $i=1,\dots,8$. The remaining operators exist only in the presence of
external sources that are absent in the ALP model.

$i$ | $O_{i}$ | $\Gamma_{i}$ | $i$ | $O_{i}^{\theta}$ | $\Gamma_{i}^{\theta}$
---|---|---|---|---|---
1 | $\braket{L^{2}}^{2}$ | $\frac{3}{32}$ | 1 | $-(\partial^{\mu}D_{\mu}\theta)\braket{P}$ | 0
2 | $\braket{L^{\mu}L^{\nu}}\braket{L_{\mu}L_{\nu}}$ | $\frac{3}{16}$ | 2 | $-(D_{\mu}\theta)\braket{L^{\mu}S}$ | $-\frac{1}{4}$
3 | $\braket{L^{4}}$ | 0 | 3 | $(D_{\mu}\theta)\braket{L^{\mu}L^{2}}$ | $0$
4 | $\braket{L^{2}}\braket{S}$ | $\frac{1}{8}$ |  |  |
5 | $\braket{L^{2}S}$ | $\frac{3}{8}$ |  |  |
6 | $\braket{S}^{2}$ | $\frac{11}{144}$ |  |  |
7 | $-\braket{P}^{2}$ | 0 |  |  |
8 | $\frac{1}{2}\braket{S^{2}-P^{2}}$ | $\frac{5}{48}$ |  |  |
Table 2.2: Operators in the $\mathcal{O}(p^{4})$ QCD Lagrangian (2.58)
relevant for the SM extended by a light ALP. We use the short-hand notation
$L^{2}\equiv L_{\nu}L^{\nu}$. The coefficients $\Gamma_{i}$ and
$\Gamma_{i}^{\theta}$ are taken from Gasser:1984gg  and Herrera-
Siklody:1996tqr , respectively.

In the presence of an ALP the assumptions mentioned above are generally not
fulfilled, since $\theta\neq 0$ and $\braket{a_{\mu}}\neq 0$ for generic ALP
couplings to gluons and quarks. Therefore, an enlargement of the basis is
needed. The additional operators contain $\theta$ and $\braket{a_{\mu}}$ in
the invariant combination $D_{\mu}\theta$ in (2.16). An exhaustive derivation
of such terms has been performed in Herrera-Siklody:1996tqr . For the purposes
of this work, we can restrict ourselves to parity-conserving operators linear
in $D_{\mu}\theta$, since we consistently work to first order in $1/f$ (with
$f\gg 4\pi F$ being the ALP decay constant). There are only three such
operators, denoted by $O_{i}^{\theta}$, whose definition is also given in
Table 2.2. They correspond to $O_{53}$, $O_{46}$, and $O_{31}$ in the notation
of Herrera-Siklody:1996tqr . All in all, our $\mathcal{O}(p^{4})$ QCD
Lagrangian then takes the form

$\displaystyle\mathcal{L}_{\rm
QCD}^{(p^{4})}=\sum_{i=1}^{8}L_{i}\,O_{i}+\sum_{i=1}^{3}L_{i}^{\theta}\,O_{i}^{\theta}\,.$
(2.58)

The bare low-energy constants $L_{i}^{(\theta)}$ are customarily written as

$\displaystyle\begin{aligned}
L_{i}^{(\theta)}&=L_{i,r}^{(\theta)}(\mu)+\lambda\,\Gamma_{i}^{(\theta)}\,,\end{aligned}$
(2.59)

where $L_{i,r}^{(\theta)}(\mu)$ are the renormalized couplings, also called
low-energy constants, and the quantity

$\displaystyle\lambda=\frac{\mu^{d-4}}{32\pi^{2}}\left[\frac{2}{d-4}-(\ln
4\pi-\gamma_{E}+1)\right],$ (2.60)

where $d=4-2\epsilon$ denotes the number of spacetime dimensions, absorbs the
$1/\epsilon$ poles in the dimensional regularization scheme. Note that the
couplings $L_{i}$ and $L_{i}^{\theta}$ have mass dimension
$[L_{i}^{(\theta)}]=(d-4)=-2\epsilon$, and we have added an auxiliary scale
$\mu$ in the definition of $\lambda$ to account for this fact. The
coefficients $\Gamma_{i}^{(\theta)}$ are the one-loop anomalous dimensions
entering the evolution equations for the renormalized couplings. One obtains

$\displaystyle\frac{d}{d\ln\mu}\,L_{i,r}^{(\theta)}(\mu)=-\frac{\Gamma_{i}^{(\theta)}}{16\pi^{2}}\,.$
(2.61)

The values for the coefficients $\Gamma_{i}^{(\theta)}$ as computed in
Gasser:1984gg ; Herrera-Siklody:1996tqr  are shown in Table 2.2.

$10^{3}\,L_{i,r}$ | Ref. Bijnens:2014lea  | Ref. Dowdall:2013rya
---|---|---
$10^{3}\,L_{4,r}(\mu)$ | $\phantom{-}0.0\pm 0.3$ | $0.00\pm 0.34$
$10^{3}\,L_{5,r}(\mu)$ | $\phantom{-}1.2\pm 0.1$ | $1.19\pm 0.25$
$10^{3}\,L_{7,r}(\mu)$ | $-0.3\pm 0.2$ | –
$10^{3}\,L_{8,r}(\mu)$ | $\phantom{-}0.5\pm 0.2$ | $0.55\pm 0.15$
Table 2.3: Phenomenological values of the renormalized QCD low-energy
constants $L_{i,r}(\mu)$ relevant to our calculation, at the scale
$\mu=m_{\rho}$.

While no estimates of $L^{\theta}_{i,r}$ are available in the literature, the
values of the low-energy constants $L_{i,r}$ with $i=4,5,7,8$ relevant to our
analysis have been determined – with uncertainties ranging from about 10% to
more than 100% – from fits to low-energy data Bijnens:2014lea  and using
lattice QCD FlavourLatticeAveragingGroupFLAG:2021npn . For our
phenomenological estimates in Section 4 we employ two data sets: the values of
the $\mathcal{O}(p^{4})$ fit reported in Table 1 of Bijnens:2014lea , and the
most recent lattice results obtained by the HPQCD collaboration from a
simulation with $n_{f}=2+1+1$ dynamical quark flavors Dowdall:2013rya . Both
parameter sets are provided at the scale $\mu=m_{\rho}=770$ MeV and are
summarized in Table 2.3. The value of $L_{7,r}$ is not available from the
lattice. Given the good agreement of the phenomenological values and with the
estimates $L_{4}^{\infty}=0$, $L_{5}^{\infty}=1.0\cdot 10^{-3}$,
$L_{7}^{\infty}=-0.175\cdot 10^{-3}$, $L_{8}^{\infty}=0.375\cdot 10^{-3}$
obtained in the large-$N_{c}$ approximation Cirigliano:2003yq , we use
$L_{7,r}=(-0.175\pm 0.2)\cdot 10^{-3}$ with a conservative error for the
second parameter set.

### 2.7 Weak Lagrangian at $\bm{\mathcal{O}(G_{F}\,p^{4})}$

As discussed in Section 2.4, the weak chiral Lagrangian involves operators
transforming as $(8_{L},1_{R})$ or $(27_{L},1_{R})$ under chiral
transformations, where the octet operators receive a strong dynamical
enhancement. Consequently, in this work we focus on the octet operators only.
We start from the octet basis constructed in Ecker:1992de  under the
assumptions $\theta=0$, $\braket{a_{\mu}}=0$ and $\braket{v_{\mu}}=0$, which
consists of 37 operators $W_{i}^{(8)}$. (This basis is the reduced version of
a redundant set of operators originally presented in Kambor:1989tz .) We
collect the operators relevant for our calculations in Table 2.4.

$i$ | $W_{i}^{(8)}$ | $Z_{i}$
---|---|---
5 | $\braket{\lambda_{6}\\{S,L^{2}\\}}$ | $\frac{3}{2}$
6 | $\braket{\lambda_{6}L_{\mu}}\braket{SL^{\mu}}$ | $-\frac{1}{4}$
8 | $\braket{\lambda_{6}L^{2}}\braket{S}$ | $-\frac{1}{2}$
9 | $i\braket{\lambda_{6}[P,L^{2}]}$ | $\frac{3}{4}$
10 | $\braket{\lambda_{6}S^{2}}$ | $\frac{2}{3}$
11 | $\braket{\lambda_{6}S}\braket{S}$ | $-\frac{13}{18}$
12 | $-\braket{\lambda_{6}P^{2}}$ | $-\frac{5}{12}$
13 | $-\braket{\lambda_{6}P}\braket{P}$ | $0$
20 | $-\frac{i}{2}\braket{\lambda_{6}\big{[}\\{W_{\mu\nu},L^{\nu}\\},l^{\mu}\big{]}}$ | $\frac{3}{4}$
21 | $\braket{\lambda_{6}\big{[}[S,L^{\mu}],l_{\mu}\big{]}}$ | $\frac{5}{6}$
22 | $-\braket{\lambda_{6}\big{[}iD_{\mu}S+\frac{1}{2}[L_{\mu},S],l^{\mu}\big{]}}$ | $\frac{5}{6}$
23 | $i\braket{\lambda_{6}\big{[}\\{L^{\mu},P\\},l_{\mu}\big{]}}$ | $0$
Table 2.4: Operators entering the $\mathcal{O}(p^{4})$ weak chiral Lagrangian
(2.63) that are relevant for $K^{-}\to\pi^{-}a$ decay. The coefficients
$Z_{i}$ are taken from Ecker:1992de .

To the best of our knowledge, the weak octet basis in the general case, where
$\theta$, $\braket{a_{\mu}}$ and $\braket{v_{\mu}}$ can be non-zero, has not
yet been derived in the literature. Working again to linear order in
$D_{\mu}\theta$, we find that in this case there are 12 additional CP
invariant octet operators, which we choose in the form:

$\displaystyle\begin{aligned}
W^{\theta\,(8)}_{1}&=(D_{\mu}\theta)\braket{\lambda_{6}\\{L^{\mu},S\\}},\qquad&W^{\theta\,(8)}_{7}&=(D_{\mu}\theta)\braket{\lambda_{6}L_{\nu}}\braket{W^{\mu\nu}},\\\
W^{\theta\,(8)}_{2}&=i(D_{\mu}\theta)\braket{\lambda_{6}[L^{\mu},P]},\qquad&W^{\theta\,(8)}_{8}&=i(D_{\mu}\theta)\braket{\lambda_{6}[L_{\nu},W^{\mu\nu}]},\\\
W^{\theta\,(8)}_{3}&=(D_{\mu}\theta)\braket{\lambda_{6}L^{\mu}}\braket{S},\qquad&W^{\theta\,(8)}_{9}&=(D_{\mu}\theta)\braket{\lambda_{6}D_{\nu}W^{\nu\mu}},\\\
W^{\theta\,(8)}_{4}&=(D_{\mu}\theta)\braket{\lambda_{6}\\{L^{\mu},L^{2}\\}},\qquad&W^{\theta\,(8)}_{10}&=i(\partial^{\mu}D_{\mu}\theta)\braket{\lambda_{6}S},\\\
W^{\theta\,(8)}_{5}&=(D_{\mu}\theta)\braket{\lambda_{6}L^{\mu}}\braket{L^{2}},\qquad&W^{\theta\,(8)}_{11}&=(\partial^{\mu}D_{\mu}\theta)\braket{\lambda_{6}P},\\\
W^{\theta\,(8)}_{6}&=D_{\mu}\theta\braket{\lambda_{6}L_{\nu}L^{\mu}L^{\nu}},\qquad&W^{\theta\,(8)}_{12}&=i(\partial^{\mu}D_{\mu}\theta)\braket{\lambda_{6}L^{2}}.\end{aligned}$
(2.62)

The complete $\mathcal{O}(p^{4})$ weak octet Lagrangian thus reads

$\displaystyle\mathcal{L}_{\rm
weak}^{(p^{4})}=\frac{F^{2}}{2}\left(\sum_{i=1}^{37}E_{i}\,W_{i}^{(8)}+\sum_{i=1}^{12}E_{i}^{\theta}\,W_{i}^{\theta\,(8)}\right),$
(2.63)

where the couplings $E_{i}$ and $E_{i}^{\theta}$ are proportional to the Fermi
constant $G_{F}$.

$i$ | $W^{\theta\,(8)}_{i}$ | $Z_{i}^{\theta}$ | $T_{i}^{\theta}$
---|---|---|---
$1$ | $(D_{\mu}\theta)\braket{\lambda_{6}\\{L^{\mu},S\\}}$ | $\frac{1}{2}$ | $\frac{3}{4}$
$2$ | $i(D_{\mu}\theta)\braket{\lambda_{6}[L^{\mu},P]}$ | $-\frac{1}{2}$ | $0$
$3$ | $(D_{\mu}\theta)\braket{\lambda_{6}L^{\mu}}\braket{S}$ | $-\frac{3}{4}$ | $\frac{1}{2}$
$8$ | $i(D_{\mu}\theta)\braket{\lambda_{6}[L_{\nu},W^{\mu\nu}]}$ | $0$ | $0$
$9$ | $(D_{\mu}\theta)\braket{\lambda_{6}D_{\nu}W^{\nu\mu}}$ | $0$ | $0$
$10$ | $i(D_{\mu}\theta)\braket{\lambda_{6}D^{\mu}S}$ | $0$ | $0$
$11$ | $(\partial^{\mu}D_{\mu}\theta)\braket{\lambda_{6}P}$ | $0$ | $0$
$12$ | $i(\partial^{\mu}D_{\mu}\theta)\braket{\lambda_{6}L^{2}}$ | $0$ | $0$
Table 2.5: $W_{i}^{\theta\,(8)}$ operators entering the $\mathcal{O}(p^{4})$
weak Lagrangian in (2.63) contributing to the process $K^{-}\to\pi^{-}a$. The
determination of the $Z_{i}^{\theta}$ and $T_{i}^{\theta}$ values is discussed
in Section 3.2.

Under renormalization, the operators $W_{i}^{(8)}$ present in the SM in
general require the new-physics operators $W_{i}^{\theta\,(8)}$ as
counterterms to remove $1/\epsilon$ poles in Green’s functions involving an
external ALP. On the other hand, the new-physics operators only mix among
themselves. For the structure of the $1/\epsilon$ pole terms in the
corresponding couplings $E_{i}$ and $E_{i}^{\theta}$, this implies

$\displaystyle\begin{aligned}
E_{i}&=E_{i,r}(\mu)+\lambda\,G_{8}Z_{i}\,;\qquad&i&=1,\dots,37\,,\\\
E_{i}^{\theta}&=E_{i,r}^{\theta}(\mu)+\lambda\left(G_{8}Z_{i}^{\theta}+G_{8}^{\theta}\,T_{i}^{\theta}\right);\quad&i&=1,\dots,12\,.\end{aligned}$
(2.64)

The $Z_{i}$ factors have been computed in Ecker:1992de , whereas the
coefficients $Z_{i}^{\theta}$ and $T_{i}^{\theta}$ will be derived here for
the first time. Unfortunately, no phenomenological estimates for the
renormalized low-energy constants $E_{i,r}$ and $E_{i,r}^{\theta}$ exist in
the literature.

## 3 Calculation of the $K^{-}\to\pi^{-}a$ decay amplitude

### 3.1 Amplitude at leading order

With the general chiral Lagrangian described in the previous section at hand,
we can now move forward to the calculation of the physical amplitude for the
process of interest, $K^{-}\to\pi^{-}a$. We work at first order in $1/f$,
where $f$ is the decay constant of the ALP, and to order $G_{F}$. This setup
features two sources of flavor violation: the SM weak interactions, and
flavor-violating ALP couplings. We neglect contributions from the product of
the two.

Figure 3.1: Feynman graphs contributing to the $K^{-}\to\pi^{-}a$ decay
amplitude at tree level. The ALP is represented by a dashed line. The black
square  denotes an insertion of a vertex from the $\mathcal{O}(p^{2})$ QCD
Lagrangian in (2.17), while the black dot $\bullet$ refers to a vertex from
the $\mathcal{O}(p^{2})$ weak Lagrangian in (2.39).

At LO in the chiral expansion, it is sufficient to sum all tree-level Feynman
diagrams. These are shown in Figure 3.1. In diagram $T_{1}$ the flavor-
changing $s\to d$ transition occurs via a flavor-changing ALP–quark
interaction, while in $T_{2-4}$ the flavor change originates from the CKM
matrix. We do not draw explicitly vertices involving mixing between neutral
states, since these have been removed via the rotations in (2.32). Adding up
the four diagrams we obtain the LO result

$\displaystyle\mathcal{A}_{\rm LO}=\mathcal{A}^{\rm FV}_{\rm
LO}+\mathcal{A}^{\rm{weak},8}_{\rm LO}+\mathcal{A}^{\rm{weak},8^{\prime}}_{\rm
LO}+\mathcal{A}^{\rm{weak},8\theta}_{\rm
LO}+\mathcal{A}^{\rm{weak},27,1/2}_{\rm
LO}+\mathcal{A}^{\rm{weak},27,3/2}_{\rm LO}\,.$ (3.1)

Expressed in terms of the mass ratios

$\displaystyle x=\frac{m_{a}^{2}}{m_{K}^{2}}\,,\qquad
y=\frac{m_{\pi}^{2}}{m_{K}^{2}}\,,$ (3.2)

the contribution involving flavor-changing ALP couplings reads

$\displaystyle i\mathcal{A}^{\rm FV}_{\rm
LO}=-[k_{q}+k_{Q}]_{23}\,\frac{m_{K}^{2}}{2f}\,(1-y)\,.$ (3.3)

The contributions from the three octet weak-interaction operators are

$\displaystyle\begin{aligned} i\mathcal{A}^{\rm{weak},8}_{\rm
LO}&=\frac{G_{8}F^{2}\,m_{K}^{2}}{4f}\,\frac{1}{4-3x-y}\\\
&\quad\times\bigg{\\{}\,16c_{GG}\,(1-x)(1-y)+2c_{uu}^{a}\,(1-y)(4-4x-y)\\\
&\hskip
36.98866pt+c^{a}_{dd}\,(4-x-3x^{2}-5y+4xy+y^{2})+c^{a}_{ss}\,(4-3x+3x^{2}-5y+y^{2})\\\\[5.69054pt]
&\hskip
36.98866pt+(c^{v}_{dd}-c^{v}_{ss})\,(4-7x+3x^{2}+3y-2xy-y^{2})\\\\[2.84526pt]
&\quad-\frac{8\varepsilon^{(2)}}{\sqrt{3}}\,\frac{1-y}{4-3x-y}\,\bigg{[}16c_{GG}\,(1-x)(1-y)-c^{a}_{uu}\,\frac{x(1-y)(4-5x+y)}{y-x}\\\
&\hskip
36.98866pt+c^{a}_{dd}\,x(1-x)\,\frac{4-9x+5y}{y-x}-c^{a}_{ss}\,x(8-9x+y)\bigg{]}\bigg{\\}}\\\
i\mathcal{A}^{\rm{weak},8^{\prime}}_{\rm
LO}&=-\frac{G_{8}^{\prime}F^{2}\,m_{K}^{2}}{2f}\,\bigg{\\{}(c^{v}_{dd}-c^{v}_{ss})-(c^{a}_{dd}-c^{a}_{ss})\,(1-y)^{2}\,,\\\
&\hskip
83.93553pt+\frac{4\varepsilon^{(2)}}{\sqrt{3}}\,(1-y)\,\Big{[}(c^{v}_{dd}-c^{v}_{ss})+(c^{a}_{dd}-c^{a}_{ss})\,(1-y)^{2}\Big{]}\bigg{\\}}\,,\\\
i\mathcal{A}^{\rm{weak},8\theta}_{\rm
LO}&=-\frac{G_{8}^{\theta}F^{2}\,m_{K}^{2}}{2f}\left(2c_{GG}+\braket{c^{a}}\right)(1-y)\,.\end{aligned}$
(3.4)

Finally, the contributions from the two 27-plet weak-interaction operators
take the form

$\displaystyle\begin{aligned} i\mathcal{A}^{\rm{weak},27,1/2}_{\rm
LO}&=\frac{G_{27}^{1/2}F^{2}\,m_{K}^{2}}{4f}\,\frac{1}{4-3x-y}\\\
&\quad\times\bigg{\\{}\,8c_{GG}\,(1-y)(4-x-3y)+4c_{uu}^{a}\,(1-y)(4-x-y)\\\
&\hskip 34.14322pt+3c^{a}_{dd}\,(4+x-x^{2}-5y+y^{2})\\\\[5.69054pt] &\hskip
34.14322pt-c^{a}_{ss}\,(28-x-3x^{2}-35y+4xy+7y^{2})\\\\[5.69054pt] &\hskip
34.14322pt+(c^{v}_{dd}-c^{v}_{ss})\,(4-7x+3x^{2}+3y-2xy-y^{2})\\\\[2.84526pt]
&\quad-\frac{8\varepsilon^{(2)}}{\sqrt{3}}\,\frac{1-y}{4-3x-y}\,\bigg{[}4c_{GG}\,(1-y)(4-x-5y)\\\
&\hskip 34.14322pt-2c^{a}_{uu}\,x(1-y)\,\frac{12-5x-7y}{y-x}\\\ &\hskip
34.14322pt+c^{a}_{dd}\,x\,\frac{24-38x+9x^{2}-10y+20xy-5y^{2}}{y-x}\\\ &\hskip
34.14322pt-c^{a}_{ss}\,x(28-9x-19y)\bigg{]}\bigg{\\}}\,,\\\
i\mathcal{A}^{\rm{weak},27,3/2}_{\rm
LO}&=\frac{G_{27}^{3/2}F^{2}\,m_{K}^{2}}{4f}\,\frac{1}{4-3x-y}\\\
&\quad\times\bigg{\\{}\,8c_{GG}\,(1-x)(1-y)-4c_{uu}^{a}\,(1-y)\,\frac{x-x^{2}-4y+3xy+y^{2}}{y-x}\\\
&\hskip
34.14322pt-3c^{a}_{dd}\,\frac{y(4-5y+y^{2})-xy(4-3y)+x^{2}(1+y)-x^{3}}{y-x}\\\\[2.84526pt]
&\hskip 34.14322pt-c^{a}_{ss}\,(4-x-3x^{2}-5y+4xy+y^{2})\\\\[5.69054pt]
&\hskip
34.14322pt+(c^{v}_{dd}-c^{v}_{ss})\,(4-7x+3x^{2}+3y-2xy-y^{2})\\\\[2.84526pt]
&\quad-\frac{8\varepsilon^{(2)}}{\sqrt{3}}\,\frac{1-y}{4-3x-y}\,\bigg{[}-4c_{GG}\,(1-y)\,\frac{12-11x+x^{2}-13y+9xy+2y^{2}}{y-x}\\\
&\hskip
34.14322pt-2c^{a}_{uu}\,x(1-y)\,\frac{6-5x-y}{y-x}+c^{a}_{dd}\,x\,\frac{11y-2y^{2}-7xy-11x+9x^{2}}{y-x}\\\
&\hskip
34.14322pt+c^{a}_{ss}\,x\,\frac{12+x-9x^{2}-25y+17xy+4y^{2}}{y-x}\bigg{]}\bigg{\\}}\,.\end{aligned}$
(3.5)

In these expressions all quark masses have been converted into the charged
meson masses $m_{K}^{2}\equiv m_{K^{-}}^{2}$ and $m_{\pi}^{2}\equiv
m_{\pi^{-}}^{2}$. In the isospin-conserving limit $m_{u}=m_{d}=\hat{m}$, the
amplitude $\mathcal{A}^{\rm{weak},8}_{\rm LO}$ has already been calculated in
Bauer:2021wjo , while the amplitudes $\mathcal{A}^{\rm{weak},27,1/2}_{\rm LO}$
and $\mathcal{A}^{\rm{weak},27,3/2}_{\rm LO}$ have been derived in
Bauer:2021mvw . Here we have included first-order corrections in the isospin-
breaking parameter

$\varepsilon^{(2)}=\frac{\sqrt{3}}{4}\,\frac{m_{d}-m_{u}}{m_{s}-\hat{m}}\,,\qquad\hat{m}=\frac{m_{u}+m_{d}}{2}\,,$
(3.6)

including the appropriate modifications of the rotation angles in (2.33). As
correctly anticipated in Bauer:2021wjo , these isospin-breaking effects are
suppressed by $1/m_{s}$ and hence yield corrections to the LO expressions that
are below the 1% level. In the results above, we have already imposed the
condition (2.11) on the parameters entering the chiral rotation. As already
noted in Bauer:2021wjo , the results do not exhibit any residual dependence on
the $\kappa_{q}$ parameters, as expected for consistency reasons. This applies
also to the contributions computed here for the first time.

### 3.2 Amplitude at next-to-leading order

In this section we discuss the $\mathcal{O}(p^{4})$ NLO result. For the
reasons explained in Section 2.4, we do not consider $\mathcal{O}(p^{4})$
corrections for the 27-plets and the weak mass term. Additionally, due to the
negligible impact of isospin-breaking corrections on the LO result, we compute
all $\mathcal{O}(p^{4})$ contributions in the isospin-conserving limit.

Figure 3.2: Feynman graphs contributing to the mesons self-energy at NLO in
the chiral expansion. Filled (empty) squared vertices denote QCD $p^{2}$
($p^{4}$) vertices.

We include one-loop diagrams with zero or one insertion of
$\mathcal{O}(G_{F}p^{2})$, along with arbitrary insertions from the
$\mathcal{O}(p^{2})$ QCD Lagrangian. Furthermore, we consider tree-level
graphs with either a $\mathcal{O}(G_{F}p^{4})$ insertion plus an arbitrary
number of $\mathcal{O}(p^{2})$ QCD insertions or, alternatively one insertion
from the $\mathcal{O}(p^{4})$ QCD Lagrangian and the $\mathcal{O}(G_{F}p^{2})$
weak Lagrangian, accompanied by an arbitrary number of insertions from the
$\mathcal{O}(p^{2})$ QCD Lagrangian. Furthermore, we account for external leg
corrections and incorporate NLO corrections to the LO meson masses. Therefore,
we write the total amplitude in the form

$\mathcal{A}_{\rm NLO}=(\sqrt{Z_{\pi}Z_{K}}-1)\,\mathcal{A}_{\rm
LO}^{\prime}+\mathcal{A}_{\rm LO}^{\Delta
m^{2}}+\mathcal{A}_{\text{1-loop}}^{(p^{4})}+\mathcal{A}_{\rm
tree}^{(p^{4})}\,.$ (3.7)

Now we describe each of these contributions in detail, starting from the
factor $\sqrt{Z_{\pi}Z_{K}}$, which represents the corrections from wave-
function renormalization. The factors $Z_{\pi,K}$ are defined as

$\displaystyle
Z_{i}=1-\left.\frac{d\Sigma_{i}(p^{2})}{dp^{2}}\right|_{p^{2}=m_{i}^{2}}\,,$
(3.8)

where $\Sigma_{i}(p^{2})$ is the self-energy of the $i$-th meson, obtained by
adding the one-loop 1-particle irreducible contribution with an insertion of a
$O(p^{2})$ QCD vertex, plus the tree-level contribution with an $O(p^{4})$ QCD
vertex, see Figure 3.2. In agreement with Gasser:1984gg , we find

$\displaystyle\begin{aligned}
Z_{K}&=1-\frac{1}{24\pi^{2}F^{2}}\left[(5m_{K}^{2}+m_{\pi}^{2})\left(\frac{1}{\epsilon}+1\right)-\frac{3}{2}\,m_{K}^{2}\ln\frac{\mu^{2}}{m_{K}^{2}}-\frac{3}{4}\,m_{\pi}^{2}\ln\frac{\mu^{2}}{m_{\pi}^{2}}\right.\\\
&\hskip
82.51282pt\left.-\left(m_{K}^{2}-\frac{m_{\pi}^{2}}{4}\right)\ln\frac{3\mu^{2}}{4m_{K}^{2}-m_{\pi}^{2}}\right]\\\
&\hskip
21.33955pt-\frac{16}{F^{2}}\left[m_{\pi}^{2}\,L_{4,r}(\mu)+m_{K}^{2}\left(2L_{4,r}(\mu)+L_{5,r}(\mu)\right)\right],\\\\[5.0pt]
Z_{\pi}&=1-\frac{1}{24\pi^{2}F^{2}}\left[(2m_{K}^{2}+4m_{\pi}^{2})\left(\frac{1}{\epsilon}+1\right)-m_{K}^{2}\ln\frac{\mu^{2}}{m_{K}^{2}}-2m_{\pi}^{2}\ln\frac{\mu^{2}}{m_{\pi}^{2}}\right]\\\
&\hskip
21.33955pt-\frac{16}{F^{2}}\left[2m_{K}^{2}\,L_{4,r}(\mu)+m_{\pi}^{2}\left(L_{4,r}(\mu)+L_{5,r}(\mu)\right)\right].\end{aligned}$
(3.9)

The second term in (3.7) is finite and comprises corrections in the masses and
the decay constant. The relation between the meson masses at LO and NLO is
given by

$\displaystyle m_{i,\rm{NLO}}^{2}=m_{i}^{2}+\Delta m_{i}^{2}\,,$ (3.10)

where $\Delta m_{i}^{2}=-\Sigma_{i}(m_{i}^{2})$. Since we have expressed all
quark masses in terms of the charged kaon and pion masses, it is sufficient to
compute these two mass shifts. We find

$\displaystyle\begin{aligned} \Delta
m_{\pi^{-}}^{2}&=\frac{m_{\pi}^{2}}{F^{2}}\,\bigg{[}-\frac{m_{\pi}^{2}}{16\pi^{2}}\ln\frac{\mu^{2}}{m_{\pi}^{2}}+\frac{4m_{K}^{2}-m_{\pi}^{2}}{144\pi^{2}}\,\ln\frac{3\mu^{2}}{4m_{K}^{2}-m_{\pi}^{2}}\\\
&\quad+16\left[(2m_{K}^{2}+m_{\pi}^{2})\left(2L_{6,r}(\mu)-L_{4,r}(\mu)\right)+m_{\pi}^{2}\left(2L_{8,r}(\mu)-L_{5,r}(\mu)\right)\right]\bigg{]}\,,\\\\[2.84526pt]
\Delta
m_{K^{-}}^{2}&=\frac{m_{K}^{2}}{F^{2}}\,\bigg{[}-\frac{4m_{K}^{2}-m_{\pi}^{2}}{72\pi^{2}}\,\ln\frac{3\mu^{2}}{4m_{K}^{2}-m_{\pi}^{2}}\\\
&\quad+16\left[(m_{\pi}^{2}+2m_{K}^{2})\left(2L_{6,r}(\mu)-L_{4,r}(\mu)\right)+m_{K}^{2}\left(2L_{8,r}(\mu)-L_{5,r}(\mu)\right)\right]\bigg{]}\,,\end{aligned}$
(3.11)

in agreement with Gasser:1984gg . In the LO result (3.1) we thus replace
$m_{i}^{2}=m_{i,\rm{NLO}}^{2}-\Delta m_{i}^{2}$ for $i=\pi,K$.

Figure 3.3: Feynman diagrams contributing to the $K^{-}\to\pi^{-}a$ decay
amplitude at NLO in the chiral expansion, and generated by flavor-violating
ALP interactions. The black square  denotes the insertion of a vertex from the
$\mathcal{O}(p^{2})$ QCD Lagrangian in (2.17), while the empty square
$\square$ refers to a vertex from the $\mathcal{O}(p^{4})$ QCD Lagrangian in
(2.58).

In principle, also the decay constant $F$ can be expressed in terms of the NLO
pion decay constant $F_{\pi}$ as

$\displaystyle F=F_{\pi}-\Delta F_{\pi}\,,$ (3.12)

where Gasser:1984gg ; Cirigliano:2003gt

$\displaystyle\begin{aligned} \Delta
F_{\pi}&=\frac{1}{16\pi^{2}F_{\pi}}\left[m_{K}^{2}\ln\frac{\mu^{2}}{m_{K}^{2}}+2m_{\pi}^{2}\ln\frac{\mu^{2}}{m_{\pi}^{2}}\right]\\\
&\quad+\frac{8}{F_{\pi}}\left[2m_{K}^{2}\,L_{4,r}(\mu)+m_{\pi}^{2}\left(L_{4,r}(\mu)+L_{5,r}(\mu)\right)\right].\end{aligned}$
(3.13)

Note that this is a finite renormalization, and the scale dependence on the
right-hand side cancels between the logarithms and the renormalized low-energy
constants. In other words, both $F_{\pi}$ and $F$ are physical parameters.
Here we choose to work in terms of $F$ instead of $F_{\pi}$, since $\Delta
F_{\pi}$ would induce large NLO corrections.

Figure 3.4: One-loop Feynman diagrams contributing to the $K^{-}\to\pi^{-}a$
decay amplitude at NLO in the chiral expansion, and generated by flavor-
conserving ALP interactions via insertions of the $\mathcal{O}(p^{2})$ chiral
Lagrangians only. The flavor change is mediated by an insertion of a vertex
from the $\mathcal{O}(p^{2})$ weak chiral Lagrangian in (2.39) and shown by a
black dot $\bullet$. The black square  refers to a vertex from the
$\mathcal{O}(p^{2})$ QCD chiral Lagrangian in (2.17).

By substituting (3.10) in the LO amplitude (3.1), effectively we split it up
in two pieces,

$\mathcal{A}_{\rm LO}(m_{i}^{2})=\mathcal{A}_{\rm
LO}(m_{i,\rm{NLO}}^{2})+\mathcal{A}_{\rm LO}^{\Delta
m_{i}^{2}}\equiv\mathcal{A}_{\rm LO}^{\prime}+\mathcal{A}_{\rm LO}^{\Delta
m_{i}^{2}}\,.$ (3.14)

The first one is formally of $\mathcal{O}(p^{2})$ and is strictly speaking the
“true” LO result, while the second is part of the $\mathcal{O}(p^{4})$
correction. Both these quantities are trivially independent of the
$\kappa_{q}$ parameters. In the following we will drop the prime when
referring to the LO result. The third term in (3.7),
$\mathcal{A}_{\text{1-loop}}^{(p^{4})}$, comprises all 1-particle irreducible
one-loop graphs built out of $\mathcal{O}(p^{2})$ vertices, including at most
one weak vertex. These leads to 21 distinct topologies, displayed in Figure
3.3 ($D_{1-2}$) and Figure 3.4 ($D_{3-21}$), each with various internal
states, for a total of 89 distinct graphs. Finally, the last term in (3.7),
$\mathcal{A}_{\rm tree}^{(p^{4})}$, contains tree-level diagrams with one
$\mathcal{O}(p^{4})$ insertion and arbitrarily many $\mathcal{O}(p^{2})$
insertions. The corresponding topologies are $C_{1}$ in Figure 3.3 and
$C_{2-10}$ in Figure 3.5. Only $C_{10}$ has two distinct possible virtual
states ($\pi^{0}$ and $\eta_{8}$), hence in total we have 11 distinct graphs.
Note that even though we have removed the ALP mixing with $\pi^{0}$ and
$\eta_{8}$ at LO, both are generated at NLO. Instead of performing a NLO
diagonalization of the mass and kinetic terms, we choose to keep these terms
explicitly and treat them as interactions. This is shown explicitly in graphs
$D_{11}$ and $C_{10}$.

Figure 3.5: Feynman diagrams contributing to the $K\to\pi a$ decay amplitude
at NLO in the chiral expansion, generated by flavor-conserving ALP
interactions involving one insertion of the $\mathcal{O}(p^{4})$ QCD
Lagrangian in (2.58) (empty square $\square$) or $\mathcal{O}(p^{4})$ weak
Lagrangian in (2.58) (empty dot $\Circle$). The black square  and circle
$\bullet$ are as explained in Figure 3.4.

We have checked explicitly that the sum of $D_{1-21}$ and $C_{1-10}$ are
(separately) independent of the $\kappa_{q}$ parameters. This is an important
consistency check of the correctness of our NLO calculation. Another crucial
check is the cancellation of UV divergences. We have confirmed that the UV
divergences cancel in the contribution involving the flavor-changing couplings
of the ALP, as well as in the contribution proportional to $G_{8}$ in the
limit where $\braket{a_{\mu}}=0$ and $\theta=0$ (i.e. $\braket{c^{a}}=0$ and
$c_{GG}=0$). This requires knowledge of the coefficients $\Gamma_{i}$,
$\Gamma^{\theta}_{i}$ and $Z_{i}$ collected in Tables 2.2 and 2.4. The
cancellation of the remaining poles, proportional to
$G_{8}\,(2c_{GG}+\braket{c^{a}})$ and
$G_{8}^{\theta}\,(2c_{GG}+\braket{c^{a}})$, is achieved via the operators
$W_{1-3}^{\theta\,(8)}$, choosing the anomalous dimensions as given in Table
2.5. The remaining (finite) result $\mathcal{A}_{\rm LO}+\mathcal{A}_{\rm
NLO}$ retains an explicit $\ln\mu$ dependence, which is canceled by the scale
dependence of the various low-energy constants. We have derived closed
analytic expressions for the NLO corrections, which are however too lengthy to
display them here. Instead, we provide complete expressions diagram by diagram
in a Mathematica notebook attached to the arXiv version of this article as an
ancillary file.

An unexpected aspect of our result is the complete absence of an absorptive
part of the $K^{-}\to\pi^{-}a$ decay amplitude. This is in stark contrast to
the case of the $K_{S}\to\pi^{+}\pi^{-}$ decay, in which case the one-loop
diagrams in chiral perturbation theory generate a significant strong
rescattering phase Buras:2016fys . In principle, non-zero imaginary parts
could arise from the loop diagrams $D_{4}$, $D_{8}$, $D_{9}$, $D_{20}$, and
$D_{21}$ in Figure 3.4, when the initial-state kaon in graph $L_{8}$ decays
into two pions, or when the final-state ALP couples to a two-pion state
(remaining graphs). However, the vertex factors in these diagrams ensure that
all relevant cuts vanish. In diagram $D_{8}$, the Feynman rule for the
$K^{-}\,\pi^{-}\,\pi^{0}$ vertex (black circle) is proportional to the
difference of the two pion propagators, yielding a sum of two tadpole
integrals that are real. Similarly, in diagrams $D_{4}$, $D_{20}$, and
$D_{21}$ the Feynman rule for the $a\,\pi\,\pi$ vertex is proportional to the
difference of the two pion propagators, yielding a sum of two tadpole
integrals in all cases. In diagram $D_{9}$ a contribution remains with two
propagators in the loop, which arises when the upper right pion propagator is
cancelled by the numerator structure. In this case, however, the loop contains
a $K\,\pi$ intermediate state, which cannot be on-shell, since
$m_{a}<m_{K}-m_{\pi}$ for the decay $K^{-}\to\pi^{-}a$ to be kinematically
allowed.

## 4 Phenomenological consequences

We now provide numerical estimates of the NLO effects. For our study we use
PDG values for the meson masses, $m_{K^{-}}=(493.677\pm 0.016)$ MeV and
$m_{\pi^{-}}=(139.57039\pm 0.00018)$ MeV ParticleDataGroup:2022pth , along
with the FLAG average for the pion decay constant, $F_{\pi}=(130.2\pm 0.8)$
MeV FlavourLatticeAveragingGroupFLAG:2021npn . We then compute the parameter
$F$ using the relations (3.12) and (3.13) along with the explicit values of
the low-energy constants shown in Table 2.3. For the values shown in the first
and second column, we find $F=(112\pm 9)$ MeV and $F=(109\pm 10)$ MeV,
respectively, where the uncertainty is dominated by $L_{4,r}$.

### 4.1 Flavor-violating ALP couplings

Figure 4.1: NLO correction to the contribution of flavor-violating ALP
couplings to $K^{-}\to\pi^{-}a$. The gray (lightblue) band corresponds to the
$1\,\sigma$ uncertainty estimated using the inputs of the first (second)
column of Table 2.3 for the QCD low-energy constants $L_{5,r}$.

We start by analyzing how NLO corrections affect the contribution to the
$K^{-}\to\pi^{-}a$ decay amplitude generated by the flavor-violating couplings
of the ALP, which does not involve the SM weak interactions. Including
$\mathcal{O}(p^{4})$ terms, the corresponding amplitude can be written in the
form

$\mathcal{A}_{\rm LO+NLO}^{\rm FV}=\mathcal{A}_{\rm LO}^{\rm
FV}\left(1+\delta_{\rm FV}\right),$ (4.1)

where $\mathcal{A}_{\rm LO}^{\rm FV}$ is given in (3.3), and $\delta_{\rm FV}$
describes the NLO corrections. In this case $\delta_{\rm{FV}}$ depends on a
single low-energy constant, namely $L_{5,r}$. We show $\delta_{\rm FV}$ as a
function of the ALP mass in Figure 4.1. As a check of the correctness of our
result, we have verified that, for $m_{a}=0$, $\delta_{\rm FV}$ coincides with
the NLO correction to the $K\to\pi$ vector form factor at $q^{2}=0$, usually
denoted by $f_{2}$, which amounts to $f_{2}\approx-0.023$ Leutwyler:1984je .
As shown in Figure 4.1, the NLO corrections to the contribution involving
flavor-violating ALP couplings are relatively small over the entire mass
range, reaching a maximum value of about 10% at the largest kinematically
allowed value of $m_{a}$.

### 4.2 Flavor-conserving ALP couplings

We now move on to assessing the impact of NLO corrections on the various
contributions involving the flavor-conserving ALP couplings. For simplicity,
we consider the case of a minimally flavor-violating ALP outlined at the end
of Section 2.5. In this scenario, the amplitude depends solely on three
independent ALP couplings: $c_{GG}$, $c^{a}_{uu}$, and $c^{a}_{dd}$. Focusing
on the contributions from the weak octet operators, we write

$\displaystyle\begin{aligned} i\mathcal{A}^{\rm FC}_{\rm
LO+NLO}&=\frac{F^{2}}{2f}\,(m_{K}^{2}-m_{\pi}^{2})\\\
&\quad\times\bigg{\\{}G_{8}\left[c_{GG}\,a^{c_{GG}}_{G_{8}}\,\big{(}1+\delta^{c_{GG}}_{G_{8}}\big{)}+c^{a}_{uu}\,a^{c_{uu}^{a}}_{G_{8}}\,\big{(}1+\delta^{c_{uu}^{a}}_{G_{8}}\big{)}+c^{a}_{dd}\,a^{c_{dd}^{a}}_{G_{8}}\,\big{(}1+\delta^{c_{dd}^{a}}_{G_{8}}\big{)}\right]\\\
&\hskip 34.14322pt-
G_{8}^{\theta}\,\big{(}2c_{GG}+c^{a}_{uu}+2c^{a}_{dd}\big{)}\big{(}1+\delta_{G_{8}^{\theta}}\big{)}\bigg{\\}}\,,\end{aligned}$
(4.2)

where the dimensionless coefficients $a^{g}_{G_{8}}$ contain the LO
contributions proportional to $G_{8}$ and $G_{8}^{\theta}$. These coefficients
are derived from the findings in Section 3.1. Their explicit expressions are

$\displaystyle\begin{aligned} a^{c_{GG}}_{G_{8}}&=\frac{8(1-x)}{4-3x-y}\,,\\\
a^{c^{a}_{uu}}_{G_{8}}&=\frac{4-y-4x}{4-3x-y}\,,\\\
a^{c^{a}_{dd}}_{G_{8}}&=\frac{4-2x-y}{4-x-3y}\,.\end{aligned}$ (4.3)

Figure 4.2: NLO corrections to the contributions of flavor-conserving ALP
couplings to the $K^{-}\to\pi^{-}a$ decay amplitude. As in Figure 4.1, the
gray (colored) bands correspond to the $1\,\sigma$ uncertainty estimated using
the inputs of the first (second) column of Table 2.3 for the QCD low-energy
constants.

The quantities $\delta^{c_{GG}}_{G_{8}}$, $\delta^{c_{GG}}_{G_{8}}$,
$\delta^{c^{a}_{uu}}_{G_{8}}$, $\delta^{c_{dd}}_{G_{8}}$,
$\delta_{G_{8}^{\theta}}$ represent the relative size of the NLO contributions
compared to the LO ones. These quantities depend on the ALP mass $m_{a}$, some
known quantities (the meson masses, $F$, and the QCD low-energy constants
$L_{4,r}$, $L_{5,r}$, $L_{7,r}$, and $L_{8,r}$), and 18 unknown low-energy
constants that appear in 15 independent combinations $X_{1-15}$ (with 14
originating from the weak $\mathcal{O}(p^{4})$ Lagrangian). Figure 4.2
illustrates the various $\delta_{i}$ parameters as a function of the ALP mass
for teh case where all unknown low-energy constants are set to 0. The gray
(colored) dotted lines correspond to the value of $\delta_{i}$ obtained using
the central values listed in the first (second) column of Table 2.3 as inputs
for the QCD low-energy constants. The bands signify the corresponding
$1\,\sigma$ uncertainties, which are completely dominated by the uncertainty
on the QCD low-energy constants. As demonstrated in Figure 4.2, a reduction of
about 20–30% is found in all cases. This effect is largely dominated by the
QCD low-energy constants, with the loops playing only a minor role, and is
accompanied by a large uncertainty originating from the present errors in the
QCD low-energy constants, main in $L_{4,r}$.

Contributions involving the unknown low-energy constants, which we have set to
0 in the plots, are additional sources of uncertainties. To estimate this, we
normalize all such constants to $G_{8}$ and assume the independent
combinations $X_{i}$ to be distributed according to a Gaussian centered at 0
and with a width $\sigma=0.001$. This treatment is motivated by the known
distribution of the QCD low-energy constants. We draw the $X_{i}$ parameters
$n=1000$ times for each value of $m_{a}$777We have checked that enlarging $n$
does not affect the result significantly. and take the standard deviation of
the resulting $n$-dimensional vector
$\\{\delta^{g}_{G_{8}}(m_{a},X_{i}^{1}),\dots,\delta^{g}_{G_{8}}(X_{i}^{n})\\}$.
The resulting uncertainty is of the same size as the one from the known QCD
low-energy constants and, much like the latter, is almost independent of the
value of the ALP mass in the allowed kinematical range. Explicit numerical
result for the case $m_{a}=0$, including the LO contributions from the 27-plet
operators, are collected in Table 4.1. The full amplitude can be obtained by
adding up all entries multiplied by the corresponding coefficients. For
example, the upper-left entry corresponds to the contribution
$\frac{G_{8}F^{2}}{2f}\,c_{GG}\,\big{[}\,0.46_{\rm\,LO}+(-0.11\pm 0.09_{a}\pm
0.10_{b})_{\rm\,NLO}\big{]}\,\mathrm{GeV^{2}}$ to
$\mathcal{A}(K^{-}\to\pi^{-}a)$.

| $c_{GG}$ | $c^{a}_{uu}$ | $c^{a}_{dd}$
---|---|---|---
LO [GeV2] | NLO [GeV2] | LO [GeV2] | NLO [GeV2] | LO [GeV2] | NLO [GeV2]
$\frac{G_{8}F^{2}}{2f}$ | $\phantom{-}0.46$ | $-0.11^{\pm 0.09_{a}}_{\pm 0.10_{b}}$ | $\phantom{-}0.22$ | $-0.06^{\pm 0.05_{a}}_{\pm 0.05_{b}}$ | $\phantom{-}0.22$ | $-0.06^{\pm 0.05_{a}}_{\pm 0.08_{b}}$
$\frac{G_{8}^{\theta}F^{2}}{2f}$ | $-0.45$ | $0.12\pm 0.09_{a}$ | $-0.22$ | $0.06\pm 0.05_{a}$ | $-0.45$ | $0.12\pm 0.09_{a}$
$\frac{G_{27}^{1/2}F^{2}}{2f}$ | $\phantom{-}0.86$ | - | $\phantom{-}0.45$ | - | $-0.45$ | -
$\frac{G_{27}^{3/2}F^{2}}{2f}$ | $\phantom{-}0.23$ | - | $\phantom{-}0.45$ | - | $-0.45$ | -
Table 4.1: Numerical coefficients entering the flavor-conserving contributions
to the $K^{-}\to\pi^{-}a$ decay amplitude for the case of a massless ALP,
using the QCD low-energy constants from lattice QCD Dowdall:2013rya  as
inputs. The pedices $a$ and $b$ denote the uncertainty stemming from the known
and unknown low-energy constants, respectively (see text for details).

## 5 Conclusions

The process $K^{-}\to\pi^{-}a$ is an important probe of axion-like particles
(ALPs). In this work, we have provided a comprehensive analysis of this decay
at next-to-leading order in chiral perturbation theory. We have improved upon
existing calculations by adding the complete set of one-loop diagrams as well
as the contributions from low-energy constants appearing in the QCD and weak
chiral Lagrangians at $\mathcal{O}(p^{4})$, and by including isospin-breaking
corrections in the quark-mass difference $(m_{u}-m_{d})$ at first order. This
was accomplished by consistently including the ALP in the $\mathcal{O}(p^{2})$
and $\mathcal{O}(p^{4})$ chiral Lagrangians, which we have shown to require a
non-trivial extension of the corresponding operator bases.

At next-to-leading order, corrections proportional to the flavor-violating ALP
couplings, related to the $K\to\pi$ vector form factor, are found to be small,
ranging from about $-3\%$ to $+10\%$ depending on the value of $m_{a}$, with a
small theoretical uncertainty. Conversely, the corrections proportional to the
various flavor-conserving ALP couplings amount to a reduction of about 20–30%
over the entire allowed kinematical range. However, the uncertainty on the QCD
low-energy constant $L_{4,r}$ and the presence of several unknown low-energy
constants result in an $\mathcal{O}(1)$ uncertainty on the size of the next-
to-leading order corrections. This emphasizes the necessity for more accurate
determinations of the low-energy constant $L_{4,r}$ and, possibly, of the many
unknown weak low-energy constants entering the weak chiral Lagrangian, as well
as of the new constants arising only in the presence of an ALP.

Our work establishes a rigorous theoretical framework for further studies of
the $K^{-}\to\pi^{-}a$ decay process and other related processes in the
context of chiral perturbation theory. We are confident that the methodology
and results presented here could contribute to a more robust understanding of
the weak decays involving ALPs and mesons, enhancing our ability to interpret
experimental observations and probe the nature of these elusive particles.
Looking forward, there are several directions that can be pursued. First of
all, it would be interesting to study the neutral decay mode
$K_{L}\to\pi^{0}a$ and the semileptonic decay $\pi^{-}\to
e^{-}\bar{\nu}_{e}\,a$. Additionally, a more quantitative investigation of the
effects of the $\eta^{\prime}$ meson could be undertaken. Finally, the study
of the $\mathcal{O}(p^{4})$ effects of the weak mass term in the presence of
generic external sources may lead to a more nuanced understanding of light
meson weak decays involving light new particles.

### Acknowledgments

It is a pleasure to thank Gerhard Ecker, Gino Isidori, Stefan Scherer, and
Marvin Schnubel for valuable discussion and support. The research of CC, AG
and MN was supported by the Cluster of Excellence Precision Physics,
Fundamental Interactions, and Structure of Matter (PRISMA+, EXC 2118/1) within
the German Excellence Strategy (Project-ID 39083149). CC would also like to
thank Perimeter Institute for hospitality during the completion of this work.
This research was supported in part by Perimeter Institute for Theoretical
Physics. Research at Perimeter Institute is supported by the Government of
Canada through the Department of Innovation, Science and Economic Development
and by the Province of Ontario through the Ministry of Research, Innovation
and Science. The Feynman diagrams in this paper have been drawn with the Latex
package TikZ-Feynman Ellis:2016jkw . Results for the loop graphs have been
cross-checked with Package X Patel:2015tea ; Patel:2016fam .

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