Flavour constraints on light spin-1 bosons
within a chiral Lagrangian approach

Luca Di Luzio, Gabriele Levati, Paride Paradisi, Xavier Ponce Díaz

Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Padova,
Via F. Marzolo 8, 35131 Padova, Italy
Dipartimento di Fisica e Astronomia ‘G. Galilei’, Università di Padova,
Via F. Marzolo 8, 35131 Padova, Italy

> We discuss the construction of the chiral Lagrangian for a light spin-1
> boson, here denoted as $X$, featuring both vector and axial-vector couplings
> to light $u,d,s$ quarks. Focusing on $\Delta S=1$ transitions, we show that
> there are model-independent tree-level contributions to
> $K^{\pm}\to\pi^{\pm}X$, sourced by Standard Model charged currents, which
> receive an $m^{2}_{K}/m_{X}^{2}$ enhancement from the emission of a
> longitudinally polarized $X$. This flavour observable sets the strongest to
> date model-independent bound on the diagonal axial-vector couplings of $X$
> to $u,d,s$ quarks for $m_{X}<m_{K}-m_{\pi}$, superseding the bounds arising
> from beam-dump and collider searches.

###### Contents

  1. 1 Introduction
  2. 2 $\Delta S=1$ chiral Lagrangian for spin-1 bosons
    1. 2.1 Lowest-order chiral Lagrangian
    2. 2.2 Chiral Lagrangian for weak interactions
  3. 3 $K^{\pm}\to\pi^{\pm}X$ in $\chi$PT
    1. 3.1 Tree-level contribution
    2. 3.2 One-loop contribution
  4. 4 Flavour bounds vs. beam-dump and collider searches
  5. 5 Conclusions
  6. A Details of the chiral Lagrangian construction

##  1 Introduction

The lack of new particles at the LHC may be suggestive of the fact that they
are either too heavy to be directly produced or too weakly coupled to Standard
Model (SM) particles. New Physics (NP) models containing new feebly
interacting massive particles with sub-GeV masses are currently among the most
studied NP scenarios both theoretically and experimentally. Many of these
studies were dedicated to the dark photon [1, 2], a new massive spin-1
particle which is kinetically mixed with the ordinary photon and that could
act as a portal to a dark sector. Dark photon searches have been conducted by
a number of experiments, including beam-dump [3], fixed-target [4, 5],
collider [6, 7, 8, 9, 10, 11, 12], as well as meson decay [13, 14, 15, 16, 17,
18, 19] experiments.

Moreover, comprehensive analyses aiming at probing a light spin-1 boson $X$
with general couplings to quarks and leptons have been also carried out (see
e.g. [20, 21, 22, 23]). If $X$ is coupled to SM particles through a non-
conserved current, such as the axial-vector current, processes which are
enhanced by the ratio $({\rm energy}/m_{X})^{2}$ involving the longitudinal
mode of $X$ are generally induced. The same happens if $X$ is coupled to a
tree-level conserved current which is broken by the chiral anomaly [24, 25].
These energy-enhanced contributions generally provide the dominant effects
both to the production mechanisms of $X$ in high-energy experiments, as well
as to flavor-changing neutral current processes such as
$K^{\pm}\to\pi^{\pm}X$.

The aim of this paper is to revisit the sensitivity to the above NP scenarios
of the rare decay $K^{\pm}\to\pi^{\pm}X$ induced by an underlying $s\to d$
quark transition. In order to accomplish this task, we will extend previous
studies by constructing the most general $\Delta S=1$ chiral Lagrangian up to
the order $\mathcal{O}(p^{4})$, which will enable us to account for all of the
dominant effects stemming from weak interactions. Indeed, the lowest-order
$\mathcal{O}(p^{2})$ terms of chiral perturbation theory ($\chi$PT) will
capture the weak effects to the $s\to dX$ transition discussed in [24, 25] and
arising from the one-loop exchange of the $W$-boson and up-quarks. Instead, as
we will see, weak effects stemming from the $\Delta S=1$ four-quark Lagrangian
[26], can be included only by keeping $\mathcal{O}(p^{4})$ terms in $\chi$PT.
Although subleading in the chiral expansion, the latter contributions to
$K^{\pm}\to\pi^{\pm}X$ arise already at tree level (they can be thought as
arising from initial or final state radiation of $X$ from the external quark
legs of the $\Delta S=1$ effective Lagrangian) and therefore their inclusion
appears to be mandatory.

Moreover, the tree-level weak contributions discussed in this work are model-
independent and therefore they represent a general and robust prediction of
any ultraviolet (UV) complete NP model entailing a light spin-1 boson.
Instead, the loop-induced effects discussed in [24, 25] are sensitive to the
specific UV completion responsible for the mass generation of $X$ (see e.g.
[27]).

The paper is organised as follows. In Sect. 2, we will present the general
derivation of the $\Delta S=1$ chiral Lagrangian, as well as the related
Feynman rules for spin-1 bosons up to the $\mathcal{O}(p^{4})$ order. In Sect.
3, we will compute the $K^{\pm}\to\pi^{\pm}X$ decay rate in $\chi$PT
exploiting the Feynman rules derived in Sect. 2, comparing our tree-level
effects with the results obtained at one-loop level in [24, 25]. In Sect. 4,
we will discuss our flavour bounds vs. beam-dump and collider searches as
reported in [21, 22]. Sect. 5 is dedicated to our conclusions, while more
technical details about the construction of the chiral Lagrangian are deferred
to App. A.

##  2 $\Delta S=1$ chiral Lagrangian for spin-1 bosons

The most general Lagrangian describing the lowest-order interactions of a new
spin-1 particle $X$ with SM fermions includes both vectorial and axial
couplings and, focusing on the interactions with the lightest quark flavours
$q=(u,d,s)^{T}$, it can be written as

$\mathcal{L}^{\rm
int}_{X}=g_{x}X_{\mu}\,\overline{q}\gamma^{\mu}(x_{V}+x_{A}\gamma_{5})q\,,$
(2.1)

where $g_{x}$ measures the strength of the universal coupling of $X$ to
quarks. The vectorial and axial charges, $x_{V,A}$, are defined in flavour
space and include off-diagonal entries in the 2-3 sector.

###  2.1 Lowest-order chiral Lagrangian

At energies above few GeV, the Lagrangian of Eq. (2.1) can be directly
employed to analyse the interactions of $X_{\mu}$ with quarks. Here, instead
we focus on the low-energy range below the GeV scale, where we can resort to
$\chi$PT – see e.g. [28, 29]. In order to construct our $\chi$PT in the
presence of $X_{\mu}$, we proceed as follows. Let us consider the massless QCD
Lagrangian with chiral symmetry group $G=SU(3)_{L}\times SU(3)_{R}$

$\mathcal{L}_{\text{QCD}}^{0}=-\frac{1}{4}G^{a}_{\mu\nu}G_{a}^{\mu\nu}+i\overline{q}_{L}\gamma^{\mu}\left(\partial_{\mu}+ig_{s}\frac{\lambda_{a}}{2}A^{a}_{\mu}\right)q_{L}+i\overline{q}_{R}\gamma^{\mu}\left(\partial_{\mu}+ig_{s}\frac{\lambda_{a}}{2}A^{a}_{\mu}\right)q_{R}\,,$
(2.2)

where $q=(u,d,s)^{T}$ and $\lambda_{a}$ are the Gell-Mann matrices.

Chiral symmetry-breaking terms (like mass terms or interactions with external
gauge fields other than gluons) can be implemented by introducing appropriate
spurions ($a_{\mu},\,v_{\mu},\,s,\,p$) as external source fields [28].
Therefore, the resulting Lagrangian $\mathcal{L}_{\text{QCD}}^{\text{ext}}$
reads

$\begin{split}\mathcal{L}_{\text{QCD}}^{\text{ext}}&=\mathcal{L}_{\text{QCD}}^{0}+\overline{q}\gamma^{\mu}(v_{\mu}+a_{\mu}\gamma_{5})q+\overline{q}(s-ip\gamma_{5})q\\\
&=\mathcal{L}_{\text{QCD}}^{0}+\overline{q}\gamma^{\mu}(2r_{\mu}P_{R}+2\ell_{\mu}P_{L})q+\overline{q}(s-ip\gamma_{5})q\,.\end{split}$
(2.3)

where $2r_{\mu}=v_{\mu}+a_{\mu}$ and $2\ell_{\mu}=v_{\mu}-a_{\mu}$. Its chiral
counterpart is then found to be

$\mathcal{L}_{\chi\text{PT}}^{\text{ext}}=\frac{f_{\pi}^{2}}{4}\mbox{Tr}\,{\left[D_{\mu}U^{\dagger}D^{\mu}U+U^{\dagger}\chi+\chi^{\dagger}U\right]}+\mathcal{O}(p^{4})$
(2.4)

where $U(x)=\exp\left[i\lambda_{a}\pi_{a}(x)/f_{\pi}\right]$ (with
$f_{\pi}\simeq 92$ MeV) is the mesonic matrix transforming as $U(x)\rightarrow
LU(x)R^{\dagger}$ under $SU(3)_{L}\times SU(3)_{R}$ and $\pi_{a}(x)$ are the
Goldstone boson fields of $SU(3)_{L}\times SU(3)_{R}\to SU(3)_{V}$ spontaneous
breaking. Moreover, we have defined

$D_{\mu}U=\partial_{\mu}U-ir_{\mu}U+iU\ell_{\mu}\qquad\text{and}\qquad\chi=2B_{0}\,(s+ip)\,.$
(2.5)

In our model, described by the Lagrangian of Eq. (2.1), the covariant
derivative $D_{\mu}U$ reads

$D_{\mu}U=\partial_{\mu}U-ig_{x}X_{\mu}(Q^{x}_{R}U-UQ^{x}_{L})\,,$ (2.6)

where $Q^{x}_{R/L}=Q^{x}_{V}\pm Q^{x}_{A}$, while
$Q^{x}_{V}=\mbox{diag}\,{(x_{V}^{u},\,x_{V}^{d},\,x_{V}^{s})}$ and
$Q^{x}_{A}=\mbox{diag}\,{(x_{A}^{u},\,x_{A}^{d},\,x_{A}^{s})}$.

Expanding the Lagrangian in (2.4) and keeping only the lowest order terms in
the NP coupling, we find

$\displaystyle\mathcal{L}_{\chi\text{PT}}^{\text{ext}}\supset$ $\displaystyle-
ig_{x}X_{\mu}(x_{V}^{u}-x_{V}^{s})\left(\partial^{\mu}K^{-}K^{+}-\partial^{\mu}K^{+}K^{-}\right)-iX_{\mu}g_{x}(x_{V}^{u}-x_{V}^{d})\left(\partial^{\mu}\pi^{-}\pi^{+}-\partial^{\mu}\pi^{+}\pi^{-}\right)$
$\displaystyle+\left[-ig_{x}X_{\mu}x_{V}^{32}\,\left(\partial^{\mu}K^{+}\pi^{-}-\partial^{\mu}\pi^{-}K^{+}\right)+\text{h.c.}\right]\,,$
(2.7)

with the corresponding Feynman rules given in Fig. 1 (all momenta flow from
left to right).

Note that all couplings in Eq. (2.1) are of vector type. This is due to the
fact that the matrix element of the axial-vector quark operators in Eq. (2.1)
vanishes between external pseudo-scalar meson states. Moreover, in the limit
of universal vector couplings, i.e. $x_{V}^{u}=x_{V}^{d}=x_{V}^{s}$, the
$K^{+}K^{-}X$ and $\pi^{+}\pi^{-}X$ interaction terms vanish as well, as a
result of the underlying $SU(3)_{V}$ chiral symmetry, while the
$K^{+}\pi^{-}X$ vector coupling still survives as the flavour-changing current
is not conserved.

$=ig_{x}(p_{1}+p_{2})_{\mu}(x_{V}^{u}-x_{V}^{s})$$K^{+}(p_{1})$$K^{+}(p_{2})$$X_{\mu}$
$=ig_{x}(p_{1}+p_{2})_{\mu}(x_{V}^{u}-x_{V}^{d})$$\pi^{+}(p_{1})$$\pi^{+}(p_{2})$$X_{\mu}$
$=-ig_{x}x_{V}^{32}(p_{1}^{\mu}+p_{2}^{\mu})$$K^{+}(p_{1})$$\pi^{+}(p_{2})$$X^{\mu}$
Figure 1: Feynman rules for the lowest-order chiral Lagrangian

Moreover, tree-level contributions to $\Delta S=1$ processes, such as
$K^{\pm}\to\pi^{\pm}X$, are generated only if the couplings $x_{V}$ are flavor
off-diagonal. Yet, even for flavour-diagonal couplings, irreducible flavour-
violating effects to $x_{V}$ are loop-induced by the exchange of the $W$ boson
and up-quarks (see e.g. [25]). In the following, we will show that weak
interactions provide additional sources of flavour-violation to the $\Delta
S=1$ chiral Lagrangian, already at tree level, when we include higher-order
terms in the momentum expansion corresponding to four-quark operators.

###  2.2 Chiral Lagrangian for weak interactions

In the SM, at energies above the scale of chiral symmetry breaking, $\Delta
S=1$ transitions are induced by the effective four-fermion Lagrangian [26]

$\mathcal{L}^{\Delta S=1}_{\rm
SM}=G\sum_{i=1}^{10}C_{i}(\mu)O_{i}(\mu)\qquad\text{with }\qquad
G\equiv-\frac{G_{F}}{\sqrt{2}}V_{ud}V^{*}_{us}\,,$ (2.8)

where

$\begin{array}[]{ll}Q_{1}=4(\overline{s}_{L}\gamma_{\mu}d_{L})(\overline{u}_{L}\gamma_{\mu}u_{L}),&\qquad\qquad
Q_{2}=4(\overline{s}_{L}\gamma_{\mu}u_{L})(\overline{u}_{L}\gamma_{\mu}d_{L}),\\\
Q_{3}=4(\overline{s}_{L}\gamma_{\mu}d_{L})(\overline{q}_{L}\gamma_{\mu}q_{L}),&\qquad\qquad
Q_{4}=4(\overline{s}^{\alpha}_{L}\gamma_{\mu}d^{\beta}_{L})(\overline{q}^{\beta}_{L}\gamma_{\mu}q^{\alpha}_{L}),\\\
Q_{5}=4(\overline{s}_{L}\gamma_{\mu}d_{L})\sum_{q}(\overline{q}_{R}\gamma_{\mu}q_{R}),&\qquad\qquad
Q_{6}=4(\overline{s}^{\alpha}_{L}\gamma_{\mu}d^{\beta}_{L})\sum_{q}(\overline{q}^{\beta}_{R}\gamma_{\mu}q^{\alpha}_{R}),\\\
Q_{7}=6(\overline{s}_{L}\gamma_{\mu}d_{L})\sum_{q}e_{q}(\overline{q}_{R}\gamma_{\mu}q_{R}),&\qquad\qquad
Q_{8}=6(\overline{s}^{\alpha}_{L}\gamma_{\mu}d^{\beta}_{L})\sum_{q}e_{q}(\overline{q}^{\beta}_{R}\gamma_{\mu}q^{\alpha}_{R}),\\\
Q_{9}=6(\overline{s}_{L}\gamma_{\mu}d_{L})\sum_{q}e_{q}(\overline{q}_{L}\gamma_{\mu}q_{L}),&\qquad\qquad
Q_{10}=6(\overline{s}^{\alpha}_{L}\gamma_{\mu}d^{\beta}_{L})\sum_{q}e_{q}(\overline{q}^{\beta}_{L}\gamma_{\mu}q^{\alpha}_{L}),\\\
\end{array}$ (2.9)

$q={u,d,s}$, $e_{u}=2/3$ and $e_{d}=e_{s}=-1/3$; $\alpha$ and $\beta$ are
colour indices which, if unspecified, are assumed to be contracted between the
two quarks in the same current.

The construction of the chiral counterpart to Eq. (2.9) proceeds in two steps:

  * •

In the first step, one constructs the chiral structures describing the product
of two fermionic currents. These structures must possess the same chiral
transformation properties of the corresponding quark currents and are obtained
by exploiting the quark-hadron duality between the Lagrangians of Eqs. (2.3)
and (2.4). At low energies, one has

$\int\mathcal{D}q\,\mathcal{D}\overline{q}\,\mathcal{D}G_{\mu}\,\exp\left(i\int
d^{4}x\,\mathcal{L}_{\text{QCD}}^{\text{ext}}\right)=\int\mathcal{D}U\exp\left(i\int
d^{4}x\,\mathcal{L}_{\chi\text{PT}}^{\text{ext}}\right)\,,$ (2.10)

and taking the functional derivatives of the QCD and the $\chi$PT actions with
respect to the external sources one can readily find the chiral counterparts
to the various Dirac structures. For instance, up to order
$\mathcal{O}(p^{2})$ one finds

$\begin{split}&\overline{q}^{i}_{L}\gamma^{\mu}q^{j}_{L}=\frac{\delta
S_{\text{QCD}}}{\delta(\ell_{\mu})_{ij}}\equiv\frac{\delta
S_{\chi\text{PT}}}{\delta(\ell_{\mu})_{ij}}=\frac{i}{2}f_{\pi}^{2}(D^{\mu}U^{\dagger}U)_{ji}=-\frac{1}{2}f_{\pi}^{2}(L^{\mu})_{ji}\,,\\\
&\overline{q}^{i}_{R}\gamma^{\mu}q^{j}_{R}=\frac{\delta
S_{\text{QCD}}}{\delta(r_{\mu})_{ij}}\equiv\frac{\delta
S_{\chi\text{PT}}}{\delta(\ell_{\mu})_{ij}}=\frac{i}{2}f_{\pi}^{2}(D^{\mu}UU^{\dagger})_{ji}=-\frac{1}{2}f_{\pi}^{2}(R^{\mu})_{ji}\,,\\\
&\overline{q}^{i}_{L}q_{R}^{j}=-\frac{\delta
S_{\text{QCD}}}{\delta(s-ip)_{ij}}\equiv-\frac{\delta
S_{\chi\text{PT}}}{\delta(s-ip)_{ij}}=-\frac{B_{0}}{2}f_{\pi}^{2}U_{ji}\,,\\\
&\overline{q}^{i}_{R}q_{L}^{j}=-\frac{\delta
S_{\text{QCD}}}{\delta(s+ip)_{ij}}\equiv-\frac{\delta
S_{\chi\text{PT}}}{\delta(s+ip)_{ij}}=-\frac{B_{0}}{2}f_{\pi}^{2}(U^{\dagger})_{ji}\,,\end{split}$
(2.11)

where in the previous expressions we have defined the chiral currents
$L_{\mu}$ and $R_{\mu}$

$L_{\mu}\equiv iU^{\dagger}D_{\mu}U=-iD_{\mu}U^{\dagger}U\,,\qquad
R_{\mu}\equiv iUD_{\mu}U^{\dagger}=-iD_{\mu}UU^{\dagger}\,.$ (2.12)

  * •

In the second step, one decomposes the product of quark currents into
irreducible representations of the flavour algebra by defining appropriate
projectors. These are to be applied as well to the chiral realisation of the
quark currents in order to obtain the desired operators in the chiral
Lagrangian, classified according to the irreducible representation of the
flavour algebra they belong to (see e.g. [30, 31, 32]). Further details are
discussed in App. A.

After carrying out the program outlined above, we finally reproduce the
$\Delta S=1$ chiral Lagrangian of Ref. [30], which takes the following simple
form

$\begin{split}\mathcal{L}^{\Delta
S=1}_{\text{eff}}=Gf_{\pi}^{4}&\left\\{g_{27}\left(L^{3}_{\mu,\,2}L^{\mu,\,1}_{1}+\frac{2}{3}L^{1}_{\mu,\,2}L^{\mu,\,3}_{1}-\frac{1}{3}L^{3}_{\mu,\,2}\mbox{Tr}\,{\left[L^{\mu}\right]})\right)+g_{8}^{S}\,L^{3}_{\mu,\,2}\mbox{Tr}\,{\left[L^{\mu}\right]}\right.\\\
&+\left.g_{8}\left(\mbox{Tr}\,{\left[\lambda
L_{\mu}L^{\mu}\right]}+e^{2}g_{\text{ew}}f_{\pi}^{2}\mbox{Tr}\,\left[\lambda
U^{\dagger}QU\right]\right)\right\\}\,,\end{split}$ (2.13)

where $\lambda\equiv\frac{1}{2}(\lambda_{6}-i\lambda_{7})$ is responsible for
the $s\rightarrow d$ flavour transition and we have specialised
$Q=\frac{1}{3}\text{diag}(2,-1,-1)$ to be the charge matrix for quarks. Out of
the pieces making up Eq. (2.13), the first one transforms in the
$(27_{L},1_{R})$ representation of the flavour group, while the second and the
third ones transform in the $(8_{L},1_{R})$ and $(8_{L},8_{R})$
representation, respectively. Clearly, no singlet term can have any effect on
$\Delta S=1$ transitions. The coefficients $g_{27}$, $g_{8}$, $g_{8}^{S}$ and
$g_{\text{ew}}$ are functions of non-perturbative effective parameters, as
well as of the Wilson coefficients of the weak operators, see Eq. (2.8). Their
values are found to be

$\displaystyle g_{8}=3.07\pm
0.14\qquad\,\,\,\,\,\text{\cite[cite]{[\@@bibref{Number}{Pich:2021yll}{}{}]}\,,}$
(2.14) $\displaystyle g_{8}^{S}=-1.17\pm
0.37\qquad\text{\cite[cite]{[\@@bibref{Number}{Gerard:2005yk}{}{}]}\,,}$
(2.15) $\displaystyle g_{27}=0.29\pm
0.02\qquad\,\,\,\text{\cite[cite]{[\@@bibref{Number}{Pich:2021yll}{}{}]}\,,}$
(2.16) $\displaystyle g_{\text{ew}}=-1.0\pm
0.3\qquad\,\,\,\text{\cite[cite]{[\@@bibref{Number}{Ecker:2000zr}{}{}]}\,.}$
(2.17)

Expanding the Lagrangian of Eq. (2.13) and keeping only the contributions
relevant for our analysis, we find

$\begin{split}\mathcal{L}^{\Delta
S=1}_{\text{eff}}\supset&\,\,\frac{2}{3}f^{2}g_{27}G\left(2\partial^{\mu}K^{+}\partial_{\mu}\pi^{-}+g_{x}X_{\mu}\left[i\partial^{\mu}K^{+}\pi^{-}(4x_{A}^{u}-x_{A}^{d}-3x_{A}^{s}+2x_{V}^{u}-2x_{V}^{d})\right.\right.\\\
&\left.\left.-i\partial^{\mu}\pi^{-}K^{+}(4x_{A}^{u}-3x_{A}^{d}-x_{A}^{s}+2x_{V}^{u}-2x_{V}^{d})+\text{h.c.}\right]\right)\\\
&+2f^{2}g_{8}^{S}Gg_{x}\,(x_{A}^{u}+x_{A}^{d}+x_{A}^{s})\,X_{\mu}\left[i\,\left(\partial^{\mu}K^{+}\pi^{-}-\partial^{\mu}\pi^{-}K^{+}\right)+\text{h.c.}\right]\\\
&+2f^{2}g_{8}G\left(\partial^{\mu}K^{+}\partial_{\mu}\pi^{-}+g_{x}X_{\mu}\left[i\partial^{\mu}K^{+}\pi^{-}(x_{A}^{u}+x_{A}^{s}+x_{V}^{u}-x_{V}^{d})\right.\right.\\\
&\left.\left.-i\partial^{\mu}\pi^{-}K^{+}(x_{A}^{u}+x_{A}^{d}+x_{V}^{u}-x_{V}^{s})+\text{h.c.}\right]\right)+2f^{4}Ge^{2}g_{8}g_{\text{ew}}K^{+}\pi^{-}\,,\end{split}$
(2.18)

which includes both a $K\pi$ mixing term as well as a flavour-violating
$K^{\pm}\to\pi^{\pm}X$ interaction, as depicted in the Feynman rules of Fig.
2.

$=i2f^{2}G\left[g_{8}(f^{2}e^{2}g_{\text{ew}}+p^{2})+\frac{2}{3}g_{27}p^{2}\right]$$K^{+}(p)$$\pi^{+}(p)$
$=ig_{x}2f^{2}G\left[g_{8}(p^{\mu}_{1}(x_{A}^{u}+x_{A}^{s}+x_{V}^{u}-x_{V}^{d})+p^{\mu}_{2}(x_{A}^{u}+x_{A}^{d}+x_{V}^{u}-x_{V}^{s}))\right.$$\left.+g^{S}_{8}(p^{\mu}_{1}+p^{\mu}_{2})(x_{A}^{u}+x_{A}^{d}+x_{A}^{s})+\frac{g_{27}}{3}(p^{\mu}_{1}(4x_{A}^{u}-3x_{A}^{d}-x_{A}^{s}+2x_{V}^{u}\right.$$\left.-2x_{V}^{d})+p^{\mu}_{2}(4x_{A}^{u}-x_{A}^{d}-3x_{A}^{s}+2x_{V}^{u}-2x_{V}^{s}))\right]$$K^{+}(p_{1})$$\pi^{+}(p_{2})$$X^{\mu}$
Figure 2: Feynman rules for the chiral Lagrangian of weak interactions.

Note that, if Eq. (2.1) contains an explicit source of flavour violation, the
latter Feynman rule has to be supplemented by the last contribution displayed
in Fig. 1.

Differently from the leading-order chiral Lagrangian of Eq. (2.1), we are now
sensitive both to $x^{f}_{V}$ and $x^{f}_{A}$ couplings. Indeed, the hadronic
matrix element $\langle K|O_{i}|\pi\rangle$, where $O_{i}$ are the weak
operators defined in Eq. (2.9), receives contributions from both vector and
axial-vector currents, as there are not symmetry arguments to forbid them.
Again, for universal vector couplings, the $K^{+}\pi^{-}X$ interaction
vanishes because of the underlying $SU(3)_{V}$ chiral symmetry.

##  3 $K^{\pm}\to\pi^{\pm}X$ in $\chi$PT

In this section we will compute the decay rate of the process
$K^{\pm}\to\pi^{\pm}X$ in $\chi$PT. Exploiting the Feynman rules derived in
the previous section, we will first analyse the tree-level contributions and
then we will compare them with the results obtained at one-loop level in [24].

###  3.1 Tree-level contribution

At the tree level, the process $K^{\pm}\rightarrow\pi^{\pm}X$ is generated by
the diagrams in Fig. 3.

$K^{+}$$X$$\pi^{+}$$K^{+}$$\pi^{+}$$X$$\pi^{+}$

$K^{+}$$K^{+}$$X$$\pi^{+}$

Figure 3: Diagrams generating the tree-level transition $K^{\pm}\to\pi^{\pm}X$
in $\chi$PT.

The $X$ boson can be emitted either at the same vertex where the flavour
transition takes place (first diagram) or at a different one. In the latter
case (second and third diagrams) weak interactions prompt a flavour transition
while the leg emission of an $X$ boson occurs at a different interaction
point.

The total amplitude
$\mathcal{M}=\mathcal{M}_{8}+\mathcal{M}_{8}^{S}+\mathcal{M}_{27}+\mathcal{M}_{\rm{ew}}$
receives four independent contributions proportional to
$g_{8},\,g_{8}^{S},\,g_{27}$ and $g_{\rm{ew}}$ which are given by

$\displaystyle\mathcal{M}_{8}$
$\displaystyle=2f_{\pi}^{2}g_{8}Gg_{x}\varepsilon_{\mu}^{*}(q)\bigg{[}p^{\mu}_{1}\bigg{(}x_{A}^{u}+x_{A}^{s}+\frac{m_{\pi}^{2}}{m_{K}^{2}-m_{\pi}^{2}}(x_{V}^{d}-x_{V}^{s})\bigg{)}$
$\displaystyle\qquad\qquad\qquad\qquad~{}+p^{\mu}_{2}\bigg{(}x_{A}^{u}+x_{A}^{d}+\frac{m_{K}^{2}}{m_{K}^{2}-m_{\pi}^{2}}(x_{V}^{d}-x_{V}^{s})\bigg{)}\bigg{]}\,,$
(3.1) $\displaystyle\mathcal{M}^{S}_{8}$
$\displaystyle=2f_{\pi}^{2}g_{8}^{S}Gg_{x}\varepsilon_{\mu}^{*}(q)(p_{1}+p_{2})^{\mu}(x_{A}^{u}+x_{A}^{d}+x_{A}^{s})\,,$
(3.2) $\displaystyle\mathcal{M}_{27}$
$\displaystyle=\frac{2f^{2}_{\pi}g_{27}Gg_{x}}{3}\,\varepsilon_{\mu}^{*}(q)\bigg{[}p^{\mu}_{1}\bigg{(}4x_{A}^{u}-3x_{A}^{d}-x_{A}^{s}+2\frac{m_{\pi}^{2}}{m_{K}^{2}-m_{\pi}^{2}}(x_{V}^{d}-x_{V}^{s})\bigg{)}$
$\displaystyle\qquad\qquad\qquad+p^{\mu}_{2}\bigg{(}4x_{A}^{u}-x_{A}^{d}-3x_{A}^{s}+2\frac{m_{K}^{2}}{m_{K}^{2}-m_{\pi}^{2}}(x_{V}^{d}-x_{V}^{s})\bigg{)}\bigg{]}\,,$
(3.3) $\displaystyle\mathcal{M}_{\text{ew}}$
$\displaystyle=-2f_{\pi}^{2}e^{2}g_{8}g_{\text{ew}}Gg_{x}(p_{1}+p_{2})^{\mu}\varepsilon^{*}_{\mu}(q)\frac{f_{\pi}^{2}}{m_{K}^{2}-m_{\pi}^{2}}(x_{V}^{s}-x_{V}^{d})\,.$
(3.4)

On the other hand, the decay rate can be written as

$\displaystyle\begin{split}\Gamma=\frac{1}{2m_{K}}\frac{\left|\overline{\mathcal{M}}\right|^{2}}{8\pi}\left[1-2\left(\frac{m_{X}^{2}+m_{\pi}^{2}}{m_{K}^{2}}\right)+\left(\frac{m_{X}^{2}-m_{\pi}^{2}}{m_{K}^{2}}\right)^{2}\right]^{1/2}\,,\end{split}$
(3.5)

where the total unpolarised amplitude squared is given by

$\begin{split}|\overline{\mathcal{M}}|^{2}&=\frac{g_{x}^{2}}{m_{X}^{2}\left(m_{K}^{2}-m_{\pi}^{2}\right)^{2}}[(m_{K}-m_{\pi})^{2}-m_{X}^{2}][(m_{K}+m_{\pi})^{2}-m_{X}^{2}]\bigg{[}x_{V}^{32}(m_{K}^{2}-m_{\pi}^{2})\\\
&-2g_{\text{ew}}e^{2}f^{4}Gg_{8}(x^{s}_{V}-x^{d}_{V})-f^{2}G\bigg{(}g_{8}^{S}(m_{K}^{2}-m_{\pi}^{2})(x_{A}^{u}+x_{A}^{d}+x_{A}^{s})\\\
&+\frac{2}{3}g_{27}m_{K}^{2}(x^{s}_{V}-x^{d}_{V}+2x^{d}_{A}+2x^{s}_{A}-4x^{u}_{A})+g_{8}m_{\pi}^{2}(x^{s}_{V}-x^{d}_{V}+x^{d}_{A}+x^{s}_{A}+2x^{u}_{A})\\\
&-\frac{2}{3}g_{27}m_{\pi}^{2}(x^{s}_{V}-x^{d}_{V}-2x^{d}_{A}-2x^{s}_{A}+4x^{u}_{A})-g_{8}m_{K}^{2}(x^{s}_{V}-x^{d}_{V}-x^{d}_{A}-x^{s}_{A}-2x^{u}_{A})\bigg{)}\bigg{]}^{2}\,.\end{split}$
(3.6)

Assuming generation universality of the couplings, i.e.
$x_{V,A}^{u}=x_{V,A}^{d}=x_{V,A}^{s}$, and taking the limit $m_{K}\gg
m_{X},m_{\pi}$, one can find the simple expression

$\displaystyle\begin{split}\Gamma\approx\frac{m_{K}}{2\pi}\left(\frac{m_{K}}{m_{X}}\right)^{2}G_{F}^{2}f_{\pi}^{4}\,|V_{us}|^{2}\,g_{x}^{2}\,(x_{A}^{u})^{2}\left(g_{8}+\frac{3}{4}g^{S}_{8}\right)^{2}\,.\end{split}$
(3.7)

A few important comments are in order:

  * •

In the limit of universal vector couplings, the decay rate of
$K^{\pm}\to\pi^{\pm}X$ becomes independent of these couplings as a result of
the underlying $SU(3)_{V}$ chiral symmetry.

  * •

The enhancement factor $(m_{K}/m_{X})^{2}$ in Eq. (3.7) for small $m_{X}$ is
conceptually similar to the enhancement obtained in [24, 25]. This enhancement
here is produced by the longitudinal component of the polarization vector:
$\sum\varepsilon_{\mu}^{*}(q)\varepsilon_{\nu}(q)=-\eta_{\mu\nu}+\frac{q_{\mu}q_{\nu}}{m_{X}^{2}}$.

In order to see where we stand, we write the branching ratio of
$K^{\pm}\to\pi^{\pm}X$ as

$\displaystyle\mathcal{B}(K^{+}\to\pi^{+}X)\approx\frac{\Gamma(K^{+}\to\pi^{+}X)}{\Gamma(K^{+}\to\mu^{+}\nu)}\times\mathcal{B}(K^{+}\to\mu^{+}\nu)\,,$
(3.8)

where $\Gamma(K\to\mu\nu)\approx
m_{K}m^{2}_{\mu}|V_{us}|^{2}f^{2}_{K}G^{2}_{F}/4\pi$ and
${\mathcal{B}}(K\to\mu\nu)\approx 64\%$. Moreover, we assume the equality
$f_{K}=f_{\pi}$ which holds in the $SU(3)$ chiral limit. Finally, exploiting
the E949 measurement
$\mathcal{B}(K^{+}\to\pi^{+}\nu\nu)=(1.73^{+1.15}_{-1.05})\times 10^{-10}$
[35], we obtain the $2\sigma$ level constraint
$\text{BR}(K^{+}\to\pi^{+}X)\lesssim 4\times 10^{-10}$. As a result, we find
the following bound

$\displaystyle g_{x}\,x_{A}^{u}\lesssim 3\times
10^{-6}\left(\frac{m_{X}}{0.1\,{\rm{GeV}}}\right)\,,$ (3.9)

where the charges $x_{A}^{u}$ and $x_{A}^{d}$ are typically expected to be of
order one. The above result will be fully confirmed by the numerical analysis
of Sect. 4.

###  3.2 One-loop contribution

In this section, we will calculate the one-loop contributions to the flavour-
violating process $K^{\pm}\to\pi^{\pm}X$. At the quark level, the Feynman
diagrams generating the underlying $s\to d$ transition are displayed in Fig.
4.

$s$$d$$u_{i}$$u_{i}$$X$$W$$s$$d$$s$$u_{i}$$X$$W$$s$$d$$u_{i}$$d$$X$$W$

Figure 4: Feynman diagrams contributing to the process $K^{\pm}\to\pi^{\pm}X$
at the one-loop level.

Notice that these diagrams are sensitive to different couplings of the $X$
boson to quarks. Summing up all contributions, the full amplitude reads

$\displaystyle\mathcal{M}$
$\displaystyle=g_{x}x^{\rm{eff}}_{sd}\epsilon^{*}_{\mu}\overline{s}\gamma^{\mu}(1-\gamma_{5})d=\frac{g^{2}}{128\pi^{2}}\,g_{x}\,V_{id}V^{*}_{is}\,x_{i}\,\epsilon^{*}_{\mu}\bigg{[}x_{R}^{u_{i}}\left(\frac{2}{\epsilon}+\log{\frac{\mu^{2}}{m_{i}^{2}}}-\frac{1}{2}-3\frac{(1-x_{i}+\log{x_{i}})}{(x_{i}-1)^{2}}\right)$
$\displaystyle+\frac{(x_{L}^{d}+x_{L}^{s})}{2}\left(\frac{3-3x_{i}^{2}+2(4x_{i}-1)\log{x_{i}}}{2(x_{i}-1)^{2}}-\frac{2}{\epsilon}-\log\frac{\mu^{2}}{m_{i}^{2}}\right)+x_{L}^{u_{i}}\frac{(-1+x_{i}-4)}{(x_{i}-1)}\bigg{]}$
$\displaystyle\times\overline{s}\gamma^{\mu}(1-\gamma_{5})d\,,$ (3.10)

where $x_{i}=m_{i}^{2}/m_{W}^{2}$ with $m_{i}$ being the mass of the up-type
quark running in the loop and we have defined the chiral charges as
$x_{L/R}^{f}=x_{V}^{f}\mp x_{A}^{f}$. The divergences, originating from the
non-renormalizability of our model, can be interpreted within a hard cutoff
regularization scheme as
$2/\epsilon+\log(\mu^{2}/m_{i}^{2})=\log(\Lambda^{2}/m_{i}^{2})$ where
$\Lambda$ is the UV cutoff. In specific renormalizable models, $\Lambda$ will
be identified with the mass scale of particles belonging to the NP sector
which will provide a UV completion of our model.

In the limit of universal couplings, i.e.
$x_{V,A}^{u_{i}}=x_{V,A}^{d}=x_{V,A}^{s}$, and keeping only the dominant loop
effects stemming from the exchange of the top quark, we obtain

$\displaystyle x^{\rm{eff}}_{sd}$
$\displaystyle\simeq\frac{g^{2}}{64\pi^{2}}\,V_{td}V^{*}_{ts}x_{A}^{u}f(x_{t})$
(3.11)

where

$\displaystyle
f(x_{t})=x_{t}\left[\frac{2}{\epsilon}+\log{\frac{\mu^{2}}{m_{t}^{2}}}-\frac{1}{2}-3\frac{(1-x_{t}+\log{x_{t}})}{(x_{t}-1)^{2}}\right]\,.$
(3.12)

The inclusion of the above loop effects in the decay rate of
$K^{\pm}\to\pi^{\pm}X$ can be implemented by the following replacement in Eq.
(3.6):

$x^{32}_{V}\to x^{32}_{V}-x^{\rm{eff}}_{sd}\,.$ (3.13)

As a result, we can estimate the relative size of loop effects and tree-level
ones as

$\displaystyle\frac{x^{\rm{eff}}_{sd}}{4g_{8}f_{\pi}^{2}Gx_{A}^{u}}\approx
f(x_{t})\,,$ (3.14)

where $f(x_{t})$ is a model-dependent loop function which depends on the
specific UV completion of our effective theory. Therefore, we have learned
that loop-effects have a similar size of tree-level contributions. However,
while the former suffer from sizeable uncertainties, the latter provide a
robust model-independent result. Moreover, we also remark that loop and tree-
level contributions generally depend on different couplings and therefore the
comparison in Eq. (3.14) is valid only in the universal scenario
$x_{V,A}^{u_{i}}=x_{V,A}^{d}=x_{V,A}^{s}$.

##  4 Flavour bounds vs. beam-dump and collider searches

We are ready now to exploit the results derived in the previous section, in
order to explore the capability of the process $K^{\pm}\to\pi^{\pm}X$ to
unveil new light vector bosons. We are going to use the DarkCast package [21,
22], which enables us to derive bounds on vector and axial couplings of models
entailing new spin-1 states by imposing current and future experimental
constraints on several processes. In Fig. 5, we show the bounds in the
$(m_{X},g_{x})$ plane arising from a variety of beam-dump and collider
searches [22] as well as from the flavour changing process
$K^{\pm}\to\pi^{\pm}X$ discussed in this paper. The three plots refer to the
benchmark models dubbed as axial, chiral and 2HDM [22] which differ for the
values of the $x_{V,A}$ charges, see Table 1.

| $x^{e}_{V}$ | $x^{\nu}_{V}$ | $x^{u,c,t}_{V}$ | $x^{d,s,b}_{V}$ | $x^{e}_{A}$ | $x^{\nu}_{A}$ | $x^{u,c,t}_{A}$ | $x^{d,s,b}_{A}$
---|---|---|---|---|---|---|---|---
Axial | 0 | 1/4 | 0 | 0 | $-1$ | $-1/4$ | 1 | $-1$
Chiral | $-1$ | 0 | 1 | 1 | $-1$ | 0 | 1 | $-1$
2HDM | 0.044 | 0.05 | 1.021 | 0.015 | $-0.1$ | 0.05 | $-0.95$ | $-0.1$
Table 1: Charges of the SM fermions under $X$ boson interactions for the
models considered in Ref. [22] where, for simplicity, flavor universal
couplings have been assumed.

The bounds from the process $K^{\pm}\to\pi^{\pm}X$ are obtained by employing
the tree-level prediction of Sect. 3.1 and exploiting the measurement of
$\text{BR}(K^{+}\to\pi^{+}\nu\nu)=(1.73^{+1.15}_{-1.05})\times 10^{-10}$ by
the E949 experiment at BNL [35]. In particular, we impose the $2\sigma$ bound
$\text{BR}(K^{+}\to\pi^{+}X)\lesssim 4\times 10^{-10}$. Remarkably, in all
scenarios of Fig. 5, the process $K^{\pm}\to\pi^{\pm}X$ sets the strongest to
date model-independent bound in the $(m_{X},g_{x})$ plane for
$m_{X}<m_{K}-m_{\pi}$.

Figure 5: The dark shaded area represents the tree-level
$K^{\pm}\to\pi^{\pm}X$ bound obtained in this work. Limits from beam-dump and
collider searches are obtained with DarkCast [22] and are shown for the
purpose of comparison for the three benchmark models given in Table 1.

##  5 Conclusions

Extensions of the SM entailing new feebly interacting massive particles with
sub-GeV masses are currently among the most studied scenarios of NP. In
particular, comprehensive analyses aiming to probe a light spin-1 boson $X$
featuring general couplings to quarks and leptons have been carried out in the
literature exploiting beam-dump and collider searches.

In this work, we revisited the flavour constraints to this scenario by means
of the rare decay $K^{\pm}\to\pi^{\pm}X$. In particular, we extended previous
studies by constructing the most general $\Delta S=1$ chiral Lagrangian as
induced by weak interactions up to the order $\mathcal{O}(p^{4})$.

The lowest-order $\mathcal{O}(p^{2})$ terms of our $\chi$PT capture the
effects to $K^{\pm}\to\pi^{\pm}X$ discussed in [25], which are loop-induced
and suppressed by the fifth power of the Cabibbo angle. Instead, the inclusion
of subleading $\mathcal{O}(p^{4})$ terms in the chiral expansion generates
$K^{\pm}\to\pi^{\pm}X$ already at the tree-level and the related amplitude is
only singly Cabibbo suppressed. As a result, rather surprisingly, the two
contributions turn out to be of comparable size, see Eq. (3.14).

However, while the tree-level weak effects discussed in this work are model-
independent, the loop-induced contributions of Ref. [25] are instead sensitive
to the specific UV completion accounting for the mass of the spin-1 boson.

In conclusion, we have shown that the process $K^{\pm}\to\pi^{\pm}X$ sets the
strongest to date model-independent bound on the diagonal axial-vector
couplings to $u,d,s$ quarks of a light $X$ with $m_{X}<m_{K}-m_{\pi}$,
superseding the bounds arising from beam-dump and collider searches.

## Acknowledgements

This work received funding from the European Union’s Horizon 2020 research and
innovation programme under the Marie Skłodowska-Curie grant agreement N∘
860881 and by the INFN Iniziative Specifica APINE. The work of LDL is also
supported by the project “CPV-Axion” under the Supporting TAlent in
ReSearch@University of Padova (STARS@UNIPD).

##  Appendix A Details of the chiral Lagrangian construction

In this appendix we will provide an extensive derivation of the chiral
Lagrangian describing $\Delta S=1$ transitions. The operators of Eq. (2.9)
have the form

$\mathcal{L}^{\rm EW,\,LL}_{\rm
eff}=[t_{L}]^{jl}_{ik}(\overline{q}_{L}^{i}\gamma^{\mu}q_{Lj})(\overline{q}_{L}^{k}\gamma_{\mu}q_{Ll})\,,$
(A.1)

and

$\mathcal{L}^{\rm EW,\,LR}_{\rm
eff}=[t^{\delta\delta}_{LR}]^{jl}_{ik}(\overline{q}_{L}^{i}\gamma^{\mu}q_{Lj})(\overline{q}_{R}^{k}\gamma_{\mu}q_{Rl})+[t^{\lambda\lambda}_{LR}]^{jl}_{ik}(\overline{q}_{L}^{i}\gamma^{\mu}T^{a}q_{Lj})(\overline{q}_{R}^{k}\gamma_{\mu}T^{a}q_{Rl})\,.$
(A.2)

Let us first discuss how to identify those combinations of four-quark
operators belonging to irreducible representations of the chiral group [30,
31, 32]. From the invariance under the flavour group $U(3)_{F}\equiv
U(3)_{L}\times U(3)_{R}$, we know that the $LL$ currents transform as the
81-dimensional representation of $(\overline{3}\otimes
3)\otimes(\overline{3}\otimes 3)$ of $U(3)_{L}$ which can be further
decomposed into irreducible symmetric or antisymmetric representations having
dimension $1,8,10,27$.

In particular, one has symmetric-symmetric combinations
$\mathcal{S}\mathcal{S}$ transforming in the $1\oplus 8\oplus 27$
representations and antisymmetric-antisymmetric combinations
$\mathcal{A}\mathcal{A}$ transforming as $1\oplus 8$ representations. We can
disregard the symmetric-antisymmetric and antisymmetric-symmetric combinations
transforming in the $8\oplus 10$ representations because they cannot be
generated by the operators in (A.1) which are symmetric under the simultaneous
exchange of upper indices with the lower ones, $(i,k)\leftrightarrow(j,l)$. In
any case, the particular combinations of quark currents transforming in each
one of these representations can be obtained by projecting the fundamental
structure in (A.1) on the orthonormal basis of irreducible representations of
$(\overline{3}\otimes 3)\otimes(\overline{3}\otimes 3)$:

$\\{(e^{a}_{27})^{\mathcal{S}}_{\mathcal{S}},\,(e^{a}_{8})^{\mathcal{S}}_{\mathcal{S}},\,(e_{1})^{\mathcal{S}}_{\mathcal{S}},\,(e^{a}_{10})^{\mathcal{A}}_{\mathcal{S}},\,(e^{a}_{8})^{\mathcal{A}}_{\mathcal{S}},\,(e^{a}_{10})^{\mathcal{S}}_{\mathcal{A}},\,(e^{a}_{8})^{\mathcal{S}}_{\mathcal{A}},\,(e^{a}_{8})^{\mathcal{A}}_{\mathcal{A}},\,(e_{1})^{\mathcal{A}}_{\mathcal{A}}\\}\,.$
(A.3)

This task is accomplished by making use of the tensorial product defined as

$\mathcal{T}_{1}\cdot\mathcal{T}_{2}\equiv(T_{1})^{ij}_{kl}(T_{2})^{kl}_{ij}\,,\qquad\qquad\mathcal{T}^{a\,M}_{r\,N}\equiv\mathcal{T}\cdot
e^{a\,M}_{r\,N}=T^{ij}_{kl}\,[e^{a\,M}_{r\,N}]^{kl}_{ij}\,,$ (A.4)

where we exploited the decomposition
$\mathcal{T}=\mathcal{T}^{a\,M}_{r\,N}e^{a\,M}_{r\,N}$.

Now, the fully symmetric singlet and octet basis elements are

$[(e_{1})_{\mathcal{S}}^{\mathcal{S}}]^{kl}_{ij}=\frac{1}{2\sqrt{6}}[\delta^{k}_{i}\delta_{j}^{l}+\delta^{l}_{i}\delta_{j}^{l}]\,,\qquad[(e^{a}_{8})_{\mathcal{S}}^{\mathcal{S}}]^{kl}_{ij}=\frac{1}{2\sqrt{10}}[(\lambda^{a})^{k}_{i}\delta^{l}_{j}+(\lambda^{a})^{l}_{i}\delta^{k}_{j}+(\lambda^{a})^{k}_{j}\delta^{l}_{i}+(\lambda^{a})^{l}_{j}\delta^{k}_{i}]\,,$
(A.5)

whereas the fully antisymmetric singlet and octet basis elements are

$[(e_{1})_{\mathcal{A}}^{\mathcal{A}}]^{kl}_{ij}=\frac{1}{2\sqrt{3}}[\delta^{k}_{i}\delta_{j}^{l}-\delta^{l}_{i}\delta_{j}^{l}]\,,\qquad[(e^{a}_{8})_{\mathcal{A}}^{\mathcal{A}}]^{kl}_{ij}=\frac{1}{2\sqrt{2}}[(\lambda^{a})^{k}_{i}\delta^{l}_{j}-(\lambda^{a})^{l}_{i}\delta^{k}_{j}-(\lambda^{a})^{k}_{j}\delta^{l}_{i}+(\lambda^{a})^{l}_{j}\delta^{k}_{i}]\,.$
(A.6)

The symmetric-symmetric 27-plet basis element is harder to construct and a
better strategy is to extract the corresponding component by subtracting from
a fully symmetric tensor its octet and singlet parts, namely

$(\mathcal{T}_{27})^{\mathcal{S}}_{\mathcal{S}}=\mathcal{T}^{\mathcal{S}}_{\mathcal{S}}-(\mathcal{T}_{8})^{\mathcal{S}}_{\mathcal{S}}-(\mathcal{T}_{1})^{\mathcal{S}}_{\mathcal{S}}=\mathcal{T}^{\mathcal{S}}_{\mathcal{S}}-[\mathcal{T}^{\mathcal{S}}_{\mathcal{S}}\cdot(e_{1})^{\mathcal{S}}_{\mathcal{S}}]\,(e_{1})^{\mathcal{S}}_{\mathcal{S}}-[\mathcal{T}^{\mathcal{S}}_{\mathcal{S}}\cdot(e^{a}_{8})^{\mathcal{S}}_{\mathcal{S}}]\,(e^{a}_{8})^{\mathcal{S}}_{\mathcal{S}}\,.$
(A.7)

The decomposition of the operators appearing in (A.1) can be easily performed
by projecting the quark currents onto the orthonormal basis elements. Their
chiral counterparts are then simply obtained by projecting the chiral
equivalent of quark currents in equation (2.11) onto the very same basis
elements.

LL currents: the fully symmetric and anti-symmetric octet Lagrangian reads

$\displaystyle\begin{split}\\!\\!\\!\\!\\!\\!\mathcal{L}_{8}^{\mathcal{S(A)}}=&~{}a_{8}^{\mathcal{S(A)}}\frac{f_{\pi}^{4}}{80}[t_{L}]^{jl}_{ik}\left(\mbox{Tr}\,{(\lambda^{a}L_{\mu})}\mbox{Tr}\,{L^{\mu}}\pm\mbox{Tr}\,{(\lambda^{a}L_{\mu}L^{\mu})}\right)\,\cdot\\\
&~{}[(\lambda^{a})^{k}_{i}\delta^{l}_{j}\pm(\lambda^{a})^{l}_{i}\delta^{k}_{j}\pm(\lambda^{a})^{k}_{j}\delta^{l}_{i}+(\lambda^{a})^{l}_{j}\delta^{k}_{i}]\end{split}$
(A.8)

In principle, one should consider also the structure
$\mbox{Tr}\,{(\lambda^{a}(U^{\dagger}\chi+\chi^{\dagger}U))}$ along with
$\mbox{Tr}\,{(\lambda^{a}L^{\mu}L_{\mu})}$. However, these additional
structures induce vacuum misalignment effects through Goldstone tadpoles and
can be rotated away by properly redefining the Goldstone fields [30].

The 27-plet Lagrangian term is finally given by

$\displaystyle\mathcal{L}_{27}$
$\displaystyle=a_{27}\frac{f_{\pi}^{4}}{8}\\{[t_{L}]^{jl}_{ik}\big{(}[(L_{\mu})^{i}_{j}(L^{\mu})^{k}_{l}+(L_{\mu})^{k}_{j}(L^{\mu})^{i}_{l}]-\frac{1}{12}\left(\mbox{Tr}\,{(L_{\mu}L^{\mu})}+\mbox{Tr}\,{L_{\mu}}\mbox{Tr}\,{L^{\mu}}\right)[\delta^{i}_{j}\delta^{k}_{l}+\delta^{i}_{l}\delta^{k}_{j}]$
$\displaystyle-\frac{1}{10}\left(\mbox{Tr}\,{(\lambda^{a}L_{\mu}L^{\mu})}+\mbox{Tr}\,{(\lambda^{a}L_{\mu})}\mbox{Tr}\,{L^{\mu}}\right)[(\lambda^{a})^{k}_{i}\delta^{l}_{j}+(\lambda^{a})^{l}_{i}\delta^{k}_{j}+(\lambda^{a})^{k}_{j}\delta^{l}_{i}+(\lambda^{a})^{l}_{j}\delta^{k}_{i}]\big{)}\\}\,.$
(A.9)

In the expressions (A.8) and (A), the parameters $a_{8}^{\mathcal{S}}$,
$a_{8}^{\mathcal{A}}$ and $a_{27}$ parametrize our ignorance about the
hadronization dynamics and are to be determined experimentally.

LR currents: Since left-handed and right-handed currents transform under
different $U(3)$ groups, these combinations correspond to the product of a
$\overline{3}\oplus 3=1\oplus 8$ representation in each chiral sector,
resulting in four possible different structures transforming as
$(1_{L},1_{R})$, $(8_{L},1_{R})$, $(1_{L},8_{R})$ and $(8_{L},8_{R})$. We
first identify the associated orthonormal basis

$\\!\\!\\!\\!\\!\\!\\!(e_{1_{L},1_{R}})^{kl}_{ij}\\!=\\!\frac{\delta^{j}_{i}\delta^{l}_{k}}{3},~{}~{}(e^{a}_{8_{L},1_{R}})^{kl}_{ij}\\!=\\!\frac{(\lambda^{a}_{L})^{j}_{i}\delta^{l}_{k}}{\sqrt{6}},~{}~{}(e^{a}_{1_{L},8_{R}})^{kl}_{ij}\\!=\\!\frac{\delta^{j}_{i}(\lambda^{b}_{R})^{l}_{k}}{\sqrt{6}},~{}~{}(e^{a,b}_{8_{L},8_{R}})^{kl}_{ij}\\!=\\!\frac{(\lambda^{a}_{L})^{j}_{i}(\lambda^{b}_{R})^{l}_{k}}{2}\,,$
(A.10)

which will be then used in order to project the appropriate structure onto the
low-energy operators possessing definite chiral transformation properties.
Exploiting the completeness relation

$T^{a}_{ij}T^{a}_{kl}=\frac{1}{2}\delta_{il}\delta_{kj}-\frac{1}{2N_{C}}\delta_{ij}\delta_{lk}\,,$
(A.11)

we can recast Eq. (A.2) in the following form

$\mathcal{L}^{\rm EW,\,LR}_{\rm
eff}=[t^{\delta\delta}_{LR}]^{jl}_{ik}(\overline{q}_{L}^{i}\gamma^{\mu}q_{Lj})(\overline{q}_{R}^{k}\gamma_{\mu}q_{Rl})-\frac{1}{2N_{C}}[t^{\lambda\lambda}_{LR}]^{jl}_{ik}(\overline{q}_{L}^{i}\gamma^{\mu}q_{Lj})(\overline{q}_{R}^{k}\gamma_{\mu}q_{Rl})\,.$
(A.12)

At this point the Fierz identity

$(\overline{q}_{L}^{i}\gamma^{\mu}q_{Lj})(\overline{q}_{R}^{k}\gamma_{\mu}q_{Rl})=-2(\overline{q}_{L}^{i}q_{Rl})(\overline{q}_{R}^{k}q_{Lj})$
(A.13)

can be used in order to identify the various operators.

The leading order chiral structure that is compatible with an $(8_{L},8_{R})$
structure reads

$\mathcal{L}_{8_{L},8_{R}}=\frac{f_{\pi}^{6}}{4}\big{(}a^{\delta\delta}_{88}[t_{LR}^{\delta\delta}]^{jl}_{ik}+a^{\lambda\lambda}_{88}[t_{LR}^{\lambda\lambda}]^{jl}_{ik}\big{)}(\lambda^{a}_{L})^{i}_{j}(\lambda^{b}_{R})^{k}_{l}\mbox{Tr}\,{(\lambda_{L}^{a}U^{\dagger}\lambda^{b}_{R}U)}+\mathcal{O}(p^{2})\,.$
(A.14)

Instead, the structures transforming as $(8_{L},1_{R})$ and $(1_{L},8_{R})$
are

$\mathcal{L}_{8}^{LR}=\frac{f_{\pi}^{4}}{6}\big{(}a^{\delta\delta}_{LR}[t_{LR}^{\delta\delta}]^{jl}_{ik}+a^{\lambda\lambda}_{LR}[t_{LR}^{\lambda\lambda}]^{jl}_{ik}\big{)}\\{\mbox{Tr}\,{(\lambda_{a}L_{\mu}L^{\mu})}(\lambda_{L}^{a})^{i}_{j}\delta^{k}_{l}+\mbox{Tr}\,{(\lambda_{a}R_{\mu}R^{\mu})}\delta^{i}_{j}(\lambda_{R}^{b})^{k}_{l}\\}\,.$
(A.15)

Also in this case the constants $a^{\delta\delta}_{88}$,
$a^{\lambda\lambda}_{88}$, $a^{\delta\delta}_{LR}$ and
$a^{\lambda\lambda}_{LR}$ parametrize our ignorance on the non-perturbative
dynamics related to the hadronization process.

Combining all above results, one obtains the $\Delta S=1$ chiral Lagrangian of
Eq. (2.13) where the coefficients $g_{27}$, $g_{8}$ and $g_{\text{ew}}$ are
functions of the non-perturbative effective parameters $a_{*}$ defined above,
as well as of the Wilson coefficients entering Eqs. (A.1) and (A.2).

As shown in [31, 32], the matching procedure requires to decompose the
operators of Eq. (2.8) into operators having well-defined chiral
transformation properties under the flavour group

$\begin{split}&Q_{1}=\frac{1}{10}Q_{S}^{(8,1),1/2}+\frac{1}{15}Q_{S}^{(27,1),1/2}+\frac{1}{3}Q_{S}^{(27,1),3/2}+\frac{1}{2}Q_{A}^{(8,1),1/2}\\\
&Q_{2}=\frac{1}{10}Q_{S}^{(8,1),1/2}+\frac{1}{15}Q_{S}^{(27,1),1/2}+\frac{1}{3}Q_{S}^{(27,1),3/2}-\frac{1}{2}Q_{A}^{(8,1),1/2}\\\
&Q_{3}=\frac{1}{2}Q_{S}^{(8,1),1/2}+\frac{1}{2}Q_{A}^{(8,1),1/2}\\\
&\tilde{Q}_{4}=\frac{1}{2}Q_{S}^{(8,1),1/2}-\frac{1}{2}Q_{A}^{(8,1),1/2}\\\
&Q_{9}=-\frac{1}{10}Q_{S}^{(8,1),1/2}+\frac{1}{10}Q_{S}^{(27,1),1/2}+\frac{1}{2}Q_{S}^{(27,1),3/2}+\frac{1}{2}Q_{A}^{(8,1),1/2}\\\
&\tilde{Q}_{10}=-\frac{1}{10}Q_{S}^{(8,1),1/2}+\frac{1}{10}Q_{S}^{(27,1),1/2}+\frac{1}{2}Q_{S}^{(27,1),3/2}+\frac{1}{2}Q_{A}^{(8,1),1/2}\,,\\\
\end{split}$ (A.16)

where $1/2$ and $3/2$ in the superscripts denote the operator isospin
properties, while $\tilde{Q}_{4}$ and $\tilde{Q}_{10}$ are the Fierzed
counterparts of $Q_{4}$ and $Q_{10}$:

$\tilde{Q}_{4}=4\sum_{q}(\overline{s}_{L}\gamma^{\mu}q_{L})(\overline{q}_{L}\gamma_{\mu}d_{L})\quad\text{and}\quad\tilde{Q}_{10}=6\sum_{q}e_{q}(\overline{s}_{L}\gamma^{\mu}q_{L})(\overline{q}_{L}\gamma_{\mu}d_{L})\,.$
(A.17)

Defining

$Q^{(27,1)}_{S}=\frac{1}{9}\big{[}Q_{S}^{(27,1),1/2}+5\,Q_{S}^{(27,1),3/2}\big{]}\,,$
(A.18)

one can then isolate in each operator the desired chiral structures
transforming in the irreducible representations of the flavour group [36].
Then, such structures can be directly translated into their $\chi$PT
counterparts.

As far as the $LR$ operators are concerned, we first need to recast the
operators in Eq. (2.9) in a form compatible with Eq. (A.2). This is achieved
by making use of the completeness relation

$\delta_{\beta\gamma}\delta_{\alpha\delta}=2T^{a}_{\alpha\gamma}T^{a}_{\beta\delta}+\frac{1}{N_{C}}\delta_{\alpha\gamma}\delta_{\beta\delta}\,,$
(A.19)

which allows us to rewrite

$\begin{split}(\overline{q}^{i}_{\alpha}\gamma^{\mu}\delta_{\beta\gamma}q^{j}_{\gamma})(\overline{q}^{k}_{\beta}\gamma_{\mu}\delta_{\alpha\delta}q^{l}_{\delta})&=2(\overline{q}^{i}_{\alpha}\gamma^{\mu}T^{a}_{\alpha\gamma}q^{j}_{\gamma})(\overline{q}^{k}_{\beta}\gamma_{\mu}T^{a}_{\beta\delta}q^{l}_{\delta})+\frac{1}{N_{C}}(\overline{q}^{i}_{\alpha}\gamma^{\mu}\delta_{\alpha\gamma}q^{j}_{\gamma})(\overline{q}^{k}_{\beta}\gamma_{\mu}\delta_{\beta\delta}q^{l}_{\delta})\\\
&\equiv
2Q^{\lambda\lambda}_{ijkl}+\frac{1}{N_{C}}Q^{\delta\delta}_{ijkl}\,.\end{split}$
(A.20)

The chiral counterparts of the operators on the right-hand side are well
known. Then, from

$C_{5(7)}Q_{5(7)}+C_{6(8)}Q_{6(8)}=\bigg{(}C_{5(7)}+\frac{C_{6(8)}}{N_{C}}\bigg{)}Q_{\delta\delta}+2C_{6(8)}Q_{\lambda\lambda}$
(A.21)

and

$Q_{5}=Q_{5}^{(8,1)}\qquad\text{and}\qquad
Q_{7}=\frac{1}{2}Q^{(8,8),3/2}_{S}+\frac{1}{2}Q^{(8,8),1/2}_{A}\,,$ (A.22)

we can proceed with the matching procedure obtaining the following results:

$\displaystyle g_{27}$ $\displaystyle=$
$\displaystyle\frac{3}{5}a_{27}(\mu)\big{(}C_{1}+C_{2}+\frac{3}{2}C_{9}+\frac{3}{2}C_{10}\big{)}(\mu)$
$\displaystyle g_{8}$ $\displaystyle=$
$\displaystyle\frac{1}{10}a_{8}^{\mathcal{S}}(\mu)\big{(}C_{1}+C_{2}+5C_{3}+5C_{4}-C_{9}-C_{10}\big{)}(\mu)+$
$\displaystyle-\frac{1}{2}a_{8}^{\mathcal{A}}(\mu)\big{(}C_{1}-C_{2}+C_{3}-C_{4}+C_{9}-C_{10}\big{)}(\mu)$
$\displaystyle+4\\{a_{LR}^{\delta\delta}(\mu)\big{(}C_{5}+\frac{C_{6}}{N_{C}}\big{)}(\mu)+2a_{LR}^{\lambda\lambda}(\mu)C_{6}(\mu)\\}$
$\displaystyle g^{S}_{8}$ $\displaystyle=$
$\displaystyle\frac{1}{10}a_{8}^{\mathcal{S}}(\mu)\big{(}C_{1}+C_{2}+5C_{3}+5C_{4}-C_{9}-C_{10}\big{)}(\mu)+$
$\displaystyle+\frac{1}{2}a_{8}^{\mathcal{A}}(\mu)\big{(}C_{1}-C_{2}+C_{3}-C_{4}+C_{9}-C_{10}\big{)}(\mu)$
$\displaystyle e^{2}g_{8}\,g_{\text{ew}}$ $\displaystyle=$ $\displaystyle
6\\{a_{88}^{\delta\delta}(\mu)\big{(}C_{7}+\frac{C_{8}}{N_{C}}\big{)}(\mu)+2a_{88}^{\lambda\lambda}(\mu)C_{8}(\mu)\\}\,.$

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