# Estimates for rare three-body decays of the Omega baryon using chiral
symmetry and the $\Delta I=1/2$ rule

Cornelis J. G. Mommers  Institutionen för fysik och astronomi, Uppsala
universitet, Box 516, S-75120 Uppsala, Sweden  Stefan Leupold
<EMAIL_ADDRESS>Institutionen för fysik och astronomi, Uppsala
universitet, Box 516, S-75120 Uppsala, Sweden

###### Abstract

We study rare three-body decays of the Omega baryon using SU(3) chiral
perturbation theory, the successful effective field theory of quantum
chromodynamics at low energies. At leading order, we calculate the branching
fractions of the decay $\Omega^{-}\to\Xi\pi\pi$ for all possible combinations
of pions. For one channel we find an order-of-magnitude discrepancy between
theory and experiment. This tension is known to exist in the non-relativistic
limit, and we confirm that it remains in the relativistic calculation. Fairly
independent of the values of the low-energy constants we establish lower
limits for the branching fractions of these three-body Omega decays, which
reaffirm the gap between theory and experiment. We point out that this
discrepancy is closely tied to the $\Delta I=1/2$ selection rule. In turn,
this means that the three-body decays constitute an interesting tool to
scrutinize the selection rule. Using next-to-leading order calculations we
also provide predictions for the decay
$\Omega^{-}\to\Xi^{0}\mu^{-}\bar{\nu}_{\mu}$. We show that fully-differential
distributions will provide access to low-energy constants needed in the axial-
vector transitions from a decuplet to octet baryon. Since data for all of
these rare three-body Omega decays are scarce (fully differential data are
nonexistent), we recommend that they be remeasured at running and upcoming
experiments, such as BESIII, LHCb, Belle-II, and PANDA.

##  I Introduction and summary

The $\Omega^{-}$ baryon acquires a special role in the lowest-lying spin-3/2
decuplet of flavor SU(3). While the decays of all other decuplet states are
dominated by the strong interaction, the $\Omega^{-}$ baryon can only decay
weakly. Weak decays of hadrons are interesting for at least two reasons.
First, they can be used as a tool to study CP violation (see Refs. [1, 2, 3,
4] and references therein). Here, the overarching context is the search for
physics beyond the Standard Model. Second, the electroweak interactions
constitute an important probe to study the structure of hadrons. Here, the
overarching context is the study of the poorly-understood non-perturbative
sector of the strong interaction.

Most weakly-decaying baryons are spin-1/2 particles, as photon radiation is
usually possible, which de-excites the higher spin state. However, when the
minimal quark content is pure in flavor ($qqq$ with $q=u,d,s,c,\ldots$) the
Pauli principle together with the color-white structure demand a completely
symmetric spin state, i.e. spin 3/2 for the ground-state baryon. The lowest-
lying $uuu$ ($\Delta^{++}$) and $ddd$ ($\Delta^{-}$) states are heavy enough
to allow for the strong decay into a nucleon and a Goldstone boson (a pion).
But, the $\Omega^{-}$ baryon, with its three strange quarks, is too light to
decay into a cascade ($\Xi$) and an antikaon [5]. Pure flavor baryons in the
charm or bottom sector are expected to exist, but have not yet been found.
Thus, the $\Omega^{-}$ baryon is special. Its decays provide information about
the interplay of the strong and weak interaction [6, 7, 8, 9, 10] that is
complementary to the weak decays of the ground state spin-1/2 octet baryons.

In fact, there are (at least) two puzzles related to the decays of the
$\Omega^{-}$ baryon. According to the present data situation, the two-body
decays $\Omega^{-}\to\Xi^{-}\pi^{0},\,\Xi^{0}\pi^{-}$ seem to violate the much
celebrated isospin selection, or $\Delta I=1/2$ rule [11, 5] and the three-
body decay $\Omega^{-}\to\Xi^{-}\pi^{-}\pi^{+}$ seems to show no trace of the
$\Xi(1530)$ resonance as an intermediate state in the $\Xi\pi$ channel [12].

From a theory point of view, the main purpose of this paper is to show that
these two puzzles are interrelated. We should point out right away that our
purpose is not to solve the two puzzles. Instead, we will discuss the
consequences of the $\Delta I=1/2$ rule for the three-body decays
$\Omega^{-}\to\Xi^{-}\pi^{+}\pi^{-},\,\Xi^{-}\pi^{0}\pi^{0}$ and
$\Xi^{0}\pi^{-}\pi^{0}$. Partly, the discussion serves to sharpen the contrast
between the current experimental situation and the current understanding of
the weak decays of hadrons. Based on our findings, we shall argue that
differential data on the three-body decays $\Omega^{-}\to\Xi\pi\pi$ would
offer much more information than the two-body decays, if it really turns out
that the $\Delta I=1/2$ rule is grossly violated in the decays of the
$\Omega^{-}$ baryon.

Unfortunately, experimental opportunities to study $\Omega^{-}$ decays, and in
extension their potential violation of the $\Delta I=1/2$ rule, have long been
absent. This absence is over now, as $\Omega^{-}$ baryons (and other hyperons)
are and will be abundantly produced in several running and upcoming
experiments, such as BESIII, LHCb, Belle-II and PANDA [13, 14, 15, 16, 17, 18,
19, 20]. These experiments offer the possibility to study weak baryon decays
in significantly more detail. Therefore, from an experimental point of view,
we regard our paper as a timely reminder of previous discussions, where we
also deepen the present discourse by interrelating two- and three-body decays.
Our work is meant as a motivation to study, in particular, the differential
distributions of three-body decays, which provide access to the low-energy
constants (LECs) of the effective field theory for the strong and weak
interaction in the flavor SU(3) sector. Besides leading-order (LO)
calculations for $\Omega^{-}\to\Xi\pi\pi$ in a relativistic framework of
chiral perturbation theory ($\chi$PT), we will also provide predictions based
on next-to-leading order (NLO) calculations for the so-far unobserved decay
$\Omega^{-}\to\Xi^{0}\mu^{-}\bar{\nu}_{\mu}$. Such semileptonic decays give
access to LECs that are SU(3)-flavor related [21, 22] to the production of
$\Delta$ baryons in neutrino-nucleon and neutrino-nucleus reactions [23, 24,
25, 26, 27].

Let us now briefly review the isospin selection rule, after which we will be
in a position to summarize the scope of this paper. In the Standard Model,
weak decays of hadrons are driven by an effective four-quark operator of left-
handed quarks [28, 3]. For the decays of strange quarks, this transition
operator can be decomposed into a left-handed octet and a left-handed 27-plet
of flavor SU(3). Assuming octet dominance leads to the $\Delta I=1/2$
selection rule for nonleptonic weak decays of strange hadrons, like kaons and
hyperons. In principle, all possibilities caused by the octet and 27-plet can
be encoded in the LECs of a corresponding hadronic effective field theory
(chiral perturbation teory) [3, 8]. Yet, if the number of LECs is large and
the data situation poor, the predictive power will be rather limited. In this
case, it is interesting to figure out if some LECs are smaller than others.
For a corresponding situation in the strong-interaction sector of mesons see
the discussions in Refs. [29, 30, 31]. How the situation can change over the
years with better data and improvements in theory can be deduced from Ref.
[32].

Coming back to the weak non-leptonic decays of hadrons, one needs better (e.g.
more differential) data to determine the LECs or some more microscopic input
(lattice QCD, hadronic models, quark models) to reduce the number of LECs by
focusing on the numerically dominant ones. Another way of describing the
microscopic situation is to consider the strange quark that decays. It might
either flip directly to a down quark (pictured, for instance, by the famous
penguin diagram from Ref. [33]) or it might produce a down, an up and an anti-
up quark. In the first case, the isospin changes definitely by $\Delta I=1/2$.
In the second case, the three quarks might form an isospin 1/2 or 3/2
combination. In some sense the $\Delta I=1/2$ selection rule is based on the
simplest possible case. Of course, it is the (poorly understood) hadron
structure that dictates which case of transition happens with which
probability amplitude. Therefore, the weak decays offer an opportunity to
explore QCD in the non-perturbative regime.

Phenomenologically, the $\Delta I=1/2$ rule is in general well satisfied, see
Refs. [34, 8, 3, 4] and references therein. For instance, $\Delta I=1/2$
implies that the ratios $\Gamma\left(\Lambda\to
p\pi^{-}\right)/\Gamma\left(\Lambda\to n\pi^{0}\right)$ and
$\Gamma\left(\Xi^{-}\to\Lambda\pi^{-}\right)/\Gamma\left(\Xi^{0}\to\Lambda\pi^{0}\right)$
should both have a value close to $2$, while $\Delta I=3/2$ implies that the
ratios should have a value close to $0.5$. The measured values [5], both
approximately $1.77$, clearly favor octet dominance.

However, according to the current experimental status, the selection rule is
violated to an unexpectedly large extent by the decays $\Omega^{-}\to\Xi\pi$.
The $\Delta I=1/2$ rule implies

$\displaystyle\left.\frac{\Gamma\left(\Omega^{-}\to\Xi^{0}\pi^{-}\right)}{\Gamma\left(\Omega^{-}\to\Xi^{-}\pi^{0}\right)}\right|_{\rm
theory}\approx 2$ (1)

(while pure $\Delta I=3/2$ implies that the ratio would be approximately equal
to $0.5$). Measurements give approximately [11, 5]

$\displaystyle\left.\frac{\Gamma\left(\Omega^{-}\to\Xi^{0}\pi^{-}\right)}{\Gamma\left(\Omega^{-}\to\Xi^{-}\pi^{0}\right)}\right|_{\rm
exp.}\approx 2.74\,,$ (2)

which suggests an unexpectedly strong admixture of a $\Delta I=3/2$ amplitude,
interfering positively for $\Xi^{0}\pi^{-}$ and negatively for
$\Xi^{-}\pi^{0}$. If this finding is confirmed by new measurements, the
current effective-field-theory picture about the interplay of the strong and
the weak interaction will receive significant modifications.

On the theoretical side we can follow two lines of thought. On the one hand,
we can explore the consequences of abandoning the $\Delta I=1/2$ rule. This
has been done in Ref. [8], with the conclusion that more data are needed to
make phenomenologically useful predictions. On the other hand, we can posit
the $\Delta I=1/2$ rule to be true and look for more disagreements between
theory and experiment. This work follows the second line of reasoning. In
particular, we look at three-body decays, as differential data give much more
information than just comparing two numbers (2 versus 2.74 for the previously
discussed ratio).

We will present new calculations for the decays $\Omega^{-}\to\Xi\pi\pi$ (for
all combinations of pions). These calculations will be kept at LO, as there
are not enough data to pin down the LECs that appear at NLO. Moreover, given
the scanty data, an NLO calculation is of little added value. In the context
of heavy-baryon $\chi$PT, the decay width for
$\Omega^{-}\to\Xi^{-}\pi^{+}\pi^{-}$ has already been calculated in Ref. [10].
We will extend this calculation by including more decays, calculating fully
differential distributions and supplanting the heavy-baryon approximation with
a fully relativistic formalism. We will put special emphasis on the $\Delta
I=1/2$ rule and use it, together with chiral symmetry, to provide model-
independent lower bounds for the branching fractions. What makes our lower
bounds robust is that chiral symmetry breaking relates the decay channels
$\Xi\pi\pi$ and $\Xi\pi$. We will find an order-of-magnitude disagreement
between current experimental data and our robust lower bound for the measured
channel $\Omega^{-}\to\Xi^{-}\pi^{+}\pi^{-}$. Even though all building blocks
to establish this lower bound have existed in the literature, we are not aware
that it has been spelled out that the failure to understand the current data
of $\Omega^{-}\to\Xi^{-}\pi^{+}\pi^{-}$ provides a further challenge to the
$\Delta I=1/2$ rule. In turn, this means that such three-body decays
$\Omega^{-}\to\Xi\pi\pi$ could be used to scrutinize our understanding of the
strong and weak interaction. The general theme of this work is that three-body
decays $\Omega^{-}\to\Xi\pi\pi$, $\Omega^{-}\to\Xi\mu\bar{\nu}_{\mu}$, and
$\Omega^{-}\to\Xi e\bar{\nu}_{e}$ (see also Ref. [22]) offer the opportunity
to determine LECs of the effective field theory of QCD at the low energies
probed by hyperon decays.

The rest of this work is structured in the following way: in the next section
we specify the relativistic chiral Lagrangian that describes at LO the strong
and weak interactions of the baryon decuplet, the baryon octet and the
Goldstone boson octet. We supplement this Lagrangian by the subleading NLO
terms that we need in the present work. We also specify our power counting
[21, 22], i.e. we give a meaning to the phrase “NLO”. In Sec. III we utilize
the LO Lagrangian to present the decays $\Omega^{-}\to\Xi\pi\pi$ in great
detail. In Sec. IV, we provide NLO predictions for the semi-leptonic decay
$\Omega^{-}\to\Xi^{0}\mu^{-}\bar{\nu}_{\mu}$. A brief outlook is presented in
Sec. V. Some technical aspects that would interrupt the main line of reasoning
have been relegated to appendices.

##  II The SU(3) Chiral Lagrangian

The dominant parts of the interactions of octet and decuplet baryons with
Goldstone bosons and external fields at low energies are provided by the
following LO chiral SU(3) Lagrangian [21, 35, 36, 37, 38]:

$\displaystyle\mathcal{L}_{\text{baryon}}^{\left(1\right)}$ $\displaystyle=$
$\displaystyle\operatorname{tr}\left[\bar{B}\left(i\not{D}-m_{\left(8\right)}\right)B\right]+\bar{T}^{\mu}_{abc}\left[i\gamma_{\mu\nu\alpha}\left(D^{\alpha}T^{\nu}\right)^{abc}-\gamma_{\mu\nu}m_{\left(10\right)}\left(T^{\nu}\right)^{abc}\right]-\frac{H_{A}}{2}\bar{T}^{\mu}_{abc}\gamma_{\nu}\gamma_{5}\left(u^{\nu}\right)^{c}_{d}T^{abd}_{\mu}$
(3)
$\displaystyle{}+\frac{D}{2}\operatorname{tr}\left(\bar{B}\gamma^{\mu}\gamma_{5}\left\\{u_{\mu},B\right\\}\right)+\frac{F}{2}\operatorname{tr}\left(\bar{B}\gamma^{\mu}\gamma_{5}\left[u_{\mu},B\right]\right)+\frac{h_{A}}{2\sqrt{2}}\left(\epsilon^{ade}\bar{T}^{\mu}_{abc}\left(u_{\mu}\right)^{b}_{d}B^{c}_{e}+\epsilon_{ade}\bar{B}^{e}_{c}\left(u^{\mu}\right)^{d}_{b}T^{abc}_{\mu}\right)\,,$

where tr denotes a flavor trace. In passing, we have introduced totally
antisymmetric products of gamma matrices,

$\displaystyle\gamma^{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[\gamma^{\mu},\gamma^{\nu}\right]=-i\sigma^{\mu\nu},$
(4) $\displaystyle\gamma^{\mu\nu\alpha}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left\\{\gamma^{\mu\nu},\gamma^{\alpha}\right\\}=i\epsilon^{\mu\nu\alpha\beta}\gamma_{\beta}\gamma_{5}.$
(5)

In agreement with Ref. [39] our conventions are
$\gamma_{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$ and
$\epsilon_{0123}=-1$.

This Lagrangian is supplemented by the corresponding LO chiral Lagrangian for
the Goldstone bosons [40, 41, 29, 42, 43]:

$\displaystyle\mathcal{L}^{\left(2\right)}_{\text{meson}}=\frac{F_{\pi}^{2}}{4}\operatorname{tr}\left(u_{\mu}u^{\mu}+\chi_{+}\right)\,.$
(6)

Finally, we add the terms that describe the non-leptonic weak decays at the
respective LO for baryons and Goldstone bosons [34, 3]:

$\displaystyle\mathcal{L}_{\text{weak}}^{\left(0\right)}$ $\displaystyle=$
$\displaystyle
h_{C}\bar{T}^{\mu}_{abc}\left[u\left(h+h^{\dagger}\right)u^{\dagger}\right]^{c}_{d}T_{\mu}^{abd}$
(7)
$\displaystyle{}+h_{D}\operatorname{tr}\left(\bar{B}\left\\{u\left(h+h^{\dagger}\right)u^{\dagger},B\right\\}\right)$
$\displaystyle{}+h_{F}\operatorname{tr}\left(\bar{B}\left[u\left(h+h^{\dagger}\right)u^{\dagger},B\right]\right)$
$\displaystyle{}+\frac{1}{4}h_{\pi}F_{\pi}^{2}\operatorname{tr}\left[u\left(h+h^{\dagger}\right)u^{\dagger}u_{\mu}u^{\mu}\right]\,.$

In the above Lagrangians we have packaged the mesons into

$\Phi=\begin{pmatrix}\pi^{0}+\frac{1}{\sqrt{3}}\eta&\sqrt{2}\pi^{+}&\sqrt{2}K^{+}\\\
\sqrt{2}\pi^{-}&-\pi^{0}+\frac{1}{\sqrt{3}}\eta&\sqrt{2}K^{0}\\\
\sqrt{2}K^{-}&\sqrt{2}\bar{K}^{0}&-\frac{2}{\sqrt{3}}\eta\end{pmatrix}.$ (8)

In our convention $\Phi^{a}_{b}$ is the entry in the $a$th row and $b$th
column. Additionally [42, 43],111Note that in this parametrization there is a
sign difference to [30].

$u^{2}=U=\exp\left(i\Phi/F_{\pi}\right).$ (9)

These Goldstone boson fields can be combined into a source-like term
$u_{\mu}=iu^{\dagger}\left(\nabla_{\mu}U\right)u^{\dagger}=u^{\dagger}_{\mu}$.

The spin-1/2 octet baryons are encoded in

$B=\begin{pmatrix}\frac{1}{\sqrt{2}}\Sigma^{0}+\frac{1}{\sqrt{6}}\Lambda&\Sigma^{+}&p\\\
\Sigma^{-}&-\frac{1}{\sqrt{2}}\Sigma^{0}+\frac{1}{\sqrt{6}}\Lambda&n\\\
\Xi^{-}&\Xi^{0}&-\frac{2}{\sqrt{6}}\Lambda\end{pmatrix}.$ (10)

Spin-3/2 decuplet baryons are collected in a totally flavor-symmetric tensor
$T^{abc}$, where

$\displaystyle T^{111}=\Delta^{++},\quad$ $\displaystyle
T^{112}=\frac{1}{\sqrt{3}}\Delta^{+},$ (11) $\displaystyle
T^{122}=\frac{1}{\sqrt{3}}\Delta^{0},\quad$ $\displaystyle
T^{222}=\Delta^{-},$ $\displaystyle
T^{113}=\frac{1}{\sqrt{3}}\Sigma^{\ast+},\quad$ $\displaystyle
T^{123}=\frac{1}{\sqrt{6}}\Sigma^{\ast 0},\quad
T^{223}=\frac{1}{\sqrt{3}}\Sigma^{\ast-},$ $\displaystyle
T^{133}=\frac{1}{\sqrt{3}}\Xi^{\ast 0},\quad$ $\displaystyle
T^{233}=\frac{1}{\sqrt{3}}\Xi^{\ast-},\quad T^{333}=\Omega^{-}.$

The chirally-covariant derivative acting on these fields is defined as

$\displaystyle D^{\mu}B$ $\displaystyle=$
$\displaystyle\partial^{\mu}B+\left[\Gamma^{\mu},B\right],$ (12)
$\displaystyle\left(D^{\mu}T\right)^{abc}$ $\displaystyle=$
$\displaystyle\partial^{\mu}T^{abc}+\left(\Gamma^{\mu}\right)^{a}_{d}T^{dbc}+\left(\Gamma^{\mu}\right)^{b}_{d}T^{adc}$
(13) $\displaystyle\quad+\left(\Gamma^{\mu}\right)^{c}_{d}T^{abd},$
$\displaystyle\left(D^{\mu}\bar{T}\right)_{abc}$ $\displaystyle=$
$\displaystyle\partial^{\mu}\bar{T}_{abc}-\left(\Gamma^{\mu}\right)^{d}_{a}\bar{T}_{dbc}-\left(\Gamma^{\mu}\right)^{d}_{b}\bar{T}_{adc}$
(14) $\displaystyle\quad-\left(\Gamma^{\mu}\right)^{d}_{c}\bar{T}_{abd},$
$\displaystyle\nabla_{\mu}U$ $\displaystyle=$
$\displaystyle\partial_{\mu}U-i\left(v_{\mu}+a_{\mu}\right)U$ (15)
$\displaystyle\quad+iU\left(v_{\mu}-a_{\mu}\right),$

where

$\displaystyle\Gamma_{\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\big{\\{}u^{\dagger}\left[\partial_{\mu}-i\left(v_{\mu}+a_{\mu}\right)\right]u$
(16)
$\displaystyle\quad+u\left[\partial_{\mu}-i\left(v_{\mu}-a_{\mu}\right)\right]u^{\dagger}\big{\\}}.$

The external sources can be used to mediate the interactions with the fields
that do not couple to the strong interaction [41]. In our case these are the
weak source $h$ emerging effectively from the left-handed four-quark operator,
the vector source $v_{\mu}$, the axial-vector source $a_{\mu}$, the scalar
source $s$ and the pseudoscalar source $p$. We can combine the latter two into
$\chi=2B_{0}\left(s+ip\right)$, from which we define
$\chi_{\pm}=u^{\dagger}\chi u^{\dagger}\pm u\chi^{\dagger}u$.

For the nonleptonic weak decays one replaces the elements of the octet matrix
$h$ by $h_{ab}=\delta_{a2}\delta_{3b}$ or equivalently
$h+h^{\dagger}\mapsto\lambda_{6}$ [3]. This choice for $h$ selects the $s\to
d$ transitions [34] consistent with the $\Delta I=1/2$ rule. We can introduce
interactions with electromagnetism via the replacement [42, 43]

$v_{\mu}\mapsto eA_{\mu}\begin{pmatrix}\frac{2}{3}&0&0\\\ 0&-\frac{1}{3}&0\\\
0&0&-\frac{1}{3}\end{pmatrix},$ (17)

where $A_{\mu}$ is the photon field and $e$ the proton charge. For semi-
leptonic weak decays mediated by $W$-bosons we replace [42, 43]

$v_{\mu}-a_{\mu}\mapsto-\frac{g_{w}}{\sqrt{2}}W_{\mu}^{+}\begin{pmatrix}0&V_{ud}&V_{us}\\\
0&0&0\\\ 0&0&0\end{pmatrix}+\text{H.c.},$ (18)

with the $W$-boson field $W_{\mu}^{+}$, the Cabibbo-Kobayashi-Maskawa matrix
elements $V_{ud}$ and $V_{us}$ [44], and the weak gauge coupling $g_{w}$.

The pion-decay constant is $F_{\pi}=92.4$ MeV, and $H_{A}$, $D$, $F$, $h_{A}$,
$B_{0}$, $h_{C}$, $h_{D}$, $h_{F}$, and $h_{\pi}$ are other LECs. Those needed
for our purposes will be specified below in Sec. III.

The octet and decuplet baryon masses in the chiral limit are
$m_{\left(8\right)}$ and $m_{\left(10\right)}$, respectively. In practice, we
take the physical masses for all the states. Formally, this can be achieved by
adding NLO mass-splitting terms [21] to the strong Lagrangian (3). We will not
provide those terms explicitly, but we shall specify the elements of the NLO
Lagrangian [22] that are needed for the semi-leptonic decays:

$\displaystyle\mathcal{L}_{\text{baryon}}^{\left(2\right)}$ $\displaystyle=$
$\displaystyle
ic_{M}\epsilon_{ade}\bar{B}^{e}_{c}\gamma_{\mu}\gamma_{5}\left(f_{+}^{\mu\nu}\right)^{d}_{b}T^{abc}_{\nu}$
(19)
$\displaystyle{}+ic_{E}\epsilon_{ade}\bar{B}^{e}_{c}\gamma_{\mu}\left(f_{-}^{\mu\nu}\right)^{d}_{b}T^{abc}_{\nu}+\text{H.c.}$

The new LECs are $c_{M}$ and $c_{E}$. The field strengths $f_{\pm}^{\mu\nu}$
[30, 32] are given by

$f_{\pm}^{\mu\nu}=uF_{L}^{\mu\nu}u^{\dagger}\pm u^{\dagger}F_{R}^{\mu\nu}u,$
(20)

with

$F^{\mu\nu}_{R,L}=\partial^{\mu}\left(v^{\nu}\pm
a^{\nu}\right)-\partial^{\nu}\left(v^{\mu}\pm a^{\mu}\right)-i\left[v^{\mu}\pm
a^{\mu},v^{\nu}\pm a^{\nu}\right].$ (21)

With respect to chiral transformations [43, 45] $L\in\text{SU(3)}_{L}$,
$R\in\text{SU(3)}_{R}$ and $h_{V}\in\text{SU(3)}_{V}$, the fields transform
as222We remark that the transformation properties for $U$ and $u$ given in
Refs. [21, 22] are incorrect. The authors mixed conventions of Ref. [45] and
Ref. [43]; see also Appendix C.

$\displaystyle U$ $\displaystyle\mapsto$ $\displaystyle RUL^{\dagger},\quad
u\mapsto Ruh_{V}^{\dagger}=h_{V}uL^{\dagger},$ $\displaystyle u_{\mu}$
$\displaystyle\mapsto$ $\displaystyle h_{V}u_{\mu}h_{V}^{\dagger},\quad
B\mapsto h_{V}Bh_{V}^{\dagger},$ $\displaystyle h$ $\displaystyle\mapsto$
$\displaystyle LhL^{\dagger},\quad uhu^{\dagger}\mapsto
h_{V}\left(uhu^{\dagger}\right)h_{V}^{\dagger},$ $\displaystyle T^{abc}_{\mu}$
$\displaystyle\mapsto$
$\displaystyle\left(h_{V}\right)^{a}_{d}\left(h_{V}\right)^{b}_{e}\left(h_{V}\right)^{c}_{f}T^{def}_{\mu},$
$\displaystyle\bar{T}^{\mu}_{abc}$ $\displaystyle\mapsto$
$\displaystyle\left(h_{V}^{\dagger}\right)^{d}_{a}\left(h_{V}^{\dagger}\right)^{e}_{b}\left(h_{V}^{\dagger}\right)^{f}_{c}\bar{T}^{\mu}_{def}.$
(22)

Finally, let us further specify our formalism and our power counting. Our
Lagrangians are fully relativistic and we shall carry out all calculations in
a relativistic manner. Conceptually, this only makes sense if it is clear how
loop corrections are handled in the relativistic framework. This requires us
to know how important each diagram is in our power-counting scheme. And, it
requires that all diagrams contributing to a certain order in the power
counting can be calculated. Otherwise, predictive power would be lost.
Historically, it was not clear how to achieve these requirements. As a
consequence the first systematic scheme that was established was the heavy-
baryon formalism. Previously, decays of $\Omega^{-}$ baryons had been carried
out in this non-relativistic framework [34, 9, 8, 10]. Meanwhile, though, it
has been established how to perform relativistic calculations that follow a
consistent power counting scheme. Therefore it makes sense that we perform our
calculations in a fully relativistic way. For a review of all these
developments from heavy-baryon to relativistic chiral formulations, we refer
to Ref. [43]. We will come back to the power counting in Sec. IV.

##  III Non-leptonic decays to cascades and pions

###  III.1 The two-body decays $\Omega^{-}\to\Xi\pi$ and the consequences of
chiral symmetry

The simplest microscopic process compatible with the $\Delta I=1/2$ rule is
the decay of a strange to a down quark. One starts with an $\Omega^{-}$ state
and its minimal quark content $sss$. After the decay one obtains a state with
quark content $dss$ and spin 3/2. The strong interaction can add extra quark-
antiquark pairs. The corresponding multi-hadron states have overlap with the
$dss$ configuration, but there is also a low-lying single-hadron state (three-
quark state) with quark content $dss$ and spin 3/2: the $\Xi(1530)$ resonance
[5], denoted by $\Xi^{*}$ in Eq. (11). Sharing the very same flavor multiplet,
it sits close by in mass to the decaying $\Omega^{-}$ state. Therefore, the
$\Xi(1530)$ will contribute significantly to the decay process
$\Omega^{-}\to\Xi\pi$ where the final state emerges as a strong decay process
from a virtual $\Xi(1530)$.

This picture of a hadron fluctuating first into another hadron, which is also
applied to the weak decays of spin-1/2 hyperons [34] and kaons [3], lies at
the heart of the Lagrangian (7). For the case at hand this concerns the term
$\sim h_{C}$.

Figure 1: Leading-order diagram for the decay $\Omega\to\Xi\pi$. Circles
indicate a strong vertex and squares indicate a weak vertex.

These considerations lead to the LO diagram shown in Fig. 1 and to the LO
formula (see also [34] for the corresponding heavy-baryon calculation)

$\displaystyle\Gamma\left(\Omega^{-}\to\Xi^{0}\pi^{-}\right)=2\Gamma\left(\Omega^{-}\to\Xi^{-}\pi^{0}\right)$
(23) $\displaystyle=$ $\displaystyle\frac{p_{\text{cm}}}{8\pi
m_{\Omega}^{2}}\frac{h_{A}^{2}h_{C}^{2}}{432F_{\pi}^{2}}\frac{1}{m_{\Omega}^{2}\left(m_{\Omega}-m_{\Xi^{\ast}}\right)^{2}}$
$\displaystyle\times$
$\displaystyle\left(m_{\Omega}^{2}-2m_{\Omega}m_{\Xi}-m_{\pi}^{2}+m_{\Xi}^{2}\right)$
$\displaystyle\times$
$\displaystyle\left(m_{\Omega}^{2}+2m_{\Omega}m_{\Xi}-m_{\pi}^{2}+m_{\Xi}^{2}\right)^{2}\,,\phantom{m}$

where

$\displaystyle p_{\text{cm}}$ $\displaystyle=$
$\displaystyle\frac{1}{2m_{\Omega}}\sqrt{\left(m_{\Omega}+m_{\pi}\right)^{2}-m_{\Xi}^{2}}$
(24) $\displaystyle\times$
$\displaystyle\sqrt{\left(m_{\Omega}-m_{\pi}\right)^{2}-m_{\Xi}^{2}}.$

Here one sees how the $\Xi^{*}=\Xi(1530)$ resonance enhances the process by
the factor $1/(m_{\Omega}-m_{\Xi^{*}})\sim 1/M_{K}^{2}$ in the amplitude.

In passing, we note that the process depicted in Fig. 1 describes the parity-
conserving p-wave amplitude. The parity-violating d-wave is phase-space
suppressed [34, 9]. More generally, it should be stressed that Eq. (23) is a
result valid at LO. There are chiral corrections that are, however, hard to
control beyond leading-log accuracy [34, 9] due to a plethora of LECs in the
weak NLO Lagrangian that complements the LO structures of Eq. (7).
Nevertheless, the dominant contribution is given by Eq. (23).

Of course, all this is compatible with the $\Delta I=1/2$ rule, which amounts
to the prediction
$\Gamma\left(\Omega^{-}\to\Xi^{0}\pi^{-}\right)/\Gamma\left(\Omega^{-}\to\Xi^{-}\pi^{0}\right)=2$.
As already stressed this is in disagreement with the current data. The results
have been used to question the data [34, 9] or to speculate on interesting new
aspects related to $\Delta I=3/2$ processes [7]. In view of this, our
perspective is that additional experimental verification of the present
results is necessary. In addition, one learns even more from three-body decays
to which we turn next.

As happens very often in low-energy QCD, the interesting aspect is chiral
symmetry breaking. It relates in a model-independent way the transition
amplitudes $A\to B$ to processes with $n$ additional soft pions, $A\to
B+n\pi_{\text{soft}}$ [46]. In other words, given the amplitude $A\to B$, we
can predict without any new parameters the strength for the amplitude $A\to
B+n\pi$ up to chiral corrections. For strong-interaction processes the number
of pions must be even to conserve parity. But, for weak processes one predicts
a parity-breaking amplitude $A\to B\pi$ from a parity-conserving amplitude
$A\to B$, and a parity-conserving amplitude from a parity-breaking one.
Diagramatically the processes of Fig. 1 and Fig. 2, diagram 3 are intimately
related.

Taking the previous considerations together, the $\Delta I=1/2$ selection rule
and chiral symmetry relate the strength of the parity-conserving part of the
amplitude of $\Omega^{-}\to\Xi\pi$ to the parity-violating part of the
amplitude of $\Omega^{-}\to\Xi\pi\pi$. We will use this connection to provide
an estimate for the lower limit of the ratio of decay widths for the processes
$\Omega^{-}\to\Xi\pi\pi$ and $\Omega^{-}\to\Xi\pi$.

Technically, the additional soft pion emerges from the appearance of
$u=\exp(i\Phi/(2F_{\pi}))$ in Eq. (7). We are by no means the first who use
this Lagrangian. Yet, to the best of our knowledge it has never been pointed
out before that the $\Delta I=1/2$ selection rule can be used for an otherwise
model independent prediction of the ratio
$\Gamma(\Omega\to\Xi\pi\pi)/\Gamma(\Omega\to\Xi\pi)$. Instead, it has been
stressed in the literature how uncertain the determination of $h_{C}$ and
other parameters is. But one can easily overestimate the uncertainties if one
does not resort to some constraints provided by chiral symmetry.

Though chiral perturbation theory might not always converge well [34], the
order of magnitude of our lower-limit prediction should be regarded as
correct. Therefore, we interpret a gross violation of this prediction not
merely as a challenge for chiral perturbation theory but rather as a challenge
of the $\Delta I=1/2$ selection rule.

###  III.2 The decay $\Omega^{-}\to\Xi^{-}\pi^{+}\pi^{-}$

Meanwhile, the concepts of current algebra and partially conserved axial-
vector current [46] are encompassed and systematized by the use of the chiral
Lagrangians [40, 41, 29, 45, 34, 42, 3, 43, 32]. We will determine some
corrections to the picture that we have just described. However, the small
size of these corrections will support our claims.

We start with the decay $\Omega^{-}\to\Xi^{-}\pi^{+}\pi^{-}$. We work with
kinematic variables
$m^{2}\left(\Xi^{-}\pi^{\pm}\right)=\left(p_{\Xi}+p_{\pm}\right)^{2}$.
Alternatively, we may choose $\cos\theta$ as a second variable, instead of
$m^{2}\left(\Xi^{-}\pi^{-}\right)$. Here, $\theta$ is the angle between
$\bm{p}_{\Xi}$ and $\bm{p}_{-}$ in the frame where
$\bm{p}_{\Xi}+\bm{p}_{+}=\bm{0}$. The relationship between $\cos\theta$ and
$m^{2}\left(\Xi^{-}\pi^{-}\right)$ is

$m^{2}\left(\Xi^{-}\pi^{-}\right)=\left(E^{\ast}_{\Xi}+E^{\ast}_{-}\right)^{2}-\bm{p}_{\Xi}^{2}-\bm{p}_{-}^{2}-2\lvert\bm{p}_{\Xi}\rvert\lvert\bm{p}_{-}\rvert\cos\theta,$
(25)

where

$\displaystyle E^{\ast}_{\Xi}$ $\displaystyle=$
$\displaystyle\frac{m^{2}\left(\Xi^{-}\pi^{+}\right)-m_{\pi}^{2}+m_{\Xi}^{2}}{2m\left(\Xi^{-}\pi^{+}\right)},$
$\displaystyle E^{\ast}_{-}$ $\displaystyle=$
$\displaystyle\frac{-m^{2}\left(\Xi^{-}\pi^{+}\right)+m_{\Omega}^{2}-m_{\pi}^{2}}{2m\left(\Xi^{-}\pi^{+}\right)},$
$\displaystyle\lvert\bm{p}_{\Xi}\rvert$ $\displaystyle=$
$\displaystyle\frac{\lambda^{1/2}\left(m^{2}\left(\Xi^{-}\pi^{+}\right),m_{\Xi}^{2},m_{\pi}^{2}\right)}{2m\left(\Xi^{-}\pi^{+}\right)},$
$\displaystyle\lvert\bm{p}_{-}\rvert$ $\displaystyle=$
$\displaystyle\frac{\lambda^{1/2}\left(m^{2}\left(\Xi^{-}\pi^{+}\right),m_{\Omega}^{2},m_{\pi}^{2}\right)}{2m\left(\Xi^{-}\pi^{+}\right)},$
(26)

are energies and momenta in the frame where $\bm{p}_{\Xi}+\bm{p}_{+}=\bm{0}$.
Here, $\lambda\left(a,b,c\right)$ is the Källén function, defined as

$\lambda\left(a,b,c\right)=a^{2}+b^{2}+c^{2}-2\left(ab+bc+ca\right).$ (27)

Let us estimate the LECs. There are several conventions for the LECs in use in
the literature. In appendix C we give an overview. The values given here are
used throughout the paper. Standard values are $D=0.80$ and $F=0.46$ [47]. The
parameter $h_{\pi}$ is related to kaon decays [34, 10, 48, 3] and is
$h_{\pi}=-3.12\times 10^{-7}$. The value for $h_{A}$ is determined from fits
to different two-body decay channels of decuplet baryons [22]. It lies within
the range $1.9\leq h_{A}\leq 2.9$. The value derived solely from cascade
decays is $h_{A}=2.0$. There are no simple observables to pin down $H_{A}$. To
that end, we average estimates from large-$N_{C}$ [37, 38, 49, 50]
considerations and use $H_{A}=2.0$. For a given $h_{A}$ we determine the
magnitude of $h_{C}$ by fitting to the decay $\Omega^{-}\to\Xi^{0}\pi^{-}$,
whose decay width is proportional to $h_{A}^{2}h_{C}^{2}$; cf. Eq. (23).
Therefore, $h_{C}$ lies between $3.5\times 10^{-8}\text{ GeV}\leq\lvert
h_{C}\rvert\leq 5.4\times 10^{-8}\text{ GeV}$. From $h_{A}=2.0$ we get $\lvert
h_{C}\rvert=5.1\times 10^{-8}$ GeV.

With the LECs fixed, we can calculate the double-differential decay width,
given by [5]

$\frac{{\rm d}^{2}\Gamma(\Omega^{-}\to\Xi^{-}\pi^{+}\pi^{-})}{{\rm
d}m^{2}\left(\Xi^{-}\pi^{+}\right){\rm
d}m^{2}\left(\Xi^{-}\pi^{-}\right)}=\frac{\langle\lvert\mathcal{M}\rvert^{2}\rangle}{32\left(2\pi
m_{\Omega}\right)^{3}},$ (28)

where $\langle\lvert\mathcal{M}\rvert^{2}\rangle$ denotes the spin-averaged
Feynman matrix element. An explicit form of the Feynman matrix element is
given in appendix A. In appendix B we provide some technical notes on deriving
interaction vertices. When calculating the decay width we use Mathematica [51]
and FeynCalc [52, 53, 54] to take traces.

(a) Diagram 1

(b) Diagram 2

(c) Diagram 3

(d) Diagram 4

Figure 2: Diagrams contributing to the decay
$\Omega^{-}\to\Xi^{-}\pi^{+}\pi^{-}$. Circles indicate a strong vertex and
squares indicate a weak vertex.

As can be seen in Fig. 2, the LO Lagrangians (3), (6), (7) provide four tree-
level diagrams contributing to the decay. In line with the discussions in
Subsec. III.1, the numerically dominant term of the decay width comes from
diagram 3 and is proportional to $h_{A}^{2}h_{C}^{2}$. In Fig. 3 we see it is
the well-known $\Xi\left(1530\right)$ resonance (see also Ref. [10]), which
completely overwhelms all other features in the phase space. This matches with
what is known from heavy-baryon calculations [10]. Relativistic corrections to
the branching fraction are on the order of ten percent as compared to the
heavy-baryon results [55].

Figure 3: The fully-differential distribution for the decay
$\Omega\to\Xi^{-}\pi^{+}\pi^{-}$ at leading order, normalized to the total
decay width. The $\Xi\left(1530\right)$ resonance overwhelms all other
features of the distribution.

Our results do not match with what has been seen in the most recent experiment
[12]. There, no trace of the $\Xi(1530)$ resonance has been observed. Data
seem to be compatible with statistical phase space. As discussed in Ref. [10],
the decay width is sensitive to the values of the LECs. Therefore, we resort
to the ratio of three- and two-body decays to obtain a robust, model-
independent result. We will see that the discrepancy is not resolved by tuning
parameters.

In the spin-averaged decay width, parity-conserving processes (diagrams 1-2)
do not interfere with parity-violating processes (diagrams 3-4). To see this,
we factor the decay width as

$\Gamma\propto\operatorname{tr}\left\\{\left[\left(\text{v
type}\right)+\left(\text{c type}\right)\gamma_{5}\right]\left[\left(\text{v
type}\right)+\left(\text{c type}\right)\gamma_{5}\right]\right\\}.$ (29)

Interference terms contain several gamma matrices and exactly one $\gamma_{5}$
matrix. It follows that the trace must vanish or be proportional to a Levi-
Civita symbol. The Levi-Civita symbol must contract with the particle momenta.
Since all momenta lie in the same plane this contraction gives zero.
Therefore,

$\Gamma=\lvert\Gamma_{c}\rvert^{2}+\lvert\Gamma_{v}\rvert^{2}\geq\lvert\Gamma_{v}\rvert^{2}.$
(30)

When the LECs are varied within their physical limits, then we find in all
cases that $\lvert\Gamma_{c}\rvert^{2}$ is numerically small as compared to
$\lvert\Gamma_{v}\rvert^{2}$ (on the order of a couple of percent). Again,
this supports our claim that the dominant contribution to the three-body decay
relates to the two-body decay by chiral symmetry.

We can expand the parity-violating part as

$\displaystyle\lvert\Gamma_{v}\rvert^{2}$ $\displaystyle=$ $\displaystyle
a\left(h_{A}h_{C}\right)^{2}+b\left(h_{A}h_{\pi}\right)^{2}+ch_{A}^{2}h_{C}h_{\pi}$
(31) $\displaystyle=$ $\displaystyle
a\left(h_{A}h_{C}\right)^{2}\left(1+\frac{b}{a}\frac{h_{\pi}^{2}}{h_{C}^{2}}+\frac{c}{a}\frac{h_{\pi}}{h_{C}}\right)\,,$

where $a$, $b$ and $c$ are numbers that only depend on the kinematic factors
and the pion decay constant, but not on other coupling constants. The
dependence on the parameter combinations follows from the formulae in Appendix
A. For realistic values of the parameters $h_{\pi}$ and $h_{C}$ the last two
terms $\sim b/a$ and $\sim c/a$ stay rather small (on the order of $10\%$).
The dominant term, on the other hand, depends on the same parameter
combination as the two-body decay amplitude $\Omega^{-}\to\Xi\pi$.

This allows us to provide a robust and parameter insensitive estimate for the
lower bound of the ratio of the branching fractions of the two-and-three-body
decays,

$\frac{\mathcal{B}\left(\Omega^{-}\to\Xi^{-}\pi^{+}\pi^{-}\right)}{\mathcal{B}\left(\Omega^{-}\to\Xi^{0}\pi^{-}\right)}\gtrsim\frac{\lvert\Gamma_{v}\rvert^{2}}{\Gamma(\Omega^{-}\to\Xi^{0}\pi^{-})}\,.$
(32)

For the numerical estimate we vary the parameter $h_{C}$ in a reasonable
range. Recall, however, that our fit for $h_{C}$ only fixes its magnitude and
not its sign. So, we calculate the lower bound for both signs of $h_{C}$ and
pick whichever case gives the lowest bound (this automatically covers the
other sign choice as well). We find

$\frac{\mathcal{B}\left(\Omega^{-}\to\Xi^{-}\pi^{+}\pi^{-}\right)}{\mathcal{B}\left(\Omega^{-}\to\Xi^{0}\pi^{-}\right)}\gtrsim
2.9\times 10^{-2}.$ (33)

If we drop the terms $\sim b/a$ and $\sim c/a$ we obtain

$\frac{\mathcal{B}\left(\Omega^{-}\to\Xi^{-}\pi^{+}\pi^{-}\right)}{\mathcal{B}\left(\Omega^{-}\to\Xi^{0}\pi^{-}\right)}\gtrsim
3.1\times 10^{-2}\,,$ (34)

fairly close to Eq. (33).

If we use the measured branching fraction of the decay
$\Omega^{-}\to\Xi^{0}\pi^{+}\pi^{-}$ [12] we get

$\frac{\mathcal{B}\left(\text{HyperCP}\right)}{\mathcal{B}\left(\Omega^{-}\to\Xi^{0}\pi^{-}\right)}=\left(0.16\pm
0.03\right)\times 10^{-2}.$ (35)

There is an order-of-magnitude difference between theory and experiment. We
stress again the intimate link between this discrepancy and a possible
violation of the $\Delta I=1/2$ rule. Obviously, the three-body decay is much
better suited to explore this discrepancy, as the order-of magnitude-
difference for the three-body decay is much larger as compared to the
discrepancy for the two-body decays expressed by the difference between ratios
(1) and (2).

At present though, it is unclear whether the resolution of this tension should
come from improvements on the theory side, experimental side, or both. Within
the heavy-baryon formalism the inclusion of higher-order terms and isospin-3/2
contributions has yielded inconclusive results [10, 8]. Empirical data are
scarce. Simply put, we urge that the decay
$\Omega^{-}\to\Xi^{-}\pi^{+}\pi^{-}$ be remeasured at ongoing and upcoming
experiments, such as BESIII, LHCb, Belle-II, and PANDA. Ideally, fully
differential data will match more closely to current results, or trigger
further improvements on the theory side.

###  III.3 Other $\Omega^{-}\to\Xi\pi\pi$ decays

We can extend our reasoning to the so-far unmeasured decays
$\Omega^{-}\to\Xi^{0}\pi^{-}\pi^{0}$ and $\Omega^{-}\to\Xi^{-}\pi^{0}\pi^{0}$.
The diagrams look like Fig. 2, with obvious particle replacements. There are
three additional diagrams similar to diagrams 1-3 where the pions are swapped.
Hence, the decay width has the same structure as Eq.
$\left(\ref{eq:Gammadecomp}\right)$ and we can make predictions for the lower
bound of the ratio of branching fractions. We find

$\displaystyle\frac{\mathcal{B}\left(\Omega^{-}\to\Xi^{0}\pi^{-}\pi^{0}\right)}{\mathcal{B}\left(\Omega^{-}\to\Xi^{0}\pi^{-}\right)}$
$\displaystyle\gtrsim$ $\displaystyle 5.1\times 10^{-2},$
$\displaystyle\frac{\mathcal{B}\left(\Omega^{-}\to\Xi^{-}\pi^{0}\pi^{0}\right)}{\mathcal{B}\left(\Omega^{-}\to\Xi^{0}\pi^{-}\right)}$
$\displaystyle\gtrsim$ $\displaystyle 1.2\times 10^{-2}.$ (36)

That the former ratio is similar in size to the ratio of the decay
$\Omega^{-}\to\Xi^{-}\pi^{+}\pi^{-}$ while the latter ratio is smaller follows
from the relative sizes of the Clebsch-Gordan coefficients corresponding to
the resonance vertices.

For completeness, let us also look at the fully-differential distributions.
They are given in Fig. 4. For the decay $\Omega^{-}\to\Xi^{0}\pi^{-}\pi^{0}$
we use $m^{2}\left(\Xi^{0}\pi^{0}\right)=\left(p_{\Xi}+p_{0}\right)^{2}$ and
the angle $\theta$ between $\bm{p}_{\Xi}$ and $\bm{p}_{-}$ in the frame where
$\bm{p}_{\Xi}+\bm{p}_{0}=\bm{0}$. For the decay
$\Omega^{-}\to\Xi^{-}\pi^{0}\pi^{0}$ we use
$m^{2}\left(\Xi^{-}\pi^{0}_{2}\right)=\left(p_{\Xi}+p_{02}\right)^{2}$ and the
angle between $\bm{p}_{\Xi}$ and $\bm{p}_{01}$ in the frame where
$\bm{p}_{\Xi}+\bm{p}_{02}=\bm{0}$. Here, we label the two neutral pions with 1
and 2.

Both distributions are double-peaked, which is not entirely surprising. The
right peak is the $\Xi\left(1530\right)$ resonance. The left peak arises due
to symmetry reasons. The three new diagrams that our decays have compared to
the decay $\Omega^{-}\to\Xi^{-}\pi^{+}\pi^{-}$ are symmetric under exchange of
the pion momenta. This means that the fully-differential distribution in the
two kinematic variables should be symmetric as well. When we change one of the
kinematic variables to an angle the left peak gets smeared out. This is also
visible in the distributions.

Another hard prediction, besides the lower bounds (33) and (36), is that the
resonance peaks should be clearly visible in the fully-differential
distributions displayed in Figs. 3 and 4. In turn, high-quality differential
distributions can be used to study the relative importance of the $\Delta
I=1/2$ and $\Delta I=3/2$ amplitudes. The corresponding theoretical study
requires such an experimental input and is therefore beyond the scope of the
present work.

(a) Fully-differential distribution for the decay
$\Omega^{-}\to\Xi^{0}\pi^{-}\pi^{0}$.

(b) Fully-differential distribution for the decay
$\Omega^{-}\to\Xi^{-}\pi^{0}\pi^{0}$.

Figure 4: Fully-differential distributions for three-body Omega decays,
normalized by the total decay width. Both distributions are double peaked. The
right peak is the $\Xi\left(1530\right)$ resonance. The left peak arises due
to symmetry reasons and is slightly smeared out, as can be seen in both
figures.

##  IV Semileptonic decays

While there are numerous LECs in the weak NLO Lagrangian [56] that provides
corrections to Eq. (7), the “strong” NLO Lagrangian has been fully specified
in Ref. [21] for the strong-interaction sector plus its coupling to external
fields. We will use it to provide predictions for the decay
$\Omega^{-}\to\Xi^{0}\mu^{-}\bar{\nu}_{\mu}$.

Before turning to the technical aspects, we have to clarify what “NLO” means
in the present context. In general, we study baryon decays of type $B_{1}\to
B_{2}+X$ where $X$, denotes one or several mesons, photons, or dileptons.
Parametrically, the baryon mass is large and the mass difference between the
baryons is small. The masses of all particles contained in $X$ are small.
Consequently, in the rest frame of the decaying state $B_{1}$ the modulus of
the momenta of $B_{2}$ and $X$ is small. This qualitative statement about the
smallness of momenta does not change if we change to the rest frame of
$B_{2}$.

But how small are such momenta? Kinematics tells

$\displaystyle|\bm{p}_{2}|$ $\displaystyle=$ $\displaystyle|\bm{p}_{X}|$ (37)
$\displaystyle=$
$\displaystyle\frac{1}{2m_{1}}\left[\left((m_{1}+m_{2})^{2}-m_{X}^{2}\right)\left((m_{1}-m_{2})^{2}-m_{X}^{2}\right)\right]^{1/2}$
$\displaystyle\approx$
$\displaystyle\left[(m_{1}-m_{2})^{2}-m_{X}^{2}\right]^{1/2},$

which holds in the frame where $B_{1}$ is at rest. Here, $m_{i}$
($\bm{p}_{i}$) is the mass (three-momentum) of baryon $i$ or of the single- or
many-particle state $X$.

Before turning to our case, a small digression is in order to understand the
overall picture. If both $B_{1}$ and $B_{2}$ belong to the lowest-lying baryon
octet (10), then their mass difference is driven by the masses of the light
quarks. Given that the strange-quark mass is significantly larger than the
masses of up and down quark, we can approximate $m_{2}-m_{1}\sim m_{s}\sim
M_{K}^{2}$, as pointed out in Ref. [34]. For kinematic reasons, the final
state $X$ cannot contain kaons ($m_{1}-m_{2}<M_{K}$). One can assume that
$m_{X}$ is parametrically smaller than $M_{K}$. Either it is $\sim M_{K}^{2}$
or suppressed even further. This leads to $|\bm{p}_{2}|=|\bm{p}_{X}|\sim
m_{1}-m_{2}\sim M_{K}^{2}$. In contrast to the typical situation studied in
meson-baryon scattering, the external momenta are not as large as the
Goldstone-boson mass, here $M_{K}$, but further suppressed. Derivative
counting in the chiral Lagrangian might not go along with kaon-mass counting.
For such hyperon decays, loop momenta count as $M_{K}$ while external momenta
count as $M_{K}^{2}$ [34].

But, there is a twist for the decuplet. The main decay branch of the
$\Omega^{-}$ baryon is $\Omega^{-}\to\Lambda K^{-}$. If we put $M_{X}=M_{K}$
in Eq. (37), we obtain the “standard” scaling
$|\bm{p}_{\Lambda}|=|\bm{p}_{K}|\sim M_{K}$ – provided we use
$m_{\Omega}-m_{\Lambda}\sim M_{K}$. Definitely, we cannot use
$m_{\Omega}-m_{\Lambda}\sim M_{K}^{2}$. This would lead to a contradiction
(imaginary momentum) in Eq. (37). For all other decays of the $\Omega^{-}$
baryon there is no predefined answer about the size of the baryon mass
difference relative to the Goldstone boson mass(es). For the present work we
have decided to deal with all $\Omega^{-}$ decays on equal footing. Therefore
we assign [21, 22]

$\displaystyle m_{\rm decuplet}-m_{\rm octet}\sim M_{K},$ (38)

and as a consequence $|\bm{p}_{2}|=|\bm{p}_{X}|\sim m_{\rm decuplet}-m_{\rm
octet}\sim M_{K}$. All soft derivatives appearing in Eq. (3) and Eq. (19)
count as order ${\cal O}(M_{K})$. This is in full analogy to the small-scale
expansion $m_{\Delta}-m_{N}\sim M_{\pi}$ in the two-flavor sector [57]. With
this power counting, the NLO calculation of the decay
$\Omega^{-}\to\Xi^{0}\mu^{-}\bar{\nu}_{\mu}$ is a tree-level calculation.
Loops start to contribute only at NNLO.

Now, we turn to the technical aspects of the decay
$\Omega^{-}\to\Xi^{0}\mu^{-}\bar{\nu}_{\mu}$. The diagrams contributing at NLO
are given in Fig. 5. Our kinematic variables are
$m^{2}\left(\mu^{-}\bar{\nu}_{\mu}\right)=\left(p_{\left(\mu\right)}+p_{\left(\nu\right)}\right)^{2}$
and $\cos\theta$, where $\theta$ is the angle between
$\bm{p}_{\left(\nu\right)}$ and $\bm{p}_{\Xi}$ in the frame where
$\bm{p}_{\left(\mu\right)}+\bm{p}_{\left(\nu\right)}=\bm{0}$.

(a) Diagram 1

(b) Diagram 2

Figure 5: LO and NLO diagrams contributing to the decay
$\Omega^{-}\to\Xi^{0}\mu^{-}\bar{\nu}_{\mu}$, see also (46), (47).

At NLO the decay width depends on two new LECs, $c_{E}$ and $c_{M}$. Our
proposal is that they can be pinned down by an experimental determination of
the differential decay width. To provide some concrete estimates we use Refs.
[21, 22], where the magnitude of each of these constants has been narrowed
down to one of two possible values:

$c_{E}=0.5\text{ GeV}^{-1}\text{ or }-5\text{ GeV}^{-1}\quad c_{M}=\pm
1.92\text{ GeV}^{-1}.$

As can be seen in Fig. 6, it is possible to distinguish the four
configurations of the LECs. It is also possible to distinguish all four cases
just by looking at the angular distributions, as can be seen in Fig. 7.
However, if the data are integrated over $\cos\theta$ it is only possible to
pin down the relative sign of $c_{E}$ and $c_{M}$.

Like the analysis done in Ref. [22] for the decay
$\Omega^{-}\to\Xi^{0}e^{-}\bar{\nu}_{e}$ 333In passing we remark that the
fully-differential distribution in Ref. [22] appears to be flipped with
respect to ours. This is because the authors use the angle between the cascade
and the electron, while we use the angle between the cascade and the
neutrino., it appears that $c_{M}\mapsto-c_{M}$ is essentially equivalent to
$\cos\theta\mapsto-\cos\theta$ (forward-backward asymmetry [2]). This is not
exactly true. Closer inspection of, for instance, Fig. 6(a) and Fig. 6(c)
reveals that there is a degree of asymmetry in this equivalence. This
asymmetry arises because the leptons have finite mass. Compared to the
electronic case, this asymmetry is more pronounced because muons are heavier
than electrons.

(a) $c_{M}=1.92\text{ GeV}^{-1}$, $c_{E}=0.5\text{ GeV}^{-1}$

(b) $c_{M}=1.92\text{ GeV}^{-1}$, $c_{E}=-5\text{ GeV}^{-1}$

(c) $c_{M}=-1.92\text{ GeV}^{-1}$, $c_{E}=0.5\text{ GeV}^{-1}$

(d) $c_{M}=-1.92\text{ GeV}^{-1}$, $c_{E}=-5\text{ GeV}^{-1}$

Figure 6: Fully-differential distribution for the branching ratio of the decay
$\Omega^{-}\to\Xi^{0}\mu^{-}\bar{\nu}_{\mu}$ at NLO for different values of
the LECs $c_{E}$ and $c_{M}$. Plots are normalized by the total decay width of
the Omega baryon.

When we use $c_{M}=1.92$ GeV-1 the predicted branching fraction is lower by
approximately $30\%$ as compared to the electronic case. As a function of
$c_{E}$ it has a similar parabolic shape as in Ref. [22]. Its value is
$\mathcal{B}\left(\Omega^{-}\to\Xi^{0}\mu^{-}\bar{\nu}_{\mu}\right)=3.79\times
10^{-3}$ for $c_{E}=0.5\,$GeV-1.

##  V Outlook

For the decay $\Omega^{-}\to\Xi^{-}\pi^{+}\pi^{-}$ we have demonstrated an
order-of-magnitude discrepancy between experiment, Eq. (35), and theory, Eq.
(33), with the latter based on the $\Delta I=1/2$ selection rule. This adds to
the well-known discrepancy for the two-body decays expressed in Eqs. (1) and
(2). Therefore, we recommend the decays $\Omega^{-}\to\Xi^{-}\pi^{+}\pi^{-}$
as well as $\Omega^{-}\to\Xi^{-}\pi^{0}$ and $\Omega^{-}\to\Xi^{0}\pi^{-}$ be
remeasured at experiments such as BESIII, LHCb, Belle-II, and PANDA. Besides
these branching fractions the related interesting question is the appearance
or non-appearance of the $\Xi(1530)$ resonance peak(s) in the fully-
differential distributions of the decays $\Omega^{-}\to\Xi\pi\pi$. If the
processes are dominated by the $\Delta I=1/2$ selection rule, this resonance
should be visible. Therefore, differential distributions, even if not
normalized, would be highly welcome to understand the fate of the intermediate
$\Xi(1530)$ state. We have also calculated lower limits for the branching
fraction of the decays $\Omega^{-}\to\Xi^{0}\pi^{-}\pi^{0}$ and
$\Omega^{-}\to\Xi^{-}\pi^{0}\pi^{0}$. Lastly, we have looked at the decay
$\Omega^{-}\to\Xi^{0}\mu^{-}\bar{\nu}_{\mu}$ and shown that measurements
thereof may be used to pin down the low-energy constants $c_{M}$ and $c_{E}$.

Figure 7: The normalized angular distribution for the decay
$\Omega^{-}\to\Xi^{0}\mu^{-}\bar{\nu}_{\mu}$ at NLO. The dash-dotted blue line
corresponds to the parameter values in fig. 6(a), the dotted blue line
corresponds to fig. 6(b), the solid orange line corresponds to fig. 6(c) and
the dashed orange line corresponds to fig. 6(d).

## Acknowledgments

SL thanks L. Eklund and A. Kupść for inspiring discussions. CJGM wishes to
thank D. An, N. Salone and F. Alvarado for useful talks. This work has been
supported by the Swedish Research Council (Vetenskapsrådet) (grant number
2019-04303).

##  Appendix A Explicit matrix-element formulae

For the decays $\Omega^{-}\to\Xi\pi\pi$ the generic structure of the Feynman
matrix element is

$\mathcal{M}_{i}(\Omega^{-}\to\Xi\pi\pi)=\bar{u}(p_{\Xi})\,\beta_{i}^{\mu}\,u_{\mu}(p_{\Omega}),$
(39)

where $\beta_{i}^{\mu}$ depends on what decay $i$ we consider. Here,
$u_{\mu}(p,\sigma)$ are vector-spinors [58] describing spin-3/2 states with
momentum $p$ and polarization $\sigma$. They satisfy

$\sum_{\sigma}u_{\mu}(p,\sigma)\,\bar{u}_{\nu}(p,\sigma)=-\left(\not{p}+m\right)P^{3/2}_{\mu\nu}(p),$
(40)

with $p^{0}=\sqrt{m^{2}+\bm{p}^{2}}$ the energy of the spin-3/2 particle and
$m$ its mass. The projector on the spin-3/2 subspace [59] is defined as

$P^{3/2}_{\mu\nu}(p)=g_{\mu\nu}-\frac{1}{3}\gamma_{\mu}\gamma_{\nu}-\frac{1}{3p^{2}}\left(\not{p}\gamma_{\mu}p_{\nu}+p_{\mu}\gamma_{\nu}\not{p}\right).$
(41)

For decay 1, $\Omega^{-}\to\Xi^{-}\pi^{+}\pi^{-}$, we find

$\displaystyle\beta^{\sigma}_{1}$
$\displaystyle=\frac{h_{A}H_{A}h_{C}}{18\sqrt{2}F_{\pi}^{2}}p_{+}^{\mu}P^{T}_{\mu\nu}(k_{1},m_{\Xi^{\ast}},\Gamma_{\Xi^{\ast}})$
$\displaystyle{}\times\not{p}_{-}\gamma_{5}P_{T}^{\nu\sigma}(p_{\Omega},m_{\Xi^{\ast}})$
$\displaystyle{}+\frac{h_{A}h_{C}\left(D-F\right)}{6\sqrt{2}F_{\pi}^{2}}p^{-}_{\mu}\not{p}_{+}\gamma_{5}$
$\displaystyle{}\times
P^{B}(k_{1},m_{\Xi})\,P_{T}^{\mu\sigma}(p_{\Omega},m_{\Xi^{\ast}})$
$\displaystyle{}-\frac{ih_{A}h_{C}}{6\sqrt{2}F_{\pi}^{2}}p^{+}_{\mu}P_{T}^{\mu\sigma}(k_{1},m_{\Xi^{\ast}},\Gamma_{\Xi^{\ast}})$
$\displaystyle{}-\frac{ih_{A}h_{\pi}}{4\sqrt{2}F_{\pi}^{2}}P^{\Phi}(\tilde{k},m_{K})\,\tilde{k}^{\sigma}\left(\tilde{k}\cdot
p_{+}+p_{+}\cdot p_{-}\right).$

For decay 2, $\Omega^{-}\to\Xi^{0}\pi^{-}\pi^{0}$, we find

$\displaystyle\beta^{\sigma}_{2}$
$\displaystyle=\bigg{[}\frac{h_{A}H_{A}h_{C}}{36F_{\pi}^{2}}p_{0}^{\mu}P_{\mu\nu}^{T}(k_{1},m_{\Xi^{\ast}},\Gamma_{\Xi^{\ast}})$
(43)
$\displaystyle{}\times\not{p}_{-}\gamma_{5}P_{T}^{\nu\sigma}(p_{\Omega},m_{\Xi^{\ast}})$
$\displaystyle{}+\frac{h_{A}h_{C}\left(D-F\right)}{12F_{\pi}^{2}}p^{-}_{\mu}\not{p}_{0}\gamma_{5}$
$\displaystyle{}\times
P^{B}(k_{1},m_{\Xi})\,P_{T}^{\mu\sigma}(p_{\Omega},m_{\Xi^{\ast}})$
$\displaystyle{}+\frac{ih_{A}h_{C}}{12F_{\pi}^{2}}p^{0}_{\mu}P_{T}^{\mu\sigma}(k_{1},m_{\Xi^{\ast}},\Gamma_{\Xi^{\ast}})-\left(p_{-}\leftrightarrow
p_{0}\right)\bigg{]}$
$\displaystyle{}+\frac{ih_{A}h_{\pi}}{8F_{\pi}^{2}}P^{\Phi}(\tilde{k},m_{K})\,\tilde{k}^{\sigma}\left(\tilde{k}\cdot
p_{-}-\tilde{k}\cdot p_{0}\right).$

For decay 3, $\Omega^{-}\to\Xi^{-}\pi^{0}\pi^{0}$, we find

$\displaystyle\beta^{\sigma}_{3}$
$\displaystyle=\bigg{[}-\frac{h_{A}H_{A}h_{C}}{36\sqrt{2}F_{\pi}^{2}}p_{20}^{\mu}P_{\mu\nu}^{T}(k_{1},m_{\Xi^{\ast}},\Gamma_{\Xi^{\ast}})$
(44)
$\displaystyle{}\times\not{p}_{10}\gamma_{5}P_{T}^{\nu\sigma}(p_{\Omega},m_{\Xi^{\ast}})$
$\displaystyle{}+\frac{h_{A}h_{C}\left(D-F\right)}{12\sqrt{2}F_{\pi}^{2}}p^{10}_{\mu}\not{p}_{20}\gamma_{5}$
$\displaystyle{}\times
P^{B}(k_{2},m_{\Xi})\,P_{T}^{\mu\sigma}(p_{\Omega},m_{\Xi^{\ast}})$
$\displaystyle{}+\frac{ih_{A}h_{C}}{12\sqrt{2}F_{\pi}^{2}}p^{20}_{\mu}P_{T}^{\mu\sigma}(k_{2},m_{\Xi^{\ast}},\Gamma_{\Xi^{\ast}})+\left(p_{10}\leftrightarrow
p_{20}\right)\bigg{]}$
$\displaystyle{}+\frac{ih_{A}h_{\pi}}{8\sqrt{2}F_{\pi}^{2}}P^{\Phi}(\tilde{k},m_{K})\,\tilde{k}^{\sigma}\left(2p_{10}\cdot
p_{20}+\tilde{k}^{2}\right).$

The intermediate momenta are $k=p_{a}+p_{\Xi}$ and $\tilde{k}=p_{a}+p_{b}$,
with $p_{a}=\left(p_{+},p_{0},p_{01}\right)$ and
$p_{b}=\left(p_{-},p_{-},p_{02}\right)$ for decays 1-3, respectively. Note
that in the momentum-swapped terms also momenta in $k$ get swapped. We use the
following shorthand for the scalar, spin-1/2 and spin-3/2 propagators,

$\displaystyle P^{\Phi}(p,m)$ $\displaystyle=\frac{i}{p^{2}-m^{2}+i\epsilon},$
$\displaystyle P^{B}(p,m)$
$\displaystyle=\frac{i\left(\not{p}+m\right)}{p^{2}-m^{2}+i\epsilon},$
$\displaystyle P^{T}_{\mu\nu}(p,m,\Gamma)$
$\displaystyle=-\frac{i\left(\not{p}+m\right)}{p^{2}-m^{2}+im\Gamma}P^{3/2}_{\mu\nu}(p)$
(45)
$\displaystyle{}+\frac{2i}{3m^{2}}\left(\not{p}+m\right)\frac{p_{\mu}p_{\nu}}{p^{2}}$
$\displaystyle{}-\frac{i}{3m}\frac{p_{\mu}p^{\alpha}\gamma_{\alpha\nu}+\gamma_{\mu\alpha}p^{\alpha}p_{\nu}}{p^{2}},$

respectively. Propagators that do not explicitly contain a width $\Gamma$ in
their argument do not hit a pole. For decay 2, we approximate
$m_{\Xi^{\ast-}}\approx
m_{\Xi^{\ast+}}\approx(m_{\Xi^{\ast-}}+m_{\Xi^{\ast+}})/2=m_{\Xi^{\ast}}$,
with an equivalent approximation for $m_{\Xi}$.

The Feynman matrix element for the semileptonic decay is

$\displaystyle\mathcal{M}\left(\Omega^{-}\to\Xi^{0}\mu^{-}\bar{\nu}_{\mu}\right)$
$\displaystyle=\bar{u}(p_{\left(\mu\right)})\beta^{\prime}_{\kappa}v(p_{\left(\nu\right)})$
(46)
$\displaystyle{}\times\bar{u}(p_{\Xi})\beta^{\prime\prime\kappa\sigma}u_{\sigma}(p_{\Omega}),$

where

$\displaystyle\beta^{\prime}_{\kappa}$
$\displaystyle=-2\sqrt{2}G_{F}V_{us}\gamma_{\kappa}P_{L},$
$\displaystyle\beta^{\prime\prime\kappa\sigma}$
$\displaystyle=\frac{h_{A}}{2\sqrt{2}}g^{\kappa\sigma}+\left(c_{E}-c_{M}\gamma_{5}\right)\left(\not{q}g^{\kappa\sigma}-\gamma^{\kappa}q^{\sigma}\right)$
(47)
$\displaystyle{}-\frac{h_{A}q^{\kappa}q^{\sigma}}{2\sqrt{2}\left(q^{2}-m_{K}^{2}\right)},$

with the Fermi constant $G_{F}$ and $q=p_{\Omega}-p_{\Xi}$.

##  Appendix B Deriving vertices

In principle, it is possible to expand the chiral Lagrangians and read off all
possible interactions. However, it is much more economical to write generic
vertices involving any meson or baryon [60, 55]. To start, we collect the
different particle fields into tuples,

$\displaystyle\left(B^{i}\right)$ $\displaystyle=$
$\displaystyle\left(\Sigma^{+},\;\Sigma^{-},\;\Sigma^{0},\;p,\;\Xi^{-},\;n,\;\Xi^{0},\;\Lambda\right),$
$\displaystyle\left(\Phi^{j}\right)$ $\displaystyle=$
$\displaystyle\left(\pi^{+},\;\pi^{-},\;\pi^{0},\;K^{+},\;K^{-},\;K^{0},\;\bar{K}^{0},\;\eta\right),$
$\displaystyle\left(T^{n}\right)$ $\displaystyle=$
$\displaystyle\big{(}\Sigma^{\ast+},\;\Sigma^{\ast-},\;\Sigma^{\ast
0},\;\Delta^{+},\;\Xi^{\ast-},\;\Delta^{0},\;\Xi^{\ast 0},$
$\displaystyle{}\;\Delta^{++},\;\Delta^{-},\;\Omega^{-}\big{)}.$

Tuples for barred fields look the same.

For the $3\times 3$ traceless matrices we use the basis $b_{i}$, defined via
the Gell-Mann matrices as [29]

$\displaystyle b_{1}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\lambda_{1}+i\lambda_{2}\right),\quad
b_{2}$ $\displaystyle=b_{1}^{\dagger},\quad b_{3}=\lambda_{3},$ $\displaystyle
b_{4}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\lambda_{4}+i\lambda_{5}\right),\quad
b_{5}$ $\displaystyle=b_{4}^{\dagger},\quad$ $\displaystyle b_{6}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\lambda_{6}+i\lambda_{7}\right),\quad
b_{7}$ $\displaystyle=b_{6}^{\dagger},\quad b_{8}=\lambda_{8}.$

From the completeness relation of the Gell-Mann matrices one may show that

$\left(b_{i}\right)^{a}_{b}\left(b_{i}^{\text{T}}\right)^{c}_{d}=\left(\lambda_{i}\right)^{a}_{b}\left(\lambda_{i}\right)^{c}_{d}=2\delta^{a}_{d}\delta^{c}_{b}-\frac{2}{3}\delta^{a}_{b}\delta^{c}_{d}.$
(48)

Note the implicit sum over $i$.

The building blocks of the totally symmetric $3\times 3\times 3$ tensors are
the tensors $e_{abc}$, whose components are zero, except for the
$abc$-element, which is one. The basis elements are then

$\displaystyle t_{1}$
$\displaystyle=\frac{1}{\sqrt{3}}\left(e_{113}+e_{131}+e_{311}\right),$
$\displaystyle t_{2}$
$\displaystyle=\frac{1}{\sqrt{3}}\left(e_{223}+e_{232}+e_{322}\right),$
$\displaystyle t_{3}$
$\displaystyle=\frac{1}{\sqrt{6}}\left(e_{123}+e_{132}+e_{213}+e_{231}+e_{312}+e_{321}\right),$
$\displaystyle t_{4}$
$\displaystyle=\frac{1}{\sqrt{3}}\left(e_{112}+e_{121}+e_{211}\right),$
$\displaystyle t_{5}$
$\displaystyle=\frac{1}{\sqrt{3}}\left(e_{332}+e_{323}+e_{233}\right),$
$\displaystyle t_{6}$
$\displaystyle=\frac{1}{\sqrt{3}}\left(e_{221}+e_{212}+e_{122}\right),$
$\displaystyle t_{7}$
$\displaystyle=\frac{1}{\sqrt{3}}\left(e_{331}+e_{313}+e_{133}\right),$
$\displaystyle t_{8}$ $\displaystyle=e_{111},\quad t_{9}=e_{222},\quad
t_{10}=e_{333}.$

The fields are split from their tensor structure as

$\Phi^{a}_{b}=\Phi^{j}\left(b_{j}\right)^{a}_{b},\quad
B^{a}_{b}=\frac{1}{\sqrt{2}}B^{i}\left(b_{i}\right)^{a}_{b},\quad
T^{abc}=T^{n}\left(t_{n}\right)^{abc}.$ (49)

Note that

$\bar{B}^{a}_{b}=\frac{1}{\sqrt{2}}\bar{B}^{i}\left(b_{i}^{\text{T}}\right)^{a}_{b}.$
(50)

As an example, suppose we want to derive at LO the vertex between a decuplet
baryon, an octet antibaryon and a meson (all in-going). The relevant part of
the Lagrangian is

$\frac{h_{A}}{2\sqrt{2}}\epsilon_{ade}\bar{B}^{e}_{c}\left(u^{\mu}\right)^{d}_{b}T^{abc}_{\mu}.$

By expanding the fields we find a vertex

$V^{T\to\Phi B}_{\mu}=-\frac{h_{A}}{2\sqrt{2}F_{\pi}}c_{nij}p^{\Phi}_{\mu},$
(51)

where $p^{\Phi}_{\mu}$ is the meson momentum and $c_{nij}$ a ‘flavor factor’,

$c_{nij}=\frac{1}{\sqrt{2}}\epsilon_{ade}\left(b_{i}\right)^{e}_{c}\left(b_{j}\right)^{d}_{b}\left(t_{n}\right)^{abc}.$
(52)

A straightforward evaluation of the flavor factor reveals all possible
trilinear couplings.

##  Appendix C Remarks on chiral conventions

For the SU(3) chiral Lagrangian there are two main sets of notation in use. In
this work, we use notation type ‘A’ (also used in Refs. [29, 30, 42, 3, 43,
32, 21, 22]). In Refs. [45, 34, 8, 56, 10] notation type ‘B’ is used. It is
surprisingly messy to harmonize both types.

In notation A, upper indices transform according to the fundamental
representation and lower indices transform according to the antifundamental
representation. In notation B this is flipped. Consequently, in notation B
$\Phi^{a}_{b}$ refers to the $b$th row and $a$th column. Therefore, in
translating notation A to notation B, upper indices become lower indices and
vice versa.

From the transformation rules of the particle fields we find that fields in
notation type B gain a relative phase compared to notation type A (i.e.
$\eta_{A}=-\eta_{B},\dots$). Table 1 gives a further overview. Note that (say)
the matrix $\Phi_{A}$ contains fields of A ($\Lambda_{A},\dots$), while
$\Phi_{B}$ contains fields of B ($\Lambda_{B}=-\Lambda_{A},\dots$).

Table 1: A comparison of two common types of notation in use for the SU(3) chiral Lagrangian. Type A is used in this work and (for example) Refs. [29, 30, 42, 3, 43, 32, 21, 22] while type B is used in Refs. [45, 34, 8, 56, 10]. Type A | Type B | Type A | Type B
---|---|---|---
$B$ | $B$ | $D^{\mu}$ | $\mathcal{D}^{\mu}$
$\Phi$ | $2\pi$ | $F_{\pi}$ | $f$
$U$ | $\Sigma^{\dagger}$ | $m_{\left(8\right)}$ | $m_{B}$
$u$ | $\xi^{\dagger}$ | $m_{\left(10\right)}$ | $m_{T}$
$L/R$ | $L/R$ | $D$ | $D$
$h_{V}$ | $U$ | $F$ | $F$
$\Gamma_{\mu}\big{\rvert}_{l_{\nu},r_{\nu}=0}$ | $V_{\mu}$ | $h_{A}$ | $\sqrt{2}\mathcal{C}$
$u_{\mu}\big{\rvert}_{l_{\nu},r_{\nu}=0}$ | $2A_{\mu}$ | $H_{A}$ | $-\mathcal{H}$

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