# Phenomenology of $J^{PC}=3^{--}$ tensor mesons

Shahriyar Jafarzade1<EMAIL_ADDRESS>Institute of Physics, Jan
Kochanowski University,
ul. Uniwersytecka 7, P-25-406 Kielce, Poland.  Adrian Koenigstein2
<EMAIL_ADDRESS>Institute for Theoretical Physics,
Goethe-University,
Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany.  Francesco Giacosa1,2
<EMAIL_ADDRESS>Institute of Physics, Jan Kochanowski University,
ul. Uniwersytecka 7, P-25-406 Kielce, Poland,  Institute for Theoretical
Physics, Goethe-University,
Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany.

###### Abstract

We study the strong and radiative decays of the antiquark-quark ground state
$J^{PC}=3^{--}$ ($n^{2S+1}L_{J}=1^{3}D_{3}$) nonet {$\rho_{3}(1690)$,
$K_{3}^{\ast}(1780)$, $\phi_{3}(1850)$, $\omega_{3}(1670)$} in the framework
of an effective quantum field theory approach, based on the
$SU_{\mathrm{V}}(3)$ flavor symmetry. The effective model is fitted to
experimental data listed by the Particle Data Group. We predict numerous
experimentally unknown decay widths and branching ratios. An overall agreement
of theory (fit and predictions) with experimental data confirms the $\bar{q}q$
nature of the states and qualitatively validates the effective approach.
Naturally, experimental clarification as well as advanced theoretical
description is needed for trustworthy quantitative predictions, which is
observed from some of the decay channels. Besides conventional spin-$3$
mesons, theoretical predictions for ratios of strong and radiative decays of a
hypothetical glueball state $G_{3}(4200)$ with $J^{PC}=3^{--}$ are also
presented.

††preprint: APS/123-QED

###### Contents

  1. I Introduction
    1. I.1 Contextualization
    2. I.2 Method
    3. I.3 Organization and structure
  2. II Effective model for $J^{PC}=3^{--}$ mesons
    1. II.1 Particle content
      1. II.1.1 Symmetries
      2. II.1.2 Mesonic nonets in the model
    2. II.2 Effective mesonic interactions
      1. II.2.1 The effective action
      2. II.2.2 Decay width
      3. II.2.3 Remarks on the exclusion of scalar mesons
      4. II.2.4 Suppression of next to leading order terms
  3. III Phenomenology of the $J^{PC}=3^{--}$ nonet
    1. III.1 Decay process $W_{3}\rightarrow P+P$
    2. III.2 Decay process $W_{3}\rightarrow V_{1}+P$
    3. III.3 Decay process $W_{3}\rightarrow\gamma+P$
    4. III.4 Decay process $W_{3}\rightarrow A_{2}+P$
    5. III.5 Decay process $W_{3}\rightarrow B_{1}+P$
    6. III.6 Decay process $W_{3}\rightarrow A_{1}+P$
    7. III.7 Decay process $W_{3}\rightarrow V_{1}+V_{1}$
  4. IV Phenomenology of the $J^{PC}=3^{--}$ glueball
    1. IV.1 The effective model for glueball decays
    2. IV.2 Decay process $G_{3}\rightarrow V_{1}+P$
    3. IV.3 Decay process $G_{3}\rightarrow\gamma+P$
    4. IV.4 Decay process $G_{3}\rightarrow V_{1}+A_{1}$
    5. IV.5 Decay process $G_{3}\rightarrow B_{1}+P$
    6. IV.6 Decay process $G_{3}\rightarrow B_{1}+A_{1}$
  5. V Discussion of the construction of the model
  6. VI Conclusions
  7. A Masses and strange-nonstrange mixing angles
  8. B Extended form of the Lagrangians of the model
  9. C Coefficients for the decay channels
  10. D Unpolarized invariant decay amplitudes
  11. E Completeness relations
    1. E.1 Spin-1 fields
    2. E.2 Higher-spin fields
    3. E.3 Spin-3 fields revisited
    4. E.4 Spin-$J$ tensor degrees of freedom

##  I Introduction

The spectroscopy and the phenomenological description of conventional mesons
is important for many reasons. It allows to test quantum chromodynamics (QCD)
in the nonperturbative regime. Furthermore, it aims to the correct and
systematic understanding and assignment of experimentally measured states,
resonances and mesons [1]. It is also fundamental for the search of mesons
that go beyond the antiquark-quark ($\bar{q}q$) picture, such as glueballs,
hybrids, and multiquark states. Namely, both in the light-quark and heavy-
quark sectors, it is possible to search for “exotics” only, if the
conventional $\bar{q}q$ mesons are fully under control [2].

Consequently, our work is just an incremental contribution to this extensive
project in high-energy physics. In our work, we provide a (qualitative)
analysis of the phenomenology of the spin-$3$ mesons $\rho_{3}(1690)$,
$K_{3}^{\ast}(1780)$, $\phi_{3}(1850)$, $\omega_{3}(1670)$ with
$J^{PC}=3^{--}$ ($n^{2S+1}L_{J}=1^{3}D_{3}$) as well as the potential spin-$3$
tensor glueball $G_{3}(4200)$ [3, 4, 5]. This involves a fit of our effective
model to experimental data as well as predictions for decay widths and ratios
of the decays of the spin-$3$ mesons. Although we provide explicit values for
the decay widths and branching ratios, these should be considered only as
first estimates and mainly interpreted as qualitative results. Hence, this
study may for example help in identifying dominant decay channels etc..

###  I.1 Contextualization

In the low-energy sector of QCD, conventional mesons contain the light up
($u$), down ($d$), and strange ($s$) quarks, hence they can be grouped into
nonets: $\bar{u}d$, $\bar{u}s$, etc.. Nonets are classified by the quantum
numbers $J^{PC}$, where $J$ is the absolute value of the total spin of the
meson ($|L-S|\leq J\leq|L+S|$, where $L$ is the angular momentum and $S$ the
spin of the system), $P$ is the sign-change under parity transformations, and
$C$ the sign-change under charge conjugation. For $\bar{q}q$ states, besides
$J^{PC}$, also the older spectroscopic notation $n^{2S+1}L_{J}$, with
$L=S,P,D,\ldots$ and with $n=1,2,\ldots$, which is the radial quantum number,
is used. In the following, we shall restrict to the radial ground state nonets
with $n=1$. For a $\bar{q}q$ system, where $P=(-1)^{L+1}$ and $C=(-1)^{L+S}$,
notice that $L$ and $S$ are, strictly speaking, not well defined quantum
numbers in a relativistic setup, hence mixing of states with the same $J^{PC}$
but with different $L$ and $S$ is possible – even if it is oftentimes a
subordinate effect. However, this nomenclature for the states is still well
upheld also in presence of mixing for practical purposes.

In the low-energy regime of QCD the “classical” symmetries of QCD at high-
energy scales still play a crucial role – more precisely the spontaneous,
anomalous or explicit breaking of these symmetries. Quickly recapitulated, the
situation is as follows: gluons are “democratic”: they interact equally
strongly with right- and left-handed quarks implying that in the so-called
chiral limit (i.e. assuming negligibly small bare quark masses $m_{u}$,
$m_{d}$, and $m_{s}$) the QCD Lagrangian is invariant under
$U_{\mathrm{L}}(3)\times U_{\mathrm{R}}(3)$ transformations. However, this
symmetry of the classical action is broken due to quantum effects, which is
called the $U_{\mathrm{A}}(1)$-anomaly [6, 7, 8, 9]. Still, the baryon number
is conserved, which corresponds to the unaffected $U_{\mathrm{V}}(1)$
symmetry. The remaining $SU_{\mathrm{L}}(3)\times SU_{\mathrm{R}}(3)$ symmetry
is spontaneously broken at low energies into $SU_{\mathrm{V}}(3)$ [10, 11].
The $SU_{\mathrm{V}}(3)$ amounts to a rotation in the flavor space spanned by
$u$, $d$ and $s$ and is consequently called flavor symmetry or vector
symmetry. Hence, at the composite mesonic level, chiral partner-mesons, which
are linked through a chiral transformation, are not degenerate anymore. After
all, the residual realization of an approximate $SU_{\mathrm{V}}(3)$, which is
only explicitly broken by the bare quark masses, is the reason, why nonets of
$\bar{q}q$ states can be used to classify the low-energy spectrum of QCD [12,
13, 14].

Typical examples of very well-known mesonic nonets are the pseudoscalar
states,

$\displaystyle\\{\pi,\,K,\,\eta^{\prime}(958),\,\eta\\}\,,\vphantom{\bigg{(}\bigg{)}}$
(1)

with $J^{PC}=0^{-+}$ $(n^{2S+1}L_{J}=1^{1}S_{0})$, the vector mesons

$\displaystyle\\{\rho(770),\,K^{\ast}(892),\,\phi(1020),\,\omega(782)\\}\,,\vphantom{\bigg{(}\bigg{)}}$
(2)

with $J^{PC}=1^{--}$ $(n^{2S+1}L_{J}=1^{3}S_{1})$, the pseudovector mesons,

$\displaystyle\\{b_{1}(1235),\,K_{1,B},\,h_{1}(1415),\,h_{1}(1170)\\}\,,\vphantom{\bigg{(}\bigg{)}}$
(3)

with $J^{PC}=1^{+-}$ $(n^{2S+1}L_{J}=1^{1}P_{1})$, the axial-vector meson,

$\displaystyle\\{a_{1}(1260),\,K_{1,A},\,f_{1}(1420),\,f_{1}(1285)\\}\,,\vphantom{\bigg{(}\bigg{)}}$
(4)

with $J^{PC}=1^{++}$ $(n^{2S+1}L_{J}=1^{3}P_{1})$, and the well-established
tensor mesons,

$\displaystyle\\{a_{2}(1320),\,K_{2}^{\ast}(1430),\,f_{2}^{\prime}(1525),\,f_{2}(1270)\\}\,,\vphantom{\bigg{(}\bigg{)}}$
(5)

with $J^{PC}=2^{++}$ $(n^{2S+1}L_{J}=1^{3}P_{2})$, compare with Ref. [15] or
the quark model review by the Particle Data Group (PDG) [1].

The aim of this work is to study the $\bar{q}q$ nonet with quantum numbers
$J^{PC}=3^{--}$, resulting from $L=2$ and $S=1$, therefore
$n^{2S+1}L_{J}=1^{3}D_{3}$ states, in a relativistic effective quantum field
theory (QFT) model based on $SU_{\mathrm{V}}(3)$ flavor symmetry. The mesons
belonging to this nonet, corresponding to the lightest states with the proper
quantum numbers in the PDG [1], are:

$\displaystyle\\{\rho_{3}(1690),\,K_{3}^{\ast}(1780),\,\phi_{3}(1850),\,\omega_{3}(1670)\\}\,.$
(6)

The study of these states is very interesting for a series of reasons (for
previous studies on related subjects see also Refs. [15, 16, 17, 18, 19, 20,
21, 22] and Refs. therein):

  1. 1.

It is the only well-known mesonic ground-state nonet with $J>2$, which is
confirmed by several experiments [1].

  2. 2.

Many decay channels are known [1], which allows for further theoretical and
experimental tests of the assignment.

  3. 3.

Predictions for not-yet measured strong and radiative decay rates are
possible.

  4. 4.

The spin-$3$ ground-state nonet (6) is also measured in lattice QCD
calculations [23, 22].

This nonet is therefore tailor made for an effective QFT study of decays. In
particular, we shall answer whether and to what extend the $\bar{q}q$
assignment works, we can test validity of flavor symmetry for various decay
channels, make various “postdictions” and – most interestingly – predictions
for many decay channels. In view of the ongoing experimental efforts in hadron
physics at GlueX [24, 25, 26] and CLAS12 [27] at Jefferson Lab, at COMPASS and
LHCb at CERN [28, 29], at BESIII in Beijing [30, 31], and at the future PANDA
experiment [32] at the GSI facility, we consider a revival of interest on such
resonances valuable.

As an additional application of our approach, we also study some decays of the
$J^{PC}=3^{--}$ glueball state. This can be done by a simple modification of
the action of spin-$3$ mesons and by using the mass of approximately
$4.2\,\,$GeV found in lattice simulations [3, 4, 5] (in the quenched
approximation) as an input. Due to the flavor-singlet nature of the glueball,
only few decay channels are possible.

###  I.2 Method

From a technical point of view, we construct $SU_{\mathrm{V}}(3)$-invariant
effective actions/Lagrangians that involve the mesonic nonet (6) as well as
its various decay products consisting in the well-established $\bar{q}q$
nonets that were previously introduced. From a (Functional) Renormalization
Group (FRG) perspective, the effective actions/Lagrangians can be interpreted
as residual infrared (IR) effective actions with coupling constants that
already involve all quantum effects from higher energy scales, the ultraviolet
(UV). These coupling constants are determined via fits to experimental data,
rather than by ab initio QCD-calculations. All calculations for the decays are
therefore performed at tree-level. This approach was already implemented in
earlier works for studies of various light mesonic nonets, such as tensor
mesons [33], axial-vector, and pseudovector mesons [34], the scalar mesons
[35, 36], the pseudotensor mesons [37], and the orbitally and radially excited
vector mesons [38].

###  I.3 Organization and structure

We organize the paper as follows: in Chap. II, after a discussion of our low-
energy effective model, we present antiquark-quark nonets, their
transformation rules, and the corresponding Lagrangian of the model. It
consists of seven interaction terms. In Chap. III we show the results for the
decay widths and branching ratios of spin-$3$ mesons, while in Chap. IV we
list some branching ratios for an hypothetical heavy $3^{--}$ glueball. A
discussion of the validity of the employed effective interaction terms is
reported in Chap. V and conclusions can be found in Chap. VI. Many technical
aspects concerning the QFT treatment of $J=3$ fields can be found in the
numerous and detailed appendices.

##  II Effective model for $J^{PC}=3^{--}$ mesons

As mentioned in the introduction, in the chiral limit ($m_{u,d,s}=0$) the
$SU_{\mathrm{L}}(3)\times SU_{\mathrm{R}}(3)\times U_{\mathrm{V}}(1)$ symmetry
of QCD is spontaneously broken into flavor symmetry $SU_{\mathrm{V}}(3)$ and
baryon-number conservation – the $U_{\mathrm{V}}(1)$ symmetry. One of the
first successful models that describes this process using four-Fermi
interactions is given by Refs. [39, 40]. At the level of confined light
hadrons, this symmetry is evident: quark-antiquark mesons are clearly grouped
into nonets, some of which were already listed in the introduction [12, 13,
14]. One possibility to describe low-energy QCD makes use of effective low-
energy hadronic models. If the degrees of freedoms are only hadrons, then
confinement and color-neutrality is automatically built in. Typically, these
models are defined by a proper action that mimics the chiral symmetry of QCD
and its spontaneous and explicit breaking.

The chiral symmetry is for example the guiding principle of chiral
perturbation theory (ChPT) [41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51], in
which chiral symmetry is nonlinearly realized, as well as for linear
$\sigma$-models [52, 53, 54, 55, 56, 57], in which it is linearly realized. In
effective models the nonets usually appear in pairs of chiral partners. The
breaking of chiral symmetry generates a mass difference between them. This
dynamics can also be studied at nonzero temperature and/or nonzero baryon or
isospin chemical potentials. Within the last years, the Functional
Renormalization Group [58, 59, 60, 61, 62, 63], turned out to be one
particularly interesting framework to describe chiral symmetry breaking (and
phase transitions) in these kind of models, see for example Refs. [64, 65, 66,
67, 68, 69, 70, 71, 72, 73, 74, 75].

Hadronic models in which only the residual flavor symmetry
$SU_{\mathrm{V}}(3)$ is explicitly conserved have also been constructed for a
variety of nonets in a series of publications by one of the authors and
collaborators [35, 36, 38, 34, 33, 37]. These models can be interpreted as the
effective emerging terms of chiral models after spontaneous symmetry breaking
is worked out. When the effective action is written down, only flavor symmetry
is retained and an expansion in dominant and subdominant terms in
$1/N_{\mathrm{c}}$ is carried out. Moreover, terms that break explicitly
flavor symmetry, either because of the underlying breaking due to nonzero and
unequal quark masses ($m_{u}\approx 2\,\,$MeV, $m_{d}\approx 5\,\,$MeV and
$m_{s}\approx 93\,\,$MeV) or due to the chiral anomaly can be included.

In this work, we study the decays of the $\bar{q}q$ ground-state
$J^{PC}=3^{--}$ nonet by constructing such a model. We shall consider only the
dominant terms in the large-$N_{\mathrm{c}}$ expansion [76, 77, 78] and
neglect flavor symmetry breaking corrections, since the present level of data
accuracy does not allow for their investigations.

###  II.1 Particle content

####  II.1.1 Symmetries

Mesonic nonets that transform under the adjoint representation of the
approximate flavor symmetry $SU_{\mathrm{V}}(3)$ are the main ingredients of
the current work. Transformation rules under parity, charge conjugation and
flavor transformations of the mesonic nonets of our model are
summarized111Other mesonic nonets constructed in the same way can be found for
instance in Refs. [53, 34, 79, 37]. in the Table 1. On the other hand, the
mesonic nonets are in direct correspondence to the physical states, which is
listed in Table 2.

Table 1:  Transformation properties of the pseudoscalar $P$ (7), the vector $V_{1}$ (9), the pseudovector $B_{1}$ (11), the axial-vector $A_{1}$ (15), the rank-$2$ tensor $A_{2}$ (13), and rank-$3$ tensor $W_{3}$ (20) nonets under parity transformations $P$, charge conjugation $C$, and $SU_{\mathrm{V}}(3)$ flavor transformations $U$. Notice the position of the Lorentz indices for parity transformations, since spatial and time-like indices do not transform identically. We use the Minkowski metric in the $(\eta_{\mu\nu})=\mathrm{diag}(+1,-1,-1,-1)$ convention.  nonet | parity | charge conjugation | flavor
---|---|---|---
$J^{PC}=F_{J}^{\cdots}$ | $P$ | $C$ | $SU_{\mathrm{V}}(3)$
$0^{-+}=P$ | $-P(t,-\vec{x})$ | $(P)^{T}$ | $UPU^{\dagger}$
$1^{--}=V_{1}^{\mu}$ | $V_{1,\mu}(t,-\vec{x})$ | $-(V_{1}^{\mu})^{T}$ | $UV_{1}^{\mu}U^{\dagger}$
$1^{+-}=B_{1}^{\mu}$ | $-B_{1,\mu}(t,-\vec{x})$ | $-(B_{1}^{\mu})^{T}$ | $UB_{1}^{\mu}U^{\dagger}$
$1^{++}=A_{1}^{\mu}$ | $-A_{1,\mu}(t,-\vec{x})$ | $(A_{1}^{\mu})^{T}$ | $UA_{1}^{\mu}U^{\dagger}$
$2^{++}=A_{2}^{\mu\nu}$ | $A_{2,\mu\nu}(t,-\vec{x})$ | $(A_{2}^{\mu\nu})^{T}$ | $UA_{2}^{\mu\nu}U^{\dagger}$
$3^{--}=W_{3}^{\mu\nu\rho}$ | $W_{3,\mu\nu\rho}(t,-\vec{x})$ | $-(W_{3}^{\mu\nu\rho})^{T}$ | $UW_{3}^{\mu\nu\rho}U^{\dagger}$

Each $\bar{q}q$ nonet can be assigned to a certain microscopic antiquark-quark
current $N_{ij}\equiv(\bar{q}_{j}\,\Gamma\,q_{i})/\sqrt{2}$, where $\Gamma$ is
a combination of Dirac matrices and derivatives, which – in the
nonrelativistic limit – reduces to the corresponding nonrelativistic
configuration with the desired $L$ and $S$.

####  II.1.2 Mesonic nonets in the model

##### Pseudoscalar mesons

The first nonet of Table 1 describes the matrix of pseudoscalar mesons $P$
with quantum numbers $L=S=0$, leading to $J^{PC}=0^{-+}$
$(n^{2S+1}L_{J}=1^{1}S_{0})$. The elements are
$P_{ij}\equiv(\bar{q}_{j}\,\mathrm{i}\gamma^{5}\,q_{i})/\sqrt{2}$ and the
mesons are {$\pi$, $K$, $\eta^{\prime}(958)$, $\eta$}, where the first entry
represents the isospin $I=1$ triplet (the three pions), the second entry the
two $I=1/2$ isodoublets (the four kaons), and the last two entries the two
isoscalar states (the etas). Pseudoscalars form the basis of almost all low-
energy effective hadronic models/limits or theories of QCD, e.g. ChPT [41, 42,
43, 44, 45, 46, 47, 48, 49, 50] and $\sigma$-models [52, 54, 55, 53, 57] (for
an link between ChPT and $\sigma$-models, see Ref. [80, 68, 69, 71]).
Explicitly, the matrix $P$ reads:

$\displaystyle
P=\tfrac{1}{\sqrt{2}}\begin{pmatrix}\frac{\eta_{N}+\pi^{0}}{\sqrt{2}}&\pi^{+}&K^{+}\\\
\pi^{-}&\frac{\eta_{N}-\pi^{0}}{\sqrt{2}}&K^{0}\\\
K^{-}&\bar{K}^{0}&\eta_{S}\end{pmatrix}\,,$ (7)

where
$\eta_{N}\equiv(\bar{u}\,\mathrm{i}\gamma^{5}\,u+\bar{d}\,\mathrm{i}\gamma^{5}\,d)/\sqrt{2}$
stands for the the purely nonstrange state and
$\eta_{S}\equiv\bar{s}\,\mathrm{i}\gamma^{5}\,s$ stands for the pure strange
state. For the pions and kaons, the physical states are directly assigned to
the fields in the model. In the isoscalar sector, physical and model fields
are related by mixing222We refer to App. A for the description of the mass
terms of the corresponding Lagrangian(s) and for the derivation of the PDG-
mixing formula, which was applied in the PDG [1] in the case of ground-state
vector and pseudoscalar mesons as well as for the ground-state tensor mesons
with $J^{PC}=2^{++}$ and $J^{PC}=3^{--}$. We shall also present the link
between the singlet-octet basis used in the PDG and the strange-nonstrange one
employed in this work.,

$\displaystyle\begin{pmatrix}\eta\\\
\eta^{\prime}(958)\end{pmatrix}=\begin{pmatrix}\cos\beta_{p}&\sin\beta_{p}\\\
-\sin\beta_{p}&\cos\beta_{p}\end{pmatrix}\begin{pmatrix}\eta_{N}\\\
\eta_{S}\end{pmatrix}\,.$ (8)

Here we shall use $\beta_{p}=-43.4^{\circ}$ obtained in Ref. [81]. Note, the
rather large mixing angle and the unexpectedly high mass of the
$\eta^{\prime}(958)$ result from the chiral (or axial) anomaly
$U_{\mathrm{A}}(1)$ [82, 83]. According to a classifications introduced by two
of the authors of the present work in Ref. [84], pseudoscalar mesons together
with their chiral partners (the scalars), belong to what we call a
“heterochiral multiplet”, which allows for the constructions of chirally
anomalous mass and interaction terms.

##### Vector mesons

Next, the second entry in Table 1 refers to the $J^{PC}=1^{--}$
$(n^{2S+1}L_{J}=1^{3}S_{1})$ nonet with $L=0$ and $S=1$. These are the very
well-known vector states {$\rho(770)$, $K^{\ast}(892)$, $\omega(782)$,
$\phi(1020)$}, which are for example also included in certain extensions of
ChPT [47, 48, 49, 50] as well as in enlarged (and realistic) versions of
hadronic models [52, 53, 54, 55, 56]. The matrix $V_{1}^{\mu}$ with elements
$V_{1,ij}^{\mu}=(\bar{q}_{j}\,\gamma^{\mu}\,q_{i})/\sqrt{2}$ has the form

$\displaystyle
V_{1}^{\mu}=\tfrac{1}{\sqrt{2}}\begin{pmatrix}\frac{\omega_{1,N}^{\mu}+\rho_{1}^{0\mu}}{\sqrt{2}}&\rho_{1}^{+\mu}&K_{1}^{\ast+\mu}\\\
\rho_{1}^{-\mu}&\frac{\omega_{1,N}^{\mu}-\rho_{1}^{0\mu}}{\sqrt{2}}&K_{1}^{\ast
0\mu}\\\ K_{1}^{\ast-\mu}&\bar{K}_{1}^{\ast
0\mu}&\omega_{1,S}^{\mu}\end{pmatrix}\,,$ (9)

where $\omega_{N}$ and $\omega_{S}$ are purely nonstrange and strange states,
respectively. Similarly as before, the physical fields arise upon mixing

$\displaystyle\begin{pmatrix}\omega(782)\\\
\phi(1020)\end{pmatrix}=\begin{pmatrix}\cos\beta_{v_{1}}&\sin\beta_{v_{1}}\\\
-\sin\beta_{v_{1}}&\cos\beta_{v_{1}}\end{pmatrix}\begin{pmatrix}\omega_{1,N}\\\
\omega_{1,S}\end{pmatrix}\,,$ (10)

where the very small isoscalar-vector mixing angle
$\beta_{v_{1}}=-3.9^{\circ}$ is taken from the PDG [1]. Hence, the physical
states $\omega(782)$ and $\phi(1020)$ are dominated by nonstrange and strange
components, respectively. This is in agreement with the “homochiral” nature of
these states [84]. In fact vector and axial-vector mesons form a “homochiral
multiplet”, for which the effect of the chiral anomaly is of subleading order
and can safely be ignored.

##### Pseudovector mesons

Next, we consider $L=1$. The choice $L=1$ and $S=0$, leading to
$J^{PC}=1^{+-}$ $(n^{2S+1}L_{J}=1^{1}P_{1})$, contains the established
pseudovector mesons {$b_{1}(1235)$, $K_{1,B}$, $h_{1}(1170)$, $h_{1}(1415)$}.
The corresponding nonet with elements
$B_{1,ij}^{\mu}=(\bar{q}_{j}\,\gamma^{5}\,\partial^{\mu}\,q_{i})/\sqrt{2}$ is

$\displaystyle
B_{1}^{\mu}=\tfrac{1}{\sqrt{2}}\begin{pmatrix}\frac{h_{1,N}^{\mu}+b_{1}^{0\mu}}{\sqrt{2}}&b_{1}^{+\mu}&K_{1,B}^{+\mu}\\\
b_{1}^{-\mu}&\frac{h_{1,N}^{\mu}-b_{1}^{0\mu}}{\sqrt{2}}&K_{1,B}^{0\mu}\\\
K_{1,B}^{-\mu}&\bar{K}_{1,B}^{0\mu}&h_{1,S}^{\mu}\end{pmatrix}\,.$ (11)

The mixing angle

$\displaystyle\begin{pmatrix}h_{1}(1170)\\\
h_{1}(1415)\end{pmatrix}=\begin{pmatrix}\cos\beta_{b_{1}}&\sin\beta_{b_{1}}\\\
-\sin\beta_{b_{1}}&\cos\beta_{b_{1}}\end{pmatrix}\begin{pmatrix}h_{1,N}\\\
h_{1,S}\end{pmatrix}$ (12)

is not known. Quite interestingly, since this nonet belongs to a heterochiral
multiplet (the chiral partners are the orbitally excited vector mesons) the
mixing angle could be nonnegligible, just as for pseudoscalar mesons. Here, we
shall consider two scenarios for our model calculations: in the first, the
mixing angle $\beta_{b_{1}}$ is set to zero, in the second we consider a large
mixing similar to the pseudoscalar sector, $\beta_{b_{1}}\approx-40^{\circ}$
and similar to what two of the authors found for the pseudotensor meson nonet
in Ref. [37] and elaborated on in Ref. [84].

##### Tensor mesons

For $L=S=1$ three nonets are possible. The well-known $J^{PC}=2^{++}$
$(n^{2S+1}L_{J}=1^{3}P_{2})$ tensor states {$a_{2}(1320)$,
$K_{2}^{\ast}(1430)$, $f_{2}(1270)$, $f_{2}^{\prime}(1525)$}, with elements
$A_{2,ij}^{\mu}=[\bar{q}_{j}\,(\mathrm{i}\gamma^{\mu}\,\partial^{\nu}+\ldots)\,q_{i}]/\sqrt{2}$,
represent an almost ideal nonet of quark-antiquark states

$\displaystyle
A_{2}^{\mu\nu}=\tfrac{1}{\sqrt{2}}\begin{pmatrix}\frac{f_{2,N}^{\mu\nu}+a_{2}^{0\mu\nu}}{\sqrt{2}}&a_{2}^{+\mu\nu}&K_{2}^{\ast+\mu\nu}\\\
a_{2}^{-\mu\nu}&\frac{f_{2,N}^{\mu\nu}-a_{2}^{0\mu\nu}}{\sqrt{2}}&K_{2}^{\ast
0\mu\nu}\\\ K_{2}^{\ast-\mu\nu}&\bar{K}_{2}^{\ast
0\mu\nu}&f_{2,S}^{\mu\nu}\end{pmatrix}\,.$ (13)

The physical isoscalar-tensor states are

$\displaystyle\begin{pmatrix}f_{2}(1270)\\\
f_{2}^{\prime}(1525)\end{pmatrix}=\begin{pmatrix}\cos\beta_{a_{2}}&\sin\beta_{a_{2}}\\\
-\sin\beta_{a_{2}}&\cos\beta_{a_{2}}\end{pmatrix}\begin{pmatrix}f_{2,N}\\\
f_{2,S}\end{pmatrix}\,,$ (14)

where $\beta_{a_{2}}=5.7^{\circ}$ is the small mixing angle reported in the
PDG [1], in agreement with the fact that tensor mesons belong to a homochiral
multiplet. The decays of tensor mesons were studied in great detail in Refs.
[33, 85] and fit very well into the $\bar{q}q$ picture [15].

##### Axial-vector mesons

The choice $L=S=1$ allows also for the $J^{PC}=1^{++}$
$(n^{2S+1}L_{J}=1^{3}P_{1})$ axial-vector nonet $A_{1}$, which contains the
resonances {$a_{1}(1260)$, $K_{1,A}$, $f_{1}(1285)$, $f_{1}(1420)$} which are
linked to the vector mesons mentioned above by chiral transformations, see
e.g. Refs. [86, 87] (building a homochiral multiplet). The nonet matrix, whose
microscopic currents are
$A_{1,ij}^{\mu}\equiv\bar{q}_{j}\,\gamma^{5}\gamma^{\mu}\,q_{i}/\sqrt{2}$,
reads:

$\displaystyle
A_{1}^{\mu}=\tfrac{1}{\sqrt{2}}\begin{pmatrix}\frac{f_{1,N}^{\mu}+a_{1}^{0\mu}}{\sqrt{2}}&a_{1}^{+\mu}&K_{1,A}^{+\mu}\\\
a_{1}^{-\mu}&\frac{f_{1,N}^{\mu}-a_{1}^{0\mu}}{\sqrt{2}}&K_{1,A}^{0\mu}\\\
K_{1,A}^{-\mu}&\bar{K}_{1,A}^{0\mu}&f_{1,S}^{\mu}\end{pmatrix}\,.$ (15)

For the isoscalar sector we find,

$\displaystyle\begin{pmatrix}f_{1}(1285)\\\
f_{1}(1420)\end{pmatrix}=\begin{pmatrix}\cos\beta_{a_{1}}&\sin\beta_{a_{1}}\\\
-\sin\beta_{a_{1}}&\cos\beta_{a_{1}}\end{pmatrix}\begin{pmatrix}f_{1,N}\\\
f_{1,S}\end{pmatrix}\,.$ (16)

The mixing angle $\beta_{a_{1}}$ is expected to be small, as the homochiral
nature of the multiplet [84] and the decay properties [34] suggest. We set
$\beta_{a_{1}}=0$ for simplicity (an anyhow small mixing angle would not
affect our results very much).

It is important to note at this point, that the kaonic states $K_{1,A}$ and
$K_{1,B}$ mix. For a study of the axial-vector and pseudovector nonets, with
focus on the $K_{1}(1270)/K_{1}(1400)$ system, by using an approach similar to
the one used in this work, see Ref. [34]. The main result concerning the
kaonic mixing reads:

$\displaystyle\begin{pmatrix}K_{1}(1270)\\\
K_{1}(1400)\end{pmatrix}=\begin{pmatrix}\cos\varphi_{K}&-\mathrm{i}\sin\varphi_{K}\\\
-\mathrm{i}\sin\varphi_{K}&\cos\varphi_{K}\end{pmatrix}\begin{pmatrix}K_{1,A}\\\
K_{1,B}\end{pmatrix}\,.$ (17)

Equivalently, in terms of kets and the angle
$\theta_{K}=\varphi_{K}+90^{\circ}$ typically used in the literature [88, 86,
87]:

$\displaystyle\begin{pmatrix}|K_{1}^{+}(1270)\rangle\\\
|K_{1}^{+}(1400)\rangle\end{pmatrix}=\begin{pmatrix}\sin\theta_{K}&\cos\theta_{K}\\\
\cos\theta_{K}&-\sin\theta_{K}\end{pmatrix}\begin{pmatrix}|K_{1,A}^{+}\rangle\\\
|K_{1,B}^{+}\rangle\end{pmatrix}\,.$ (18)

The final result reads $\varphi_{K}=(56.4\pm 4.3)^{\circ}$, or equivalently
$\theta_{K}=(-33.6\pm 4.3)^{\circ}$, hence $K_{1}(1270)$ is predominantly
$K_{1,B}$ and $K_{1}(1400)$ predominantly $K_{1,A}$, yet the mixing is still
large.

##### Scalar mesons

For completeness, we recall that $L=S=1$ yields also the scalar states
$J^{PC}=0^{++}$ $(n^{2S+1}L_{J}=1^{3}P_{0})$, corresponding to the chiral
partners of the pseudoscalar mesons. The scalars are subject to an ongoing
controversial debate on the correct assignment of measured states. The
identification is still uncertain, but the set

$\displaystyle\\{a_{0}(1450),K_{0}^{\ast}(1430),f_{0}(1710),f_{0}(1370)\\}\,,$
(19)

seems to be favored (for the description of mixing with e.g. the long-searched
scalar glueball, see e.g. Refs. [35, 36, 89, 90, 91, 92, 93, 94] and Refs.
therein). The reason why this nonet is so peculiar is due to the fact that it
has the quantum numbers of the QCD vacuum, hence condensation of the scalar-
isoscalar fields should take place. The corresponding nonet $S$, whose
microscopic currents are $S_{ij}\equiv(\bar{q}_{j}\,q_{i})/\sqrt{2}$, is not
included in this work, since no decays of spin-$3$ states into scalars can be
realized in our model approach, see Sub.Sec. II.2.3.

Moreover, light scalar four-quark states (in the form and combination of
tetraquark/molecular/companion poles) below $1\,\,$GeV also exist [1].
Nevertheless, they are also not included our work.

##### Higher-spin mesons

Besides the peculiar case of the scalar nonet, all other nonets mentioned
above are well-established. Yet, what about nonets with $L=2$? For $L=2$ four
nonets can be constructed, one of which is the main subject of this work.
Thus, even if, besides spin-$3$ fields, they do not enter into the Lagrangian
of this work, since they are too heavy, they allow us to put the nonet of
mesons with $J=3$ into the correct physical framework. Hence, we briefly
discuss them.

The easiest choice is $L=2$ and $S=0,$ which implies $J^{PC}=2^{-+}$
$(n^{2S+1}L_{J}=1^{1}D_{2})$: these so-called pseudotensor states with
currents
$[\bar{q}_{j}\,(\mathrm{i}\gamma^{5}\,\partial^{\mu}\partial^{\nu}+\ldots)\,q_{i}]/\sqrt{2}$
are assigned to {$\pi_{2}(1670)$, $K_{2}(1770)$, $\eta_{2}^{\prime}(1870)$,
$\eta_{2}(1645)$} [95, 96, 97, 37]. Moreover, as recently described in Ref.
[84], the chiral anomaly may also be important for this nonet, explaining a
potential large mixing between strange and nonstrange components, leading to
the isoscalar states $\eta_{2}(1650)$ and $\eta_{2}^{\prime}(1870)$. The
details of the need for a large mixing angle can be found in Ref. [37], in
which strong decays of these mesons are studied in a similar relativistic
effective model.

For $L=2$ and $S=1$ one can construct the orbitally excited vector
$J^{PC}=1^{--}$ ($n^{2S+1}L_{J}=1^{3}D_{1}$) states {$\rho(1700)$,
$K^{\ast}(1680)$, $\phi(????)$, $\omega(1650)$}, which are the (hetero) chiral
partners of the pseudovector mesons [79, 84]. Microscopically, the elements
are $(\bar{q}_{j}\mathrm{i}\partial^{\mu}q_{i})/\sqrt{2}$. The predominately
strange member of the nonet $\phi(????)$ could not be identified yet. For
predictions of a possible $\phi(1930)$ meson see Ref. [38] and Refs. therein,
in which a flavor-symmetry-based QFT model is employed for two nonets of
excited vector mesons.

The quantum numbers $L=2$ and $S=1$ may also lead to a axial tensor nonet with
$J^{PC}=2^{--}$ $(n^{2S+1}L_{J}=1^{3}D_{2})$ with currents
$[\bar{q}_{j}\,(\gamma^{5}\,\gamma^{\mu}\,\partial^{\nu}+\ldots)\,q_{i}]/\sqrt{2}$,
which is however poorly understood. In fact, only the state $K_{2}(1820)$ is
identified and listed by the PDG [1]. This nonet builds together with the
$2^{++}$ tensor mesons a homochiral multiplet.

The latter two nonets are omitted in this work, because their members are too
heavy.

##### Spin-3 mesons

Finally and most importantly, the choice $L=2$ and $S=1$ can also lead the
nonet of $J^{PC}=3^{--}$ $(n^{2S+1}L_{J}=1^{3}D_{3})$ states
{$\rho_{3}(1690)$, $K_{3}^{\ast}(1780)$, $\phi_{3}(1850)$,
$\omega_{3}(1670)$}. The nonet with elements
$W_{ij}^{\mu\nu\rho}=[\bar{q}_{j}\,(\gamma^{\mu}\,\partial^{\nu}\partial^{\rho}+\ldots)\,q_{i}]/\sqrt{2}$
reads,

$\displaystyle
W_{3}^{\mu\nu\rho}=\tfrac{1}{\sqrt{2}}\begin{pmatrix}\frac{\omega_{3,N}^{\mu\nu\rho}+\rho_{3}^{0\mu\nu\rho}}{\sqrt{2}}&\rho_{3}^{+\mu\nu\rho}&K_{3}^{+\mu\nu\rho}\\\
\rho_{3}^{-\mu\nu\rho}&\frac{\omega_{3,N}^{\mu\nu\rho}-\rho_{3}^{0\mu\nu\rho}}{\sqrt{2}}&K_{3}^{0\mu\nu\rho}\\\
K_{3}^{-\mu\nu\rho}&\bar{K}_{3}^{0\mu\nu\rho}&\omega_{3,S}^{\mu\nu\rho}\end{pmatrix}\,.$
(20)

In this case the mixing angle

$\displaystyle\begin{pmatrix}\omega_{3}(1670)\\\
\phi_{3}(1850)\end{pmatrix}=\begin{pmatrix}\cos\beta_{w_{3}}&\sin\beta_{w_{3}}\\\
-\sin\beta_{w_{3}}&\cos\beta_{w_{3}}\end{pmatrix}\begin{pmatrix}\omega_{3,N}\\\
\omega_{3,S}\end{pmatrix}$ (21)

is $\beta_{w_{3}}=3.5^{\circ}$ [1]. The small mixing is also in agreement with
the homochiral nature of the corresponding chiral multiplet in which this
nonet is embedded [84]. In the next section we couple this nonet to the other
nonets listed in Table 1.

Table 2:  Assignment of physical resonances to $\bar{q}q$ states in the model. †The isospin $1/2$ states in the pseudovector and axial-vector sectors are strong mixtures of $K_{1}(1270)$ and $K_{1}(1400)$, see Ref. [1] and Refs. therein.  $J^{PC}$ | physical | nonet $\bar{q}q$ states | mixing
---|---|---|---
resonances | angles
$0^{-+}$ | $\pi$ | $\hphantom{-}\pi$ |
$K$ | $\hphantom{-}K$ | $\beta_{p}=-43.4^{\circ}$
$\eta$ | $\hphantom{-}\eta_{N}\cos\beta_{p}+\eta_{S}\sin\beta_{p}$ | Ref. [81]
$\eta^{\prime}(958)$ | $-\eta_{N}\sin\beta_{p}+\eta_{S}\cos\beta_{p}$ |
$1^{--}$ | $\rho(770)$ | $\hphantom{-}\rho_{1}$ |
$K^{\ast}(892)$ | $\hphantom{-}K_{1}^{\ast}$ | $\beta_{v_{1}}=-3.9^{\circ}$
$\omega(782)$ | $\hphantom{-}\omega_{1,N}\cos\beta_{v_{1}}+\omega_{1,S}\sin\beta_{v_{1}}$ | Ref. [1]
$\phi(1020)$ | $-\omega_{1,N}\sin\beta_{v_{1}}+\omega_{1,S}\cos\beta_{v_{1}}$ |
$1^{+-}$ | $b_{1}(1235)$ | $\hphantom{-}b_{1}$ |
$K_{1,B}^{\dagger}$ | $\hphantom{-}K_{1,B}$ |  $\beta_{b_{1}}$ unknown
$h_{1}(1170)$ | $\hphantom{-}h_{1,N}\cos\beta_{b_{1}}+h_{1,S}\sin\beta_{b_{1}}$ |  $0^{\circ}$ and $-40^{\circ}$
$h_{1}(1415)$ | $-h_{1,N}\sin\beta_{b_{1}}+h_{1,S}\cos\beta_{b_{1}}$ | are tested
$1^{++}$ | $a_{1}(1260)$ | $\hphantom{-}a_{1}$ |
$K_{1,A}^{\dagger}$ | $\hphantom{-}K_{1,A}$ | $\beta_{a_{1}}\simeq 0$
$f_{1}(1285)$ | $\hphantom{-}f_{1,N}\cos\beta_{a_{1}}+f_{1,S}\sin\beta_{a_{1}}$ | Refs. [34, 84]
$f_{1}(1420)$ | $-f_{1,N}\sin\beta_{a_{1}}+f_{1,S}\cos\beta_{a_{1}}$ |
$2^{++}$ | $a_{2}(1320)$ | $\hphantom{-}a_{2}$ |
$K_{2}^{\ast}(1430)$ | $\hphantom{-}K_{2}^{\ast}$ | $\beta_{a_{2}}=5.7^{\circ}$
$f_{2}(1270)$ | $\hphantom{-}f_{2,N}\cos\beta_{a_{2}}+f_{2,S}\sin\beta_{a_{2}}$ | Ref. [1]
$f_{2}^{\prime}(1525)$ | $-f_{2,N}\sin\beta_{a_{2}}+f_{2,S}\cos\beta_{a_{2}}$ |
$3^{--}$ | $\rho_{3}(1690)$ | $\hphantom{-}\rho_{3}$ |
$K_{3}^{\ast}(1780)$ | $\hphantom{-}K_{3}^{\ast}$ | $\beta_{w_{3}}=3.5^{\circ}$
$\omega_{3}(1670)$ | $\hphantom{-}\omega_{3,N}\cos\beta_{w_{3}}+\omega_{3,S}\sin\beta_{w_{3}}$ | Ref. [1]
$\phi_{3}(1850)$ | $-\omega_{2,N}\sin\beta_{w_{3}}+\omega_{2,S}\cos\beta_{w_{3}}$ |

###  II.2 Effective mesonic interactions

In this section, we present the effective mesonic interactions and discuss the
derivation of the tree level decay widths from our model.

####  II.2.1 The effective action

Using the nonets introduced in the previous subsection, we construct the
effective Lagrangian describing the strong decays of spin-$3$ tensor mesons as
follows,

$\displaystyle\mathcal{L}_{W,\text{total}}=\,$
$\displaystyle\mathcal{L}_{\text{mass}}+\mathcal{L}_{\text{kin}}+\mathcal{L}_{w_{3}pp}+\mathcal{L}_{w_{3}v_{1}p}+\vphantom{\bigg{(}\bigg{)}}$
(22)
$\displaystyle+\mathcal{L}_{w_{3}a_{2}p}+\mathcal{L}_{w_{3}a_{1}p}+\mathcal{L}_{w_{3}b_{1}p}+\mathcal{L}_{w_{3}v_{1}v_{1}}\,,\vphantom{\bigg{(}\bigg{)}}$

where
$\mathcal{L}_{\text{kin}}=\frac{1}{2}\,\mathrm{tr}\,(\partial_{\mu}P)^{2}+\ldots$
contains the usual kinetic terms and $\mathcal{L}_{\text{mass}}$ contains all
the quadratic mass terms describing the masses of all relevant nonets. In this
work, all masses are taken from the PDG [1] and assumed to be exact, while
some mixing angles are derived from the masses following the quark model
review of the PDG [1], see also App. A. The other terms describing the decays
are listed explicitly in Table 3. The quantities $g_{w_{3}\circ\circ}$ are the
coupling constants, which are fitted to experimental results,
$\varepsilon^{\mu\nu\rho\sigma}$ is the antisymmetric Levi-Civita
pseudotensor, $[\circ,\,\circ]_{-}$ stands for the commutator, and
$\\{\circ,\,\circ\\}_{+}$ for the anticommutator.

Table 3:  Effective relativistic interaction terms describing the strong decays of mesons with $J^{PC}=3^{--}$.  decay mode | interaction Lagrangians
---|---
$3^{--}\rightarrow 0^{-+}+0^{-+}$ | $\mathcal{L}_{w_{3}pp}=g_{w_{3}pp}\,\mathrm{tr}\big{[}W_{3}^{\mu\nu\rho}\,\big{[}P\,,(\partial_{\mu}\partial_{\nu}\partial_{\rho}P)\big{]}_{-}\big{]}$
$3^{--}\rightarrow 0^{-+}+1^{--}$ | $\mathcal{L}_{w_{3}v_{1}p}=g_{w_{3}v_{1}p}\,\varepsilon^{\mu\nu\rho\sigma}\,\mathrm{tr}\big{[}W_{3,\mu\alpha\beta}\,\big{\\{}(\partial_{\nu}V_{1,\rho})\,,(\partial^{\alpha}\partial^{\beta}\partial_{\sigma}P)\big{\\}}_{+}\big{]}$
$3^{--}\rightarrow 0^{-+}+2^{++}$ | $\mathcal{L}_{w_{3}a_{2}p}=g_{w_{3}a_{2}p}\,\varepsilon_{\mu\nu\rho\sigma}\,\mathrm{tr}\big{[}\tensor{W}{{}_{3}^{\mu}{}_{\alpha}{}_{\beta}}\big{[}(\partial^{\nu}A_{2}^{\rho\alpha})\,,(\partial^{\sigma}\partial^{\beta}P)\big{]}_{-}\big{]}$
$3^{--}\rightarrow 0^{-+}+1^{+-}$ | $\mathcal{L}_{w_{3}b_{1}p}=g_{w_{3}b_{1}p}\,\mathrm{tr}\big{[}W_{3}^{\mu\nu\rho}\big{\\{}B_{1,\mu}\,,(\partial_{\nu}\partial_{\rho}P)\big{\\}}_{+}\big{]}$
$3^{--}\rightarrow 0^{-+}+1^{++}$ | $\mathcal{L}_{w_{3}a_{1}p}=g_{w_{3}a_{1}p}\,\mathrm{tr}\big{[}W_{3}^{\mu\nu\rho}\big{[}A_{1,\mu}\,,(\partial_{\nu}\partial_{\rho}P)\big{]}_{-}\big{]}$
$3^{--}\rightarrow 1^{--}+1^{--}$ | $\mathcal{L}_{w_{3}v_{1}v_{1}}=g_{w_{3}v_{1}v_{1}}\,\mathrm{tr}\big{[}W_{3}^{\mu\nu\rho}\big{[}(\partial_{\mu}V_{1,\nu})\,,V_{1,\rho}\big{]}_{-}\big{]}$

All the interaction Lagrangians in Table 3 are invariant under $CPT$\- ,
Poincaré- and flavor transformations, listed in Table 1. We considered only
couplings involving a minimal number of derivatives within each single
interaction term. This strategy turned to be successful in previous works,
e.g. Refs. [33, 34, 35, 36, 37, 38]333Note, Ref. [16] included higher
derivative couplings and form factors in their approach to strong decays of
spin-$3$ mesons involving charm quarks.. Later on, in Chap. V, we shall also
argue how to justify this strategy using insights from the extended linear
sigma model (eLSM) and the Functional Renormalization Group (FRG) approaches
to low energy QCD. The explicit form of the Lagrangians (after having carried
out the traces) in Table 3 are reported in App. B. Each Lagrangian represents
the dominant contribution in the large-$N_{\mathrm{c}}$ expansion in the given
channel, hence each coupling constant scales as $1/\sqrt{N_{\mathrm{c}}}$.

####  II.2.2 Decay width

The tree level decay widths have the following general form [1, 10],

$\displaystyle\Gamma_{W_{3}\rightarrow
A+B}(m_{w_{3}},m_{a},m_{b})=\vphantom{\frac{\big{|}\vec{k}_{a,b}\big{|}}{8\uppi\,m_{w_{3}}^{2}}}$
(23) $\displaystyle=\,$
$\displaystyle\frac{\big{|}\vec{k}_{a,b}\big{|}}{8\uppi\,m_{w_{3}}^{2}}\,|\mathcal{M}|^{2}\,\kappa_{i}\,\Theta(m_{w_{3}}-m_{a}-m_{b})\,,$

where $m_{w_{3}}$ is the mass of a (decaying) spin-$3$ particle, while $m_{a}$
and $m_{b}$ are the masses of the decay products “$A$” and “$B$”, $\Theta(x)$
denotes the Heaviside step function, and the modulus of the outgoing particles
momentum has the following analytic expression,

$\displaystyle|\vec{k}_{a,b}|\equiv\frac{1}{2\,m_{w_{3}}}\sqrt{(m_{w_{3}}^{2}-m_{a}^{2}-m_{b}^{2})^{2}-4\,m_{a}^{2}\,m_{b}^{2}}\,.$
(24)

We obtain the factors $\kappa_{i}$ in Eq. (23) for the $i$-th decay channel
from the explicit forms of the Lagrangians in the tables in App. B by
considering the square of the coefficients for a given decay channel as well
as eventual sum over members of the same isospin multiplet. The $\kappa_{i}$
for all interaction Lagrangians and all relevant channels are presented in
App. C. The decay amplitudes $|\mathcal{M}|^{2}$ are derived via Feynman rules
under the use of the polarization vectors and tensors as well as their
corresponding completeness relations in App. D. The results for
$\frac{1}{7}|\mathcal{M}|^{2}$ are listed in444Since the decaying particles
are spin-$3$ mesons we multiply by the factor $\frac{1}{7}$, averaging over
the different spin states. Table 4.

Table 4:  Decay amplitudes for different decay modes.  decay mode | $\frac{1}{7}\,|\mathcal{M}|^{2}$
---|---
$3^{--}\rightarrow 0^{-+}+0^{-+}$ | $g_{w_{3}pp}^{2}\,\frac{2}{35}\,\big{|}\vec{k}_{p^{(1)},p^{(2)}}\big{|}^{6}$
$3^{--}\rightarrow 0^{-+}+1^{--}$ | $g_{w_{3}v_{1}p}^{2}\,\frac{8}{105}\,\big{|}\vec{k}_{v_{1},p}\big{|}^{6}\,m_{w_{3}}^{2}$
$3^{--}\rightarrow 0^{-+}+2^{++}$ | $g_{w_{3}a_{2}p}^{2}\,\frac{2}{105}\,\big{|}\vec{k}_{a_{2},p}\big{|}^{4}\,\frac{m_{w_{3}}^{2}}{m_{a_{2}}^{2}}\,\big{(}2\,\big{|}\vec{k}_{a_{2},p}\big{|}^{2}+7\,m_{a_{2}}^{2}\big{)}$
$3^{--}\rightarrow 0^{-+}+1^{+-}$ | $g_{w_{3}b_{1}p}^{2}\,\frac{2}{105}\,\big{|}\vec{k}_{b_{1},p}\big{|}^{4}\,\Big{(}7+3\,\frac{|\vec{k}_{b_{1},p}|^{2}}{m_{b_{1}}^{2}}\Big{)}$
$3^{--}\rightarrow 0^{-+}+1^{++}$ | $g_{w_{3}a_{1}p}^{2}\,\frac{2}{105}\,\big{|}\vec{k}_{a_{1},p}\big{|}^{4}\,\Big{(}7+3\,\frac{|\vec{k}_{a_{1},p}|^{2}}{m_{a_{1}}^{2}}\Big{)}$
$3^{--}\rightarrow 1^{--}+1^{--}$ | $g_{w_{3}v_{1}v_{1}}^{2}\,\frac{1}{105}\,\big{(}m_{v_{1}^{(1)}}^{2}\,m_{v_{1}^{(2)}}^{2}\big{)}^{-1}\,\big{|}\vec{k}_{v_{1}^{(1)},v_{2}^{(2)}}\big{|}^{2}\,\big{[}6\,|\vec{k}_{v_{1}^{(1)},v_{1}^{(2)}}\big{|}^{4}+35\,m_{v_{1}^{(1)}}^{2}\,m_{v_{1}^{(2)}}^{2}+14\,\big{|}\vec{k}_{v_{1}^{(1)},v_{1}^{(2)}}\big{|}^{2}\,\big{(}m_{v_{1}^{(1)}}^{2}+m_{v_{1}^{(2)}}^{2}\big{)}\big{]}$

####  II.2.3 Remarks on the exclusion of scalar mesons

As a final side remark, we note that we cannot couple the spin-$3$ nonet (20)
to scalar mesons (19). In fact, if we try to couple them to scalars and
pseudoscalars, the only possible CPT- and flavor-invariant interaction term,

$\displaystyle\varepsilon_{\mu\nu\rho\sigma}\,\mathrm{tr}\big{(}\partial^{\mu}\tensor{W}{{}_{3}^{\nu}{}_{\alpha}{}_{\beta}}\,\big{[}(\partial^{\rho}S),\,(\partial^{\sigma}\partial^{\alpha}\partial^{\beta}P)\big{]}_{-}\big{)}=0\,,\vphantom{\bigg{(}\bigg{)}}$
(25)

vanishes identically since it involves the contraction of antisymmetric and
symmetric tensors. Correspondingly, scalar mesons do not enter our study,
which agrees with the experimental results reported in the PDG [1]. This is
also of direct advantage for our work, since we do not have to deal with the
identification of the scalar states, which, as already mentioned, is a long-
standing and yet unsolved issue of low-energy QCD [89, 90, 91, 92, 93, 94].

####  II.2.4 Suppression of next to leading order terms

The Lagrangian terms considered in Table 3 are large-$N_{\mathrm{c}}$ (and
thus Okubo-Zweig-Iizuka) dominant and symmetric under $U_{\mathrm{V}}(3)$. As
we shall discuss later on, this approximation is expected to be acceptable in
view of the precision of the experimental data presently available. Yet, by
considering the $W_{3}B_{1}P$ interaction as an example, the Lagrangian
$\mathcal{L}_{w_{3}b_{1}p}$ can be regarded as the first term of an expansion
in large-$N_{\mathrm{c}}$ and/or symmetry breaking terms that takes the form:

$\displaystyle\mathcal{L}_{w_{3}b_{1}p}^{\text{full}}=\vphantom{\bigg{(}\bigg{)}}$
(26) $\displaystyle=\,$ $\displaystyle
g_{w_{3}b_{1}p}\,\mathrm{tr}\big{[}W^{\mu\nu\rho}\big{\\{}B_{1\mu}\,,(\partial_{\mu}\partial_{\rho}P)\big{\\}}_{+}\big{]}+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle+g_{w_{3}b_{1}p}^{(2)}\,\mathrm{tr}\big{[}W^{\mu\nu\rho}\big{]}\,\mathrm{tr}\big{[}B_{1\mu}\,(\partial_{\mu}\partial_{\rho}P)\big{]}+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle+g_{w_{3}b_{1}p}^{(3)}\,\mathrm{tr}\big{[}W^{\mu\nu\rho}\big{]}\,\mathrm{tr}\big{[}B_{1\mu}\big{]}\,\mathrm{tr}\big{[}\partial_{\mu}\partial_{\rho}P\big{]}+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle+g_{w_{3}b_{1}p}^{(4)}\,\mathrm{tr}\big{[}\hat{\delta}\,W^{\mu\nu\rho}\,\big{\\{}B_{1\mu}\,,(\partial_{\mu}\partial_{\rho}P)\big{\\}}_{+}\big{]}+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle+g_{w_{3}b_{1}p}^{(5)}\,\mathrm{tr}\big{[}\hat{\delta}\,W^{\mu\nu\rho}\big{]}\,\mathrm{tr}\big{[}B_{1\mu}\,(\partial_{\mu}\partial_{\rho}P)\big{]}+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle+g_{w_{3}b_{1}p}^{(6)}\,\mathrm{tr}\big{[}W^{\mu\nu\rho}\big{]}\,\mathrm{tr}\big{[}\hat{\delta}\,B_{1\mu}\,(\partial_{\mu}\partial_{\rho}P)\big{]}+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle+\ldots\vphantom{\bigg{(}\bigg{)}}$

The first term, reported in Table 3, is a flavor symmetric term and scales as
$g_{w_{3}b_{1}p}\propto N_{\mathrm{c}}^{-1/2},$ hence it is the dominant term
in a large-$N_{\mathrm{c}}$ expansion. The second and the third term are also
flavor symmetric, but scale as $g_{w_{3}b_{1}p}^{(2)}\propto
N_{\mathrm{\mathrm{c}}}^{-\frac{3}{2}}$ and $g_{w_{3}b_{1}p}^{(3)}\propto
N_{\mathrm{c}}^{-\frac{5}{2}}$, respectively. They involve gluon exchanges and
are therefore suppressed. The fourth term is not large-$N_{\mathrm{c}}$
suppressed, but it breaks flavor symmetry via the matrix

$\displaystyle\hat{\delta}=\mathrm{diag}\\{0,\delta_{d},\delta_{s}\\}\,.$ (27)

Isospin violation is proportional to $\delta_{d}\propto m_{d}-m_{u}$ and is
expected to be very small, yet the breaking due to the $s$ quark
$\delta_{s}\propto m_{s}-m_{u}$ can be nonnegligible following Ref. [90, 89],
it could be along the order of $g_{w_{3}b_{1}p}^{(4)}\simeq
g_{w_{3}b_{1}p}\,\mathrm{diag}\\{0,0,0.1\,-\,0.2\\}$, but the actual value for
$J=3$ mesons should be determined by an independent fit to data. It is however
expected to be sufficiently small to be neglected in this work. At this stage,
the first large-$N_{\mathrm{c}}$ correction $g_{w_{3}b_{1}p}^{(2)}$ and the
first flavor symmetric correction $g_{w_{3}b_{1}p}^{(4)}$ are expected to be
of the same intensity and should be the first to be included. Further terms
$g_{w_{3}b_{1}p}^{(n\geq 5)}$ are both large-$N_{\mathrm{c}}$ subdominant and
flavor suppressed, thus are regarded to be very small.

The same analysis can be carried out for all the interaction terms, yet it is
interesting to observe that the second and the third term would vanish
whenever the commutator is present. (Flavor-symmetry violation is expected to
be the main next-to-leading-order contribution for those interaction terms).

##  III Phenomenology of the $J^{PC}=3^{--}$ nonet

In this section we present our results for decay rates and branching ratios of
the $J^{PC}=3^{--}$ mesons. In each subsection and for each interaction
Lagrangian term, we compare the experimental data to our theoretical results.

We recall that the total decay widths of the $J^{PC}=3^{--}$ mesons under
consideration are [1],

$\displaystyle\Gamma_{\rho_{3}(1690)}^{\text{tot}}=\,$ $\displaystyle(161\pm
10)\,\,\text{MeV}\,,\vphantom{\bigg{(}\bigg{)}}$ (28)
$\displaystyle\Gamma_{K_{3}^{\ast}(1780)}^{\text{tot}}=\,$
$\displaystyle(159\pm 21)\,\,\text{MeV}\,,\vphantom{\bigg{(}\bigg{)}}$ (29)
$\displaystyle\Gamma_{\omega_{3}(1670)}^{\text{tot}}=\,$ $\displaystyle(168\pm
10)\,\,\text{MeV}\,,\vphantom{\bigg{(}\bigg{)}}$ (30)
$\displaystyle\Gamma_{\phi_{3}(1850)}^{\text{tot}}=\,$
$\displaystyle(87_{-23}^{+28})\,\,\text{MeV}\,.\vphantom{\bigg{(}\bigg{)}}$
(31)

We shall verify that the sum of all the single decay channels from our
theoretical calculations never overshoots these values.

For what concerns the accuracy of our results, whenever possible we determine
the coupling and its error via a simple fit of experimental decay width and
ratios. We ignore the experimental uncertainties for all masses of the
particles of the model and assume the masses to be exact, because their errors
are small and they are of minor importance for the overall errors. The
experimental errors for the decay widths and branching ratios as well as the
systematic errors in the model are much larger, which justifies this
approximation. Yet, the quoted errors represents a lower(!) limit of the
actual error of this work, since other indeterminacy features are inherent in
the approximations, that lead us to the effective action, where only those
terms are included, which are expected to be dominant in terms of
large-$N_{\mathrm{c}}$, flavor- and momentum-space expansions. Moreover, the
decay widths are calculated at tree level. Since the width/mass ratio for the
decaying resonances is rather small, contributions due to loops are expected
to be negligible [98, 99], if seen from a perturbative perspective. Otherwise,
if the interaction terms are interpreted as effective couplings in a full
quantum effective IR action, calculations have to be performed at tree level
anyhow.

###  III.1 Decay process $W_{3}\rightarrow P+P$

The effective interaction term describing the decay of spin-$3$ tensor mesons
into two pseudoscalar mesons has the following form

$\displaystyle\mathcal{L}_{w_{3}pp}=g_{w_{3}pp}\,\mathrm{tr}\big{[}W_{3}^{\mu\nu\rho}\,\big{[}P,\,(\partial_{\mu}\partial_{\nu}\partial_{\rho}P)\big{]}_{-}\big{]}\,.$
(32)

Its extended version is listed in App. B in Eq. (126). Correspondingly, the
tree level decay rate is

$\displaystyle\Gamma_{W_{3}\rightarrow
P^{(1)}+P^{(2)}}(m_{w_{3}},m_{p^{(1)}},m_{p^{(2)}})=\vphantom{\Bigg{(}\Bigg{)}}$
(33) $\displaystyle=\,$ $\displaystyle
g_{w_{3}pp}^{2}\,\frac{\big{|}\vec{k}_{p^{(1)},p^{(2)}}\big{|}^{7}}{140\uppi\,m_{w_{3}}^{2}}\,\kappa_{i}\,\Theta(m_{w_{3}}-m_{p^{(1)}}-m_{p^{(2)}})\,,\vphantom{\Bigg{(}\Bigg{)}}$

compare Eq. (148). The factors $\kappa_{i}$ are reported in Table 19. In order
to determine the coupling constant $g_{w_{3}pp}$, we use the above formula and
the following experimental data:

  1. 1.

For $\rho_{3}(1690)\rightarrow\pi\,\pi$ one finds $\kappa_{1}=1$. Using the
experimental result

$\displaystyle\Gamma^{\mathrm{exp}}_{\rho_{3}(1690)\rightarrow\pi\,\pi}=(38.0\pm
3.2)\,\,\text{MeV}\,,$ (34)

one obtains a first determination of the coupling constant squared and its
error. We shall denote them as $\tilde{g}_{1}^{2}$ and
$\Delta\tilde{g}_{1}^{2}$.

  2. 2.

The experimental value

$\displaystyle\Gamma^{\text{exp}}_{\rho_{3}(1690)\rightarrow\bar{K}\,K}=(2.54\pm
0.45)\,\,\text{MeV}\,,$ (35)

together with $\kappa_{2}=2\,\big{(}\frac{1}{2}\big{)}^{2}$ yields an
independent determinations of $\tilde{g}_{2}^{2}$ and
$\Delta\tilde{g}_{2}^{2}$.

  3. 3.

From

$\displaystyle\Gamma^{\text{exp}}_{K_{3}^{\ast}(1780)\rightarrow\pi\,\bar{K}}=(29.9\pm
4.3)\,\,\text{MeV}$ (36)

and
$\kappa_{3}=\big{(}\frac{1}{2}\big{)}^{2}+\big{(}\frac{\sqrt{2}}{2}\big{)}^{2}$
we get $\tilde{g}_{3}^{2}$ and $\Delta\tilde{g}_{3}^{2}$.

  4. 4.

Finally, from

$\displaystyle\Gamma^{\text{exp}}_{K_{3}^{\ast}(1780)\rightarrow\bar{K}\,\eta}=(48\pm
22)\,\,\text{MeV}$ (37)

and
$\kappa_{4}=\big{[}\frac{1}{2}\,\big{(}-\cos\beta_{p}+\sqrt{2}\sin\beta_{p}\big{)}\big{]}^{2}$
we obtain $\tilde{g}_{4}$ and $\Delta\,\tilde{g}_{4}$.

The coupling constant and its indeterminacy are evaluated using a simple
$\chi^{2}$-approach,555We are aware of the fact that some of the data might be
correlated, because it is taken from similar measurements at the same
experiments. Within the level of accuracy of our approach, there is no need to
take care of these effects.

$\displaystyle
g_{w_{3}pp}^{2}=\frac{\sum_{i=1}^{4}\frac{\tilde{g}_{i}}{\Delta\tilde{g}_{i}^{2}}}{\sum_{j=1}^{4}\frac{1}{\Delta\tilde{g}_{j}^{2}}}\,,$
$\displaystyle\Delta
g_{w_{3}pp}^{2}=\sqrt{\frac{1}{\sum_{j=1}^{4}\frac{1}{\Delta g_{j}^{2}}}}\,,$
(38)

which results in

$g_{w_{3}pp}^{2}=(1.5\pm 0.1)\cdot 10^{-10}\,\,\text{MeV}^{-4}\,.$ (39)

The comparison of theoretical and experimental results, which is obtained by
using this value for the coupling constant, is reported in Table 5. A good
overall agreement is obtained, but there is also a sizable mismatch: the
experimental value for $K_{3}^{\ast}(1780)\rightarrow\bar{K}\,\eta$ is much
larger than our theoretical prediction. Still, the experimental error is large
and a better experimental determination would be interesting. Moreover, a
noteworthy prediction concerning $\phi_{3}(1850)\rightarrow\bar{K}\,K$ is
obtained. From the theoretical large prediction, we conclude that an
experimental determination should be feasible.

Table 5:  Decays of $J^{PC}=3^{--}$ mesons into two pseudoscalars. Experimental data is taken from Ref. [1].  decay process | theory | experiment
---|---|---
$\Gamma/$MeV |  $\Gamma/$MeV
$\rho_{3}(1690)\rightarrow\pi\,\pi$ | $32.7\pm 2.3$ | $38.0\pm 3.2$
$\rho_{3}(1690)\rightarrow\bar{K}\,K$ | $4.0\pm 0.3$ | $2.54\pm 0.45$
$K_{3}^{\ast}(1780)\rightarrow\pi\,\bar{K}$ | $18.5\pm 1.3$ | $29.9\pm 4.3$
$K_{3}^{\ast}(1780)\rightarrow\bar{K}\,\eta$ | $7.4\pm 0.5$ | $48\pm 22$
$K_{3}^{\ast}(1780)\rightarrow\bar{K}\,\eta^{\prime}(958)$ | $0.021\pm 0.001$ |
$\omega_{3}(1670)\rightarrow\bar{K}\,K$ | $3.0\pm 0.2$ |
$\phi_{3}(1850)\rightarrow\bar{K}\,K$ | $18.8\pm 1.3$ | seen

###  III.2 Decay process $W_{3}\rightarrow V_{1}+P$

The interaction Lagrangian for the vector and pseudoscalar decay mode reads

$\displaystyle\mathcal{L}_{w_{3}v_{1}p}=\vphantom{\bigg{(}\bigg{)}}$ (40)
$\displaystyle=\,$ $\displaystyle
g_{w_{3}v_{1}p}\,\varepsilon^{\mu\nu\rho\sigma}\,\mathrm{tr}\big{[}W_{3,\mu\alpha\beta}\,\big{\\{}(\partial_{\nu}V_{1,\rho}),\,(\partial^{\alpha}\partial^{\beta}\partial_{\sigma}P)\big{\\}}_{+}\big{]}\,.\vphantom{\bigg{(}\bigg{)}}$

An extended version is given by Eq. (127) in App. B. In this case, the tree
level decay rate formula has the form

$\displaystyle\Gamma_{W_{3}\rightarrow
V_{1}+P}(m_{w_{3}},m_{v_{1}},m_{p})=\vphantom{\Bigg{(}\Bigg{)}}$ (41)
$\displaystyle=\,$ $\displaystyle
g_{w_{3}v_{1}p}^{2}\,\frac{\big{|}\vec{k}_{v_{1},p}\big{|}^{7}}{105\uppi}\,\kappa_{i}\,\Theta(m_{w_{3}}-m_{v_{1}}-m_{p})\,,\vphantom{\Bigg{(}\Bigg{)}}$

see also Eq. (151), where the factors $\kappa_{i}$ are reported in Table 30.
In order to define the coupling constant we proceed as in the pseudoscalar-
pseudoscalar case. We use

$\displaystyle\Gamma^{\text{exp}}_{\rho_{3}(1690)\rightarrow\omega(782)\,\pi}=\,$
$\displaystyle(25.8\pm 9.8)\,\,\text{MeV}\,,\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle\Gamma^{\text{exp}}_{K_{3}^{\ast}(1780)\rightarrow\rho(770)\,K}=\,$
$\displaystyle(49.3\pm 15.7)\,\,\text{MeV}\,,\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle\Gamma^{\text{exp}}_{K_{3}^{\ast}(1780)\rightarrow\bar{K}^{\ast}(892)\,\pi}=\,$
$\displaystyle(31.8\pm 9.0)\,\,\text{MeV}\,.\vphantom{\bigg{(}\bigg{)}}$

The coupling constant and its error are

$\displaystyle
g_{w_{3}v_{1}p}^{2}=\frac{\sum_{i=1}^{3}\frac{\tilde{g}_{i}}{\Delta\tilde{g}_{i}^{2}}}{\sum_{j=1}^{3}\frac{1}{\Delta\tilde{g}_{j}^{2}}}\,,$
$\displaystyle\Delta
g_{w_{3}v_{1}p}^{2}=\sqrt{\frac{1}{\sum_{j=1}^{3}\frac{1}{\Delta
g_{j}^{2}}}}\,,$ (42)

hence

$\displaystyle g_{w_{3}v_{1}p}^{2}=(9.2\pm 1.9)\cdot
10^{-16}\,\,\text{MeV}^{-6}$ (43)

This value leads to the results listed in Table 6. We observe that an
acceptable agreement is reached, although the
$\Gamma_{K_{3}^{\ast}(1780)\rightarrow\rho(770)\,K}$ mode is theoretically
underestimated (the experimental error is nevertheless large). Quite
remarkably, the two theoretically sizable and dominant decays
$\omega_{3}(1670)\rightarrow\rho(770)\,K$ and $\phi_{3}(1850)\rightarrow
K^{\ast}(892)\,K$ have been indeed seen in experiments, though they could not
be quantified. Their future determination would represent a test of our
approach – at least on a qualitative level.

Table 6:  Decays of $J^{PC}=3^{--}$ mesons into a pseudoscalar-vector pair. Experimental data taken from Ref. [1].  decay process | theory | experiment
---|---|---
$\Gamma/$MeV |  $\Gamma/$MeV
$\rho_{3}(1690)\rightarrow\rho(770)\,\eta$ | $3.8\pm 0.8$ | seen
$\rho_{3}(1690)\rightarrow\bar{K}^{\ast}(892)\,K$ | $3.4\pm 0.7$ |
$\rho_{3}(1690)\rightarrow\omega(782)\,\pi$ | $35.8\pm 7.4$ | $25.8\pm 9.8$
$\rho_{3}(1690)\rightarrow\phi(1020)\,\pi$ | $0.036\pm 0.007$ |
$K_{3}^{\ast}(1780)\rightarrow\rho(770)\,K$ | $16.8\pm 3.5$ | $49.3\pm 15.7$
$K_{3}^{\ast}(1780)\rightarrow\bar{K}^{\ast}(892)\,\pi$ | $27.2\pm 5.6$ | $31.8\pm 9.0$
$K_{3}^{\ast}(1780)\rightarrow\bar{K}^{\ast}(892)\,\eta$ | $0.09\pm 0.02$ |
$K_{3}^{\ast}(1780)\rightarrow\omega(782)\,\bar{K}$ | $4.3\pm 0.9$ |
$K_{3}^{\ast}(1780)\rightarrow\phi(1020)\,\bar{K}$ | $1.2\pm 0.3$ |
$\omega_{3}(1670)\rightarrow\rho(770)\,\pi$ | $97\pm 20$ | seen
$\omega_{3}(1670)\rightarrow\bar{K}^{\ast}(892)\,K$ | $2.9\pm 0.6$ |
$\omega_{3}(1670)\rightarrow\omega(782)\,\eta$ | $2.8\pm 0.6$ |
$\omega_{3}(1670)\rightarrow\phi(1020)\,\eta$ | $(7.6\pm 1.6)\cdot 10^{-6}$ |
$\phi_{3}(1850)\rightarrow\rho(770)\,\pi$ | $1.1\pm 0.2$ |
$\phi_{3}(1850)\rightarrow\bar{K}^{\ast}(892)\,K$ | $35.5\pm 7.3$ | seen
$\phi_{3}(1850)\rightarrow\omega(782)\,\eta$ | $0.015\pm 0.003$ |
$\phi_{3}(1850)\rightarrow\omega(782)\,\eta^{\prime}(958)$ | $0.003\pm 0.001$ |
$\phi_{3}(1850)\rightarrow\phi(1020)\,\eta$ | $3.8\pm 0.8$ |

In addition, we present the following ratio

$\displaystyle\frac{\Gamma_{\phi_{3}(1850)\rightarrow\bar{K}^{\ast}(892)\,K}}{\Gamma_{\phi_{3}(1850)\rightarrow\bar{K}\,K}}=1.9\pm
0.4\,,$ (44)

which does not totally contradict the PDG [1] average taken from Ref. [100],

$\displaystyle\frac{\Gamma^{\text{exp}}_{\phi_{3}(1850)\rightarrow\bar{K}^{\ast}(892)\,K}}{\Gamma^{\text{exp}}_{\phi_{3}(1850)\rightarrow\bar{K}\,K}}=0.55_{-0.45}^{+0.85}\,.$
(45)

Interestingly Ref. [101] even reports a slightly larger ratio of

$\displaystyle\frac{\Gamma^{\text{exp}}_{\phi_{3}(1850)\rightarrow\bar{K}^{\ast}(892)\,K}}{\Gamma^{\text{exp}}_{\phi_{3}(1850)\rightarrow\bar{K}\,K}}=0.8\pm
0.4\,.$ (46)

Furthermore, we would like to mention a recent lattice QCD study [22], which
also confirms our overall predictions on dominant and less dominant vector-
pseudoscalar decay channels. Reference [22] predicts (without providing
explicit errors due to large uncertainties),

$\displaystyle\Gamma^{\text{J,D}}_{\rho_{3}(1690)\rightarrow\omega(782)\,\pi}=\,$
$\displaystyle 22\,\,\text{MeV}\,,\vphantom{\bigg{(}\bigg{)}}$ (47)
$\displaystyle\Gamma^{\text{J,D}}_{\rho_{3}(1690)\rightarrow\bar{K}^{\ast}(892)\,K}=\,$
$\displaystyle 2\,\,\text{MeV}\,,\vphantom{\bigg{(}\bigg{)}}$ (48)
$\displaystyle\Gamma^{\text{J,D}}_{\omega_{3}(1670)\rightarrow\rho(770)\,\pi}=\,$
$\displaystyle 62\,\,\text{MeV}\,,\vphantom{\bigg{(}\bigg{)}}$ (49)
$\displaystyle\Gamma^{\text{J,D}}_{\omega_{3}(1670)\rightarrow\bar{K}^{\ast}(892)\,K}=\,$
$\displaystyle 2\,\,\text{MeV}\,,\vphantom{\bigg{(}\bigg{)}}$ (50)
$\displaystyle\Gamma^{\text{J,D}}_{\omega_{3}(1670)\rightarrow\omega(782)\,\eta}=\,$
$\displaystyle 1\,\,\text{MeV}\,,\vphantom{\bigg{(}\bigg{)}}$ (51)
$\displaystyle\Gamma^{\text{J,D}}_{\phi_{3}(1850)\rightarrow\bar{K}^{\ast}(892)\,K}=\,$
$\displaystyle 20\,\,\text{MeV}\,,\vphantom{\bigg{(}\bigg{)}}$ (52)
$\displaystyle\Gamma^{\text{J,D}}_{\phi_{3}(1850)\rightarrow\phi(1020)\,\eta}=\,$
$\displaystyle 3\,\,\text{MeV}\,.\vphantom{\bigg{(}\bigg{)}}$ (53)

It is quite remarkable that our (quite simple) model is able to predict at
least qualitatively rather similar results to such an advanced and
comprehensive lattice QCD study. This leads us to the conclusion, that even
for high-spin mesons with masses above $1\,\,$GeV like the conventional
$3^{--}$-nonet chiral symmetry (breaking) is the decisive guiding principle
for their phenomenology.

###  III.3 Decay process $W_{3}\rightarrow\gamma+P$

As a next step, we also present the results for the radioactive decays
$W_{3}\rightarrow\gamma\,P$, where $\gamma$ represents the photon. These can
be obtained by using vector meson dominance [102, 103, 104], which takes into
account of the photon-vector-meson mixing through the shift

$\displaystyle V_{\mu\nu}\mapsto
V_{\mu\nu}+\tfrac{eQ}{g_{\rho}}\,F_{\mu\nu}\,,$ (54)

where $V_{\mu\nu}\equiv\partial_{\mu}V_{\nu}-\partial_{\nu}V_{\mu}$ and
$Q=\mathrm{diag}\big{(}\frac{2}{3},-\frac{1}{3},\,-\frac{1}{3}\big{)}$ is the
charge quark matrix, which includes the charges of the up, down, and strange
quark. The electromagnetic field tensor is denoted as
$F_{\mu\nu}=\partial_{\mu}a_{\nu}-\partial_{\nu}a_{\mu}$, with $a_{\mu}$ being
the photon field, while $e=\sqrt{4\uppi\,\alpha}$ is the electric coupling
constant, and $g_{\rho}\simeq 5.5$ parametrizes the photon-vector-meson
transition. By applying the shift of Eq. (54) into Eq. (40) we obtain the
Lagrangian for radioactive decays

$\displaystyle\mathcal{L}_{w_{3}\gamma p}=\vphantom{\bigg{(}\bigg{)}}$ (55)
$\displaystyle=\,$ $\displaystyle
g_{w_{3}v_{1}p}\,\tfrac{e}{g_{\rho}}\,\varepsilon^{\mu\nu\rho\sigma}\,(\partial_{\nu}a_{\rho})\,\mathrm{tr}\big{[}W_{3,\mu\alpha\beta}\,\big{\\{}Q,\,(\partial^{\alpha}\partial^{\beta}\partial_{\sigma}P)\big{\\}}_{+}\big{]}\,.\vphantom{\bigg{(}\bigg{)}}$

The extended version of this Lagrangian is provided by Eq. (128) in App. B.
For the tree level decay rate formula, we obtain via Eq. (154),

$\displaystyle\Gamma_{W_{3}\rightarrow\gamma+P}(m_{w_{3}},m_{p})=\,$
$\displaystyle
g_{w_{3}v_{1}p}^{2}\,\big{(}\tfrac{e}{g_{\rho}}\big{)}^{2}\,\frac{\big{|}\vec{k}_{\gamma,p}\big{|}^{7}}{105\pi}\,\kappa_{i}^{\gamma}\,.\vphantom{\Bigg{(}\Bigg{)}}$
(56)

where one of the masses $m_{a}$ and $m_{b}$ in Eq. (24) is set to zero,
because the photon is massless. The factors $\kappa_{i}^{\gamma}$ are listed
in Table 20. (The Heaviside function is not needed, because all pseudoscalars
are lighter than the conventional spin-$3$ mesons and the photons can be
arbitrarily soft.)

Various predictions for the radiative decays $W_{3}\rightarrow\gamma\,P$ are
calculated and presented in Table 7. Because of the rather large errors in the
coupling constant (43), we round all results to integers in units of keV.
Quite large radiative decay channels are
$\omega_{3}(1670)\rightarrow\gamma\,\pi^{0}$ as well as
$K_{3}^{0}(1780)\rightarrow\gamma\,K^{0}$.

In general, these processes imply that a photoproduction of mesons with spin
$J^{PC}=3^{--}$ can take place at the ongoing GlueX [24, 25, 26] and CLAS12
[27] experiments at Jefferson Lab.

Table 7:  Theoretical predictions for the radiative decays $W_{3}\rightarrow\gamma\,P$.  decay process | theory
---|---
$\Gamma/$keV
$\rho_{3}^{\pm/0}(1690)\rightarrow\gamma\,\pi^{\pm/0}$ | $69\pm 14$
$\rho_{3}^{0}(1690)\rightarrow\gamma\,\eta$ | $157\pm 32$
$\rho_{3}^{0}(1690)\rightarrow\gamma\,\eta^{\prime}(958)$ | $20\pm 4$
$K_{3}^{\pm}(1780)\rightarrow\gamma\,K^{\pm}$ | $58\pm 12$
$K_{3}^{0}(1780)\rightarrow\gamma\,K^{0}$ | $231\pm 48$
$\omega_{3}(1670)\rightarrow\gamma\,\pi^{0}$ | $560\pm 120$
$\omega_{3}(1670)\rightarrow\gamma\,\eta$ | $19\pm 4$
$\omega_{3}(1670)\rightarrow\gamma\,\eta^{\prime}(958)$ | $1.4\pm 0.3$
$\phi_{3}(1850)\rightarrow\gamma\,\pi^{0}$ | $4\pm 1$
$\phi_{3}(1850)\rightarrow\gamma\,\eta$ | $129\pm 26$
$\phi_{3}(1850)\rightarrow\gamma\,\eta^{\prime}(958)$ | $35\pm 7$

###  III.4 Decay process $W_{3}\rightarrow A_{2}+P$

The interaction Lagrangian describing the decay of the spin-$3$ tensor mesons
into a spin-$2$ tensor and a pseudoscalar mesons has the following form

$\displaystyle\mathcal{L}_{w_{3}a_{2}p}=\vphantom{\bigg{(}\bigg{)}}$ (57)
$\displaystyle=\,$ $\displaystyle
g_{w_{3}a_{2}p}\,\varepsilon_{\mu\nu\rho\sigma}\,\mathrm{tr}\big{[}\tensor{W}{{}_{3}^{\mu}{}_{\alpha}{}_{\beta}}\,\big{[}(\partial^{\nu}A_{2}^{\rho\alpha}),\,(\partial^{\sigma}\partial^{\beta}P)\big{]}_{-}\big{]}\,.\vphantom{\bigg{(}\bigg{)}}$

The extended version is provided in App. B in Eq. (129). Using Eq. (157) the
corresponding decay formula is

$\displaystyle\Gamma_{W_{3}\rightarrow
A_{2}+P}(m_{w_{3}},m_{a_{2}},m_{p})=\vphantom{\Bigg{(}\Bigg{)}}$ (58)
$\displaystyle=\,$ $\displaystyle
g_{w_{3}a_{2}p}^{2}\,\frac{\big{|}\vec{k}_{a_{2},p}\big{|}^{5}}{420\uppi\,m_{a_{2}}^{2}}\,\big{(}2\,\big{|}\vec{k}_{a_{2},p}\big{|}^{2}+7\,m_{a_{2}}^{2}\big{)}\,\kappa_{i}\times\vphantom{\Bigg{(}\Bigg{)}}$
$\displaystyle\times\Theta(m_{w_{3}}-m_{a_{2}}-m_{p})\,,\vphantom{\Bigg{(}\Bigg{)}}$

where the $\kappa_{i}$ are listed in Table 21. For this channel, there are no
experimental reported branching ratios that allow for a direct determination
of the coupling constant. Yet, we can use the experimental ratio listed by the
PDG [1] and taken from Ref. [105],

$\displaystyle\frac{\Gamma^{\text{exp}}_{\rho_{3}(1690)\rightarrow
a_{2}(1320)\,\pi}}{\Gamma^{\text{exp}}_{\rho_{3}(1690)\rightarrow\rho(770)\,\eta}}=5.5\pm
2.0\,.$ (59)

together with the previously determined theoretical value
$\Gamma_{\rho_{3}(1690)\rightarrow\rho(770)\,\eta}=(3.8\pm 0.8)\,\,$MeV
reported in Table 6. The corresponding value for the coupling constant is

$\displaystyle g_{w_{3}a_{2}p}^{2}=(2.8\pm 1.2)\cdot
10^{-9}\,\,\text{MeV}^{-4}\,.$ (60)

Once the coupling constant is fixed, we get the results for the decay rates
reported in Table 8. The decay width
$\Gamma_{K_{3}^{\ast}(1780)\rightarrow\bar{K}^{\ast}_{2}(1430)\,\pi}$ is
safely smaller the experimental upper limit [1, 106]. Moreover, the quite
large mode $\rho_{3}(1690)\rightarrow a_{2}(1320)\,\pi$ is seen in
experiments, but also in this case no branching ratio is listed in the PDG
[1].

Table 8:  Decays of $J^{PC}=3^{--}$ mesons into a pseudoscalar-tensor pair. Experimental data taken from Ref. [1].  decay process | theory | experiment
---|---|---
$\Gamma/$MeV |  $\Gamma/$MeV
$\rho_{3}(1690)\rightarrow a_{2}(1320)\,\pi$ | $20.9\pm 8.7$ | seen
$K_{3}^{\ast}(1780)\rightarrow\bar{K}^{\ast}_{2}(1430)\,\pi$ | $5.8\pm 2.4$ | $<25.4\pm 3.4$
$K_{3}^{\ast}(1780)\rightarrow f_{2}(1270)\,\bar{K}$ | $(5.4\pm 2.2)\cdot 10^{-5}$ |

###  III.5 Decay process $W_{3}\rightarrow B_{1}+P$

The interaction Lagrangian describing the decay into a pseudovector and
pseudoscalar meson is given by

$\displaystyle\mathcal{L}_{w_{3}b_{1}p}=g_{w_{3}b_{1}p}\,\mathrm{tr}\big{[}W_{3}^{\mu\nu\rho}\,\big{\\{}B_{1,\mu},\,(\partial_{\nu}\partial_{\rho}P)\big{\\}}_{+}\big{]}\,,$
(61)

where the extended form is given in Eq. (130) in App. B. The tree level decay
rate takes the form,

$\displaystyle\Gamma_{W_{3}\rightarrow
B_{1}+P}(m_{w_{3}},m_{b_{1}},m_{p})=\vphantom{\Bigg{(}\Bigg{)}}$ (62)
$\displaystyle=\,$ $\displaystyle
g_{w_{3}b_{1}p}^{2}\,\frac{|\vec{k}_{b_{1},p}|^{5}}{420\uppi\,m_{w_{3}}^{2}}\,\Bigg{(}7+3\,\frac{\big{|}\vec{k}_{b_{1},p}\big{|}^{2}}{m_{b_{1}}^{2}}\Bigg{)}\,\kappa_{i}\times\vphantom{\Bigg{(}\Bigg{)}}$
$\displaystyle\times\Theta(m_{w_{3}}-m_{b_{1}}-m_{p})\,.\vphantom{\Bigg{(}\Bigg{)}}$

Here, we use Eq. (162) and the $\kappa_{i}$ can be found in Table 22. Also in
this case, a direct determination of the coupling constant $g_{w_{3}b_{1}p}$
is not possible due to the lack of experimental information. In order to
estimate it, we proceed as it follows.

  1. 1.

According to the PDG [1], there exists a lower limit for the ratio

$\displaystyle\frac{\Gamma_{\omega_{3}(1670)\rightarrow
b_{1}(1235)\,\pi}}{\Gamma_{\omega_{3}(1670)\rightarrow\omega(782)\,\pi\,\pi}}>0.75\,.$
(63)

The PDG [1] extracts this lower bound for the ratio from Ref. [107] and does
not list this bound as confirmed data. In Ref. [107] the authors in fact
report

$\displaystyle\frac{\Gamma_{\omega_{3}(1670)\rightarrow
b_{1}(1235)\,\pi}}{\Gamma_{\omega_{3}(1670)\rightarrow\omega(782)\,\pi\,\pi}}=1.0_{-0.25}^{+0.0}\,.$
(64)

They further argue that the decay mode $\omega_{3}(1670)\rightarrow
b_{1}(1235)\,\pi$ is the dominant contribution to the $\omega(782)\,\pi\,\pi$
final state via the subsequent decay
$b_{1}(1235)\,\rightarrow\omega(782)\,\pi$. They claim that one can assume
that

$\displaystyle\Gamma_{\omega_{3}(1670)\rightarrow
b_{1}(1235)\,\pi}\approx\Gamma_{\omega_{3}(1670)\rightarrow\omega(782)\,\pi\,\pi}$
(65)

and deviations are expected to be smaller than $10\%$.

  2. 2.

Based on this assumption we use the following approximation

$\displaystyle\frac{\Gamma_{\omega_{3}(1670)\rightarrow
b_{1}(1235)\,\pi}}{\Gamma_{\omega_{3}(1670)\rightarrow\rho(770)\,\pi}}\overset{\text{Eq.\
\eqref{saturation}}}{\approx}$ (66) $\displaystyle\overset{\text{Eq.\
\eqref{saturation}}}{\approx}\,$
$\displaystyle\frac{\Gamma_{\omega_{3}(1670)\rightarrow\omega\,\pi\,\pi}}{\Gamma_{\omega_{3}(1670)\rightarrow\rho(770)\,\pi}}\overset{\text{PDG}}{=}0.71\pm
0.27\,.$

Here, one should mention that the PDG [1] cites Ref. [108] for the second
ratio, but still excludes it from their confirmed data. Additionally, Ref.
[108] even lists a different branching ratio,

$\displaystyle\frac{\Gamma_{\omega_{3}(1670)\rightarrow\omega\,\pi\,\pi}}{\Gamma_{\omega_{3}(1670)\rightarrow\rho(770)\,\pi}}=0.47\pm
0.18\,,$ (67)

as well as

$\displaystyle\frac{\Gamma_{\omega_{3}(1670)\rightarrow
b_{1}(1235)\,\pi}}{\Gamma_{\omega_{3}(1670)\rightarrow\rho(770)\,\pi}}=0.32\pm
0.16\,,$ (68)

It is not clear to the authors of this work, how the PDG [1] extracted their
ratio from Ref. [108]. Nevertheless, we will stick to the data reported in the
PDG [1] and assume that there was some reasonable reanalysis of the data of
Ref. [108]. In this context, it might be worth mentioning, that all
experimental data concerning the $\omega_{3}(1670)$ is rather old and its mass
and total width could not be determined up to high accuracy at the time when
Refs. [107, 108] were published. Definitely, there is a need for future
experimental investigations of this state.

For what concerns our work, we use the result
$\Gamma_{\omega_{3}(1670)\rightarrow\rho(770)\,\pi}=(97\pm 20)\,\,$MeV
presented in the previous section as well as the ratio (66). We obtain the
following estimate for the coupling constant

$\displaystyle g_{w_{3}b_{1}p}^{2}\approx(0.008\pm
0.003)\,\,\text{MeV}^{-2}\,.$ (69)

We are aware that this value and the corresponding results, which are listed
in Table 9, are only first rough estimates. Still, they might again help
identifying dominant decay channels. Additionally, we consider the following
aspects for the determination and interpretation of our results.

Table 9:  Decays of $J^{PC}=3^{--}$ mesons into a pseudoscalar-pseudovector pair obtained by using the mixing angle $\beta_{b_{1}}=0$ and $\beta_{b_{1}}=-40^{\circ}$, respectively. No experimental value is listed in the PDG [1]. The decay channel $\omega_{3}(1670)\rightarrow b_{1}(1235)\,\pi$ is possibly seen.  decay process | theory for | theory for
---|---|---
$\beta_{b_{1}}=0$ | $\beta_{b_{1}}=-40^{\circ}$
$\Gamma/$MeV |  $\Gamma/$MeV
$\rho_{3}(1690)\rightarrow h_{1}(1170)\,\pi$ | $53\pm 23$ | $31\pm 13$
$\rho_{3}(1690)\rightarrow h_{1}(1415)\,\pi$ | $0$ | $0.73\pm 0.31$
$K_{3}^{\ast}(1780)\rightarrow b_{1}(1235)\,K$ | $0.55\pm 0.24$ | $0.55\pm 0.24$
$K_{3}^{\ast}(1780)\rightarrow\bar{K}_{1,B}\,\pi$ | $37\pm 16$ | $37\pm 16$
$K_{3}^{\ast}(1780)\rightarrow h_{1}(1170)\,\bar{K}$ | $1.61\pm 0.70$ | $0.03\pm 0.01$
$\omega_{3}(1670)\rightarrow b_{1}(1235)\,\pi$ | $69\pm 30$ | $69\pm 30$
$\phi_{3}(1850)\rightarrow b_{1}(1235)\,\pi$ | $1.17\pm 0.51$ | $1.17\pm 0.51$
$\phi_{3}(1850)\rightarrow\bar{K}_{1,B}\,K+\mathrm{c}.\mathrm{c}.$ | $9.1\pm 3.9$ | $9.1\pm 3.9$
$\phi_{3}(1850)\rightarrow h_{1}(1170)\,\eta$ | $0.006\pm 0.003$ | $1.1\pm 0.5$

  1. 1.

We consider $K_{1,B}\approx K_{1}(1270)$, since this is the dominant
contribution; then, the decay rate
$\Gamma_{\phi_{3}(1850)\rightarrow\bar{K}_{1,B}\,K}$ is rather small. A
disclaimer is in order: due to the large mixing between $K_{1,A}$ and
$K_{1,B}$ the results involving the identification $K_{1,B}\approx
K_{1}(1270)$ can be only considered as a first approximation. The more correct
procedure should be to consider the full mixing in Eq. (17), hence $K_{1,B}$
should be expressed as a superposition of $K_{1}(1270)$ and $K_{1}(1400)$.
Yet, this is not an easy task, since the interaction in Eq. (61) alone is not
enough. One should also take into account the interaction terms for the decays
$W_{3}\rightarrow A_{1}\,P$ in Eq. (70), which includes $K_{1,A}$, see next
section. Then, the decays into $K_{1}(1270)$ and $K_{1}(1400)$ should be
calculated by the joint Lagrangians (61) and (70) together with the mixing
(17). Furthermore, interference between the two Lagrangians is expected. This
calculation is however not possible, since the coupling constant
$g_{w_{3}a_{1}p}$ cannot be determined by the present experimental data (see
next subsection). Hence, we must limit our study to the dominant assignment
$K_{1,B}\approx K_{1}(1270)$ (in this subsection) and $K_{1,A}\approx
K_{1}(1400)$ (in the next subsection).

  2. 2.

We present the results in Table 9 for two values of the unknown mixing angle
in the isoscalar sector. In one case, we assume that the mixing angle
vanishes, $\beta_{b_{1}}\approx 0$, in the second we study
$\beta_{b_{1}}\approx-40^{\circ}$, which is a quite large and negative value
similar to the mixing angle for the pseudoscalar-isoscalars. Note, the mixing
angle $\beta_{b_{1}}$ appears in the decay channels $\rho_{3}(1690)\rightarrow
h_{1}(1170)\,\pi$, $\rho_{3}(1690)\rightarrow h_{1}(1415)\,\pi$,
$K_{3}^{\ast}(1780)\rightarrow h_{1}(1170)\,\bar{K}$, and
$\phi_{3}(1850)\rightarrow h_{1}(1170)\,\eta$, see Table 22. The only
experimentally possibly seen decay $\omega_{3}(1670)\rightarrow
b_{1}(1235)\,\pi$, see Ref. [108], corresponds to a quite large theoretical
partial decay widths and is independent of any assumption on $\beta_{b_{1}}$.

  3. 3.

The decay $\rho_{3}(1690)\rightarrow h_{1}(1170)\,\pi$ is also quite large for
both choices of the mixing angle, hence it is a potentially interesting
channel for future search. Moreover, it is also an interesting decay channel
in order to determine the value of the mixing angle $\beta_{b_{1}}$. The
remaining decay channels are pretty small, which might be an explanation why
they could not be observed in experiment.

###  III.6 Decay process $W_{3}\rightarrow A_{1}+P$

The interaction Lagrangian and the decay formula for the decay into an axial-
vector and a pseudoscalar meson are given by

$\displaystyle\mathcal{L}_{w_{3}a_{1}p}=g_{w_{3}a_{1}p}\,\mathrm{tr}\big{[}W_{3}^{\mu\nu\rho}\,\big{[}A_{1,\mu},\,(\partial_{\nu}\partial_{\rho}P)\big{]}_{-}\big{]}\,,$
(70)

and

$\displaystyle\Gamma_{W_{3}\rightarrow
A_{1}+P}(m_{w_{3}},m_{a_{1}},m_{p})=\vphantom{\Bigg{(}\Bigg{)}}$ (71)
$\displaystyle=\,$ $\displaystyle
g_{w_{3}a_{1}p}^{2}\,\frac{\big{|}\vec{k}_{a_{1},p}\big{|}^{5}}{420\uppi\,m_{w_{3}}^{2}}\,\Bigg{(}7+3\,\frac{\big{|}\vec{k}_{a_{1},p}\big{|}^{2}}{m_{a_{1}}^{2}}\Bigg{)}\,\kappa_{i}\times\vphantom{\Bigg{(}\Bigg{)}}$
$\displaystyle\times\Theta(m_{w_{3}}-m_{a_{1}}-m_{p})\,.\vphantom{\Bigg{(}\Bigg{)}}$

The extended version of the Lagrangian is given by Eq. (131) in App. B. The
$\kappa_{i}$-values for each channel are taken from Table 23 and the decay
formula was derived by using Eq. (163). As explained previously, for a first
rough estimate, we assume here that $K_{1,A}\approx K_{1}(1400)$.

Since we do not have enough experimental information for obtaining the
coupling constant $g_{w_{3}a_{1}p}$, we can only get some theoretical
predictions for ratios among different decay channels, that are reported in
Table 10. Note, the mixing angle $\beta_{a_{1}}$ does not appear in any of the
nonzero reported decay channel, as it can be read from Table 23. Anyway, the
mixing is expected to be small [84].

Table 10:  Predictions for the nonzero branching ratios of $W_{3}\rightarrow A_{1}\,P$. We choose as a reference the $a_{1}(1260)\,\pi$ decay mode.  branching ratio | theory
---|---
$\frac{\Gamma_{K_{3}^{\ast}(1780)\rightarrow K_{1,A}\,\pi}}{\Gamma_{\rho_{3}(1690)\rightarrow a_{1}(1260)\,\pi}}$ | $0.12$
$\frac{\Gamma_{K_{3}^{\ast}(1780)\rightarrow a_{1}(1260)\,K}}{\Gamma_{\rho_{3}(1690)\rightarrow a_{1}(1260)\,\pi}}$ | $0.01$

In the end, it is interesting to mention that there might be an option to size
the coupling constant by linking the present model to an underlying chiral
model, e.g. the eLSM. This approximately predicts $2\,\,$MeV for
$\rho_{3}(1690)\rightarrow a_{1}(1260)\,\pi$, see Chap. V for an estimate of
$g_{w_{3}a_{1}p}$. It seems therefore that the decays of the type
$W_{3}\rightarrow A_{1}+P$ are suppressed.

###  III.7 Decay process $W_{3}\rightarrow V_{1}+V_{1}$

The interaction Lagrangian describing the decay into two vector mesons is
given by

$\displaystyle\mathcal{L}_{w_{3}v_{1}v_{1}}=g_{w_{3}v_{1}v_{1}}\,\mathrm{tr}\big{[}W_{3}^{\mu\nu\rho}\,\big{[}(\partial_{\mu}V_{1,\nu}),\,V_{1,\rho}\big{]}_{-}\big{]}\,,$
(72)

see Eq. (132) in App. B for the extended version . The corresponding decay
rate reads

$\displaystyle\Gamma_{W_{3}\rightarrow
V_{1}^{(1)}+V_{1}^{(2)}}(m_{w_{3}},m_{v_{1}^{(1)}},m_{v_{1}^{(2)}})=\vphantom{\Bigg{(}\Bigg{)}}$
(73) $\displaystyle=\,$ $\displaystyle
g_{w_{3}v_{1}v_{1}}^{2}\,\frac{\big{|}\vec{k}_{v_{1}^{(1)},v_{1}^{(2)}}\big{|}^{3}}{840\uppi\,m_{v_{1}^{(1)}}^{2}\,m_{v_{1}^{(2)}}^{2}\,m_{w_{3}}^{2}}\,\Big{[}6\,|\vec{k}_{v_{1}^{(1)},v_{1}^{(2)}}\big{|}^{4}+\vphantom{\Bigg{(}\Bigg{)}}$
$\displaystyle+35\,m_{v_{1}^{(1)}}^{2}\,m_{v_{1}^{(2)}}^{2}+14\,\big{|}\vec{k}_{v_{1}^{(1)},v_{1}^{(2)}}\big{|}^{2}\,\big{(}m_{v_{1}^{(1)}}^{2}+m_{v_{1}^{(2)}}^{2}\big{)}\Big{]}\times\vphantom{\Bigg{(}\Bigg{)}}$
$\displaystyle\times\kappa_{i}\,\Theta(m_{w_{3}}-m_{v_{1}^{(1)}}-m_{v_{1}^{(2)}})\,,\vphantom{\Bigg{(}\Bigg{)}}$

where the $\kappa_{i}$’s are listed in Table 24. For the derivation of the
decay formula, we use Eq. (166). Also in this case, we cannot determine the
coupling constant, but we can present the branching ratios reported in Table
11.

Table 11:  Predictions for the branching ratios of $W_{3}\rightarrow V_{1}+V_{1}$. We choose as a reference the $\rho\,\rho$ decay mode.  branching ratio | theory
---|---
$\frac{\Gamma_{K_{3}^{\ast}(1780)\rightarrow\rho(770)\,K^{\ast}(892)}}{\Gamma_{\rho_{3}(1690)\rightarrow\rho(770)\,\rho(770)}}$ | $0.49$
$\frac{\Gamma_{K_{3}^{\ast}(1780)\rightarrow K^{\ast}(892)\,\omega(782)}}{\Gamma_{\rho_{3}(1690)\rightarrow\rho(770)\,\rho(770)}}$ | $0.17$
$\frac{\Gamma_{\phi_{3}(1850)\rightarrow K^{\ast}(892)\,K^{\ast}(892)}}{\Gamma_{\rho_{3}(1690)\rightarrow\rho(770)\,\rho(770)}}$ | $0.35$

Moreover, we can find an upper limit for the coupling constant by considering

$\displaystyle\frac{\Gamma_{\rho_{3}(1690)\rightarrow\pi\,\pi}}{\Gamma_{\rho_{3}(1690)\rightarrow\rho(770)\,\rho(770)}}\gtrapprox$
(74) $\displaystyle\gtrapprox\,$
$\displaystyle\frac{\Gamma_{\rho_{3}(1690)\rightarrow\pi\,\pi}}{\Gamma_{\rho_{3}(1690)\rightarrow\pi^{\pm}\,\pi^{+}\,\pi^{-}\,\pi^{0}}}\overset{\text{PDG}}{=}0.35\pm
0.11\,,$

which follows from the fact that the
$\rho_{3}(1690)\rightarrow\rho(770)\,\rho(770)$ mode is part of the
$\pi^{\pm}\,\pi^{+}\,\pi^{-}\,\pi^{0}$ decay mode (and eventually it is one of
its dominating contributors). Since the experimental value for
$\Gamma_{\rho_{3}(1690)\rightarrow\pi\,\pi}=(38.0\pm 3.2)\,\,$MeV is known, we
find

$\displaystyle g_{w_{3}v_{1}v_{1}}^{2}\lesssim 535\pm 174\,.$ (75)

Based on this information we can predict upper limits for the decay rates in
Table 12.

Table 12:  Upper limits for the decays of $J^{PC}=3^{--}$ mesons into a vector-vector pair. Two different estimates are used, see text.  decay process | theory using | theory using
---|---|---
(74) & (75) | (77) & (79)
$\Gamma/$MeV |  $\Gamma/$MeV
$\rho_{3}(1690)\rightarrow\rho(770)\,\rho(770)$ | $\lesssim 108.6\pm 35.3$ | $\lesssim 30$
$K_{3}^{\ast}(1780)\rightarrow\rho(770)\,\bar{K}^{\ast}(892)$ | $\lesssim 53.6\pm 17.4$ | $\lesssim 15$
$K_{3}^{\ast}(1780)\rightarrow\bar{K}^{\ast}(892)\,\omega(782)$ | $\lesssim 19.0\pm 6.2$ | $\lesssim 5$
$\phi_{3}(1850)\rightarrow\bar{K}^{\ast}(892)\,K^{\ast}(892)$ | $\lesssim 38.3\pm 12.5$ | $\lesssim 11$

The theoretically largest decay is
$\rho_{3}(1690)\rightarrow\rho(770)\,\rho(770)$, which has been experimentally
seen. Yet, the decay width of about $108\,\,$ MeV is surely too large, since
using a value close to this upper limit would imply that the the sum of all
the decays of the state $\rho_{3}(1690)$ would overshoot the experimental
total width of $(161\pm 10)\,\,$MeV. This is in agreement with the ratio

$\displaystyle\frac{\Gamma_{\rho_{3}(1690)\rightarrow\rho(770)\,\rho(770)}}{\Gamma_{\rho_{3}(1690)\rightarrow\pi^{\pm}\,\pi^{+}\,\pi^{-}\,\pi^{0}}}\,,$
(76)

which is smaller than $1$ for all experimental measurements [109, 110, 111,
112], which are listed by the PDG [1].

However, we can then obtain a second, more realistic estimate of these decay
channels by summing up the largest decay channels:

$\displaystyle\Gamma_{\rho_{3}}^{\text{tot}}=161\,\,\text{MeV}\approx\vphantom{\bigg{(}\bigg{)}}$
(77) $\displaystyle=\,$
$\displaystyle\Gamma_{\rho_{3}(1690)\rightarrow\pi\,\pi}+\Gamma_{\rho_{3}(1690)\rightarrow\bar{K}\,K}+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle+\Gamma_{\rho_{3}(1690)\rightarrow\rho(770)\,\eta}+\Gamma_{\rho_{3}(1690)\rightarrow\bar{K}^{\ast}(892)\,K}+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle+\Gamma_{\rho_{3}(1690)\rightarrow\omega(782)\,\pi}+\Gamma_{\rho_{3}(1690)\rightarrow\phi(1020)\,\pi}+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle+\Gamma_{\rho_{3}(1690)\rightarrow
a_{2}(1320)\,\pi}+\Gamma_{\rho_{3}(1690)\rightarrow
a_{1}(1260)\,\pi}+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle+\Gamma_{\rho_{3}(1690)\rightarrow
h_{1}(1170)\,\pi}+\Gamma_{\rho_{3}(1690)\rightarrow
h_{1}(1415)\,\pi}+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle+\Gamma_{\rho_{3}(1690)\rightarrow\rho(770)\,\rho(770)}+\Gamma_{\rho_{3}(1690)\rightarrow\text{other}}\,,\vphantom{\bigg{(}\bigg{)}}$

where $\Gamma_{\rho_{3}\rightarrow\text{other}}$ refers to all other decay
channels not listed above. Anyhow, these should be small. For
$\Gamma_{\rho_{3}(1690)\rightarrow h_{1}(1170)\,\pi}$ we use the smaller value
presented in Table 9, assuming a large mixing $\beta_{b_{1}}=-40^{\circ}$. In
this way we estimate the more reasonable upper limit

$\displaystyle\Gamma_{\rho_{3}(1690)\rightarrow\rho(770)\,\rho(770)}\lesssim
30\,\,\text{MeV}\,,$ (78)

from which follows,

$\displaystyle g_{w_{3}v_{1}v_{1}}^{2}\lesssim 148\,.$ (79)

The corresponding estimates for the other channels can be found in Table 12.

Following this estimate, we perform the sum of the predicted and sizable decay
channels for the other spin-$3$ mesons, obtaining:

$\displaystyle\Gamma^{\mathrm{sum}}_{K_{3}^{\ast}(1780)}\approx\,$
$\displaystyle 146\,\,\text{MeV}\,,\vphantom{\bigg{(}\bigg{)}}$ (80)
$\displaystyle\Gamma^{\mathrm{sum}}_{\omega_{3}(1670)}\approx\,$
$\displaystyle 175\,\,\text{MeV}\,,\vphantom{\bigg{(}\bigg{)}}$ (81)
$\displaystyle\Gamma^{\mathrm{sum}}_{\phi_{3}(1850)}\approx\,$ $\displaystyle
80\,\,\text{MeV}\,,\vphantom{\bigg{(}\bigg{)}}$ (82)

which are in agreement with the experimental values from the PDG [1],

$\displaystyle\Gamma_{K_{3}^{\ast}(1780)}^{\text{tot}}=\,$
$\displaystyle(159\pm 21)\,\,\text{MeV}\,,\vphantom{\bigg{(}\bigg{)}}$ (83)
$\displaystyle\Gamma_{\omega_{3}(1670)}^{\text{tot}}=\,$ $\displaystyle(168\pm
10)\,\,\text{MeV}\,,\vphantom{\bigg{(}\bigg{)}}$ (84)
$\displaystyle\Gamma_{\phi_{3}(1850)}^{\text{tot}}=\,$
$\displaystyle(87_{-23}^{+28})\,\,\text{MeV}\,.\vphantom{\bigg{(}\bigg{)}}$
(85)

These results demonstrate that the model is internally consistent.

##  IV Phenomenology of the $J^{PC}=3^{--}$ glueball

In this chapter, we study the branching ratios of the decays of a hypothetical
glueball with $J^{PC}=3^{--}$. Similar to all other bound states of gluons,
this glueball is not yet experimentally detected. However, lattice QCD
calculations in the quenched approximation predict its mass of approximately
$4.13\,\,$GeV [3], $4.33\,\,$GeV [4], or $4.20\,\,$GeV [5]. For general works
on glueballs, see e.g. Refs. [113, 114, 115, 116]. For model approaches to the
phenomenology of glueballs, which are similar to this one, see for example
previous publications by some of the authors [33, 37, 56, 79, 117].

Before we start our discussion, we explicitly state that all branching ratios
should be considered as first indicative results.

###  IV.1 The effective model for glueball decays

The Lagrangian describing the decays of this glueball can be obtained by
making use of the fact that each glueball is a flavor singlet – an object,
which is invariant under $SU_{\mathrm{V}}(3)$ transformations. Hence, we can
perform the simple replacement

$\displaystyle W^{\mu\nu\rho}\rightarrow
G_{3}^{\mu\nu\rho}\cdot\openone_{3\times 3}$ (86)

in the effective interaction terms, where $\openone_{3\times 3}$ is the
flavor-space-identity matrix and $G_{3}^{\mu\nu\rho}$ is the glueball field
[33, 117]. As a consequence, the interaction terms that come with a commutator
vanish and the only nonzero contributions to a residual IR-effective action
for the $3^{--}$ glueball can be derived from the former interaction terms
involving the anticommutator. Thus,

$\displaystyle\Gamma_{G_{3}(4200)\rightarrow V_{1}+P}\neq 0\,,$
$\displaystyle\Gamma_{G_{3}(4200)\rightarrow B_{1}+P}\neq 0\,.$ (87)

The corresponding Lagrangian terms are reported in Table 13.

Table 13:  Strong interaction Lagrangian terms for the hypothetical glueball state $G_{3}(4200)$ [in comparison with Eq. (22)].  decay mode | interaction Lagrangians
---|---
$G_{3}\rightarrow V_{1}+P$ | $\mathcal{L}_{g_{3}v_{1}p}=c_{g_{3}v_{1}p}\,G_{3,\mu\alpha\beta}\,\varepsilon^{\mu\nu\rho\sigma}\,\mathrm{tr}\big{[}\big{\\{}(\partial_{\nu}V_{1,\rho}),\,(\partial^{\alpha}\partial^{\beta}\partial_{\sigma}P)\big{\\}}_{+}\big{]}$
$G_{3}\rightarrow V_{1}+A_{1}$ | $\mathcal{L}_{g_{3}v_{1}a_{1}}=c_{g_{3}v_{1}a_{1}}\,G_{3,\mu\alpha\beta}\,\varepsilon^{\mu\nu\rho\sigma}\,\mathrm{tr}\big{[}\big{\\{}(\partial_{\nu}V_{1,\rho}),\,(\partial^{\alpha}\partial^{\beta}A_{1,\sigma})\big{\\}}_{+}\big{]}$
$G_{3}\rightarrow B_{1}+P$ | $\mathcal{L}_{g_{3}b_{1}p}=c_{g_{3}b_{1}p}\,G_{3}^{\mu\nu\rho}\,\mathrm{tr}\big{[}\big{\\{}B_{1,\mu},\,(\partial_{\nu}\partial_{\rho}P)\big{\\}}_{+}\big{]}$
$G_{3}\rightarrow B_{1}+A_{1}$ | $\mathcal{L}_{g_{3}b_{1}a_{1}}=c_{g_{3}b_{1}a_{1}}\,G_{3}^{\mu\nu\rho}\,\mathrm{tr}\big{[}\big{\\{}B_{1,\mu},\,(\partial_{\nu}A_{1,\rho})\big{\\}}_{+}\big{]}$

In addition, there are also decays of the type $G_{3}(4200)\rightarrow
V_{1}+A_{1}$ and $G_{3}(4200)\rightarrow B_{1}+A_{1}$ that were not included
in for the lightest conventional mesons with $J^{PC}=3^{--}$ because they were
kinematically forbidden. They are however possible for the glueball
$G_{3}(4200)$ since it is expected to be much heavier than the $3^{--}$ nonet
(6). Below we present the theoretical predictions for ratios of decays, which
can be easily calculated following the same steps of the previous sections.
The only ingredients that are needed for a determination of the branching
ratios are the masses of the particles, which are for the sake of simplicity
again assumed to be exact. For the $3^{--}$ glueball mass, we use
$m_{g_{3}}=4.2\,\,$GeV [5].

###  IV.2 Decay process $G_{3}\rightarrow V_{1}+P$

The interaction Lagrangian for decaying spin-$3$ tensor glueball into a vector
and a pseudoscalar meson reads

$\displaystyle\mathcal{L}_{g_{3}v_{1}p}=\vphantom{\bigg{(}\bigg{)}}$ (88)
$\displaystyle=\,$ $\displaystyle
c_{g_{3}v_{1}p}\,G_{3,\mu\alpha\beta}\,\varepsilon^{\mu\nu\rho\sigma}\,\mathrm{tr}\big{[}\big{\\{}(\partial_{\nu}V_{1,\rho}),\,(\partial^{\alpha}\partial^{\beta}\partial_{\sigma}P)\big{\\}}_{+}\big{]}\,.\vphantom{\bigg{(}\bigg{)}}$

An extended version is given by Eq. (133) in App. B. The corresponding decay
formula is analogous to Eq. (41),

$\displaystyle\Gamma_{G_{3}\rightarrow
V_{1}+P}(m_{g_{3}},m_{v_{1}},m_{p})=\vphantom{\Bigg{(}\Bigg{)}}$ (89)
$\displaystyle=\,$ $\displaystyle
c_{g_{3}v_{1}p}^{2}\,\frac{\big{|}\vec{k}_{v_{1},p}\big{|}^{7}}{105\uppi}\,\kappa_{i}\,\Theta(m_{g_{3}}-m_{v_{1}}-m_{p})\,.\vphantom{\Bigg{(}\Bigg{)}}$

Here, the $\kappa_{i}$’s can be taken from Table 25. We use again Eq. (24) for
the momenta of the outgoing particles throughout this chapter, while replacing
the mass of the conventional spin-$3$ mesons $m_{w_{3}}$ with the glueball
mass $m_{g_{3}}=4200\,\,$MeV. Since the coupling constant $c_{g_{3}v_{1}p}$ is
unknown, we calculate the nonzero ratios in Table 14. This is an interesting
decay channel to search for this glueball. Since the vector mesons further
decay in two (or three) pseudoscalar mesons, one should search for final
channels with three (or four) pseudoscalar mesons.

Table 14:  Predictions for the branching ratios of glueball $G_{3}(4200)\rightarrow V_{1}+P$.  branching ratio | theory
---|---
$\frac{\Gamma_{G_{3}(4200)\rightarrow\bar{K}^{\ast}(892)\,K}}{\Gamma_{G_{3}(4200)\rightarrow\rho(770)\,\pi}}$ | $1.11$
$\frac{\Gamma_{G_{3}(4200)\rightarrow\omega(782)\,\eta}}{\Gamma_{G_{3}(4200)\rightarrow\rho(770)\,\pi}}$ | $0.17$
$\frac{\Gamma_{G_{3}(4200)\rightarrow\omega(782)\,\eta^{\prime}(958)}}{\Gamma_{G_{3}(4200)\rightarrow\rho(770)\,\pi}}$ | $0.089$
$\frac{\Gamma_{G_{3}(4200)\rightarrow\phi(1020)\,\eta}}{\Gamma_{G_{3}(4200)\rightarrow\rho(770)\,\pi}}$ | $0.098$
$\frac{\Gamma_{G_{3}(4200)\rightarrow\phi(1020)\,\eta^{\prime}(958)}}{\Gamma_{G_{3}(4200)\rightarrow\rho(770)\,\pi}}$ | $0.11$

###  IV.3 Decay process $G_{3}\rightarrow\gamma+P$

Using vector meson dominance via the shift of Eq. (54) and analogous
replacements, we can also construct a radioactive decay term for the spin-$3$
glueball,

$\displaystyle\mathcal{L}_{g_{3}\gamma p}=\vphantom{\bigg{(}\bigg{)}}$ (90)
$\displaystyle=\,$ $\displaystyle c_{g_{3}\gamma
p}\,\tfrac{e}{g_{\rho}}\,G_{3,\mu\alpha\beta}\,(\partial_{\nu}a_{\rho})\,\varepsilon^{\mu\nu\rho\sigma}\,\mathrm{tr}\big{[}\big{\\{}Q,\,(\partial^{\alpha}\partial^{\beta}\partial_{\sigma}P)\big{\\}}_{+}\big{]}\,.\vphantom{\bigg{(}\bigg{)}}$

An extended version of this Lagrangian can be found in Eq. (134) in App. B.
For the decay formula one derives

$\displaystyle\Gamma_{G_{3}\rightarrow\gamma+P}(m_{g_{3}},m_{p})=\,$
$\displaystyle
c_{g_{3}v_{1}p}^{2}\,\big{(}\tfrac{e}{g_{\rho}}\big{)}^{2}\,\frac{\big{|}\vec{k}_{\gamma,p}\big{|}^{7}}{105\pi}\,\kappa_{i}^{\gamma}\,,\vphantom{\Bigg{(}\Bigg{)}}$

where the corresponding factors $\kappa_{i}^{\gamma}$ are listed in Table 26.
Similar to Eq. (56) one of the masses $m_{a}$ and $m_{b}$ in Eq. (24) has to
be set to zero, due to the masslessness of the photon.

From these formula, we calculate ratios for the radioactive decays
$\Gamma_{G_{3}\rightarrow\gamma\,P}$ w.r.t. the strong decay channel
$\Gamma_{G_{3}(4200)\rightarrow\rho(770)\,\pi}$. The results are listed in
Table 15.

Table 15:  Predictions for the branching ratios of glueball $G_{3}(4200)\rightarrow\gamma+P$ with the $G_{3}(4200)\rightarrow\rho\,\pi$ reference decay mode.  branching ratio | theory
---|---
$\frac{\Gamma_{G_{3}(4200)\rightarrow\gamma\,\pi^{0}}}{\Gamma_{G_{3}(4200)\rightarrow\rho(770)\,\pi}}$ | $129\cdot 10^{-5}$
$\frac{\Gamma_{G_{3}(4200)\rightarrow\gamma\,\eta}}{\Gamma_{G_{3}(4200)\rightarrow\rho(770)\,\pi}}$ | $37\cdot 10^{-5}$
$\frac{\Gamma_{G_{3}(4200)\rightarrow\gamma\,\eta^{\prime}(958)}}{\Gamma_{G_{3}(4200)\rightarrow\rho(770)\,\pi}}$ | $1\cdot 10^{-5}$

###  IV.4 Decay process $G_{3}\rightarrow V_{1}+A_{1}$

We also present the glueball decays into vectors and axial-vectors by
considering the Lagrangian in which $G_{3}$ couples to $V_{1}\,A_{1}$

$\displaystyle\mathcal{L}_{g_{3}v_{1}a_{1}}=\vphantom{\bigg{(}\bigg{)}}$ (91)
$\displaystyle=\,$ $\displaystyle
c_{g_{3}v_{1}a_{1}}\,G_{3,\mu\alpha\beta}\,\varepsilon^{\mu\nu\rho\sigma}\,\mathrm{tr}\big{[}\big{\\{}(\partial_{\nu}V_{1,\rho}),\,(\partial^{\alpha}\partial^{\beta}A_{1,\sigma})\big{\\}}_{+}\big{]}\,.\vphantom{\bigg{(}\bigg{)}}$

The extended version is provided in App. B in Eq. (135). The tree level decay
formula is given by

$\displaystyle\Gamma_{G_{3}\rightarrow
V_{1}+A_{1}}(m_{g_{3}},m_{v_{1}},m_{a_{1}})=\vphantom{\Bigg{(}\Bigg{)}}$ (92)
$\displaystyle=\,$ $\displaystyle
c_{g_{3}v_{1}a_{1}}^{2}\,\frac{\big{|}\vec{k}_{v_{1},a_{1}}\big{|}^{5}}{210\uppi\,m_{g_{3}}^{2}}\,\bigg{[}\big{|}\vec{k}_{v_{1},a_{1}}\big{|}^{2}\,\bigg{(}3+2\,\frac{m_{g_{3}}^{2}}{m_{a_{1}}^{2}}\bigg{)}+7\,m_{v_{1}}^{2}\bigg{]}\times\vphantom{\Bigg{(}\Bigg{)}}$
$\displaystyle\times\kappa_{i}\,\Theta(m_{g_{3}}-m_{v_{1}}-m_{a_{1}})\,.\vphantom{\Bigg{(}\Bigg{)}}$

where we used Eq. (167). The coefficients $\kappa_{i}$ are listed in Table 27.
Results are listed in Table 16.

Table 16:  Predictions for the branching ratios of glueball $G_{3}(4200)\rightarrow V_{1}+A_{1}$.  branching ratio | theory
---|---
$\frac{\Gamma_{G_{3}(4200)\rightarrow\bar{K}^{\ast}(892)\,K_{1,A}}}{\Gamma_{G_{3}(4200)\rightarrow\rho(770)\,a_{1}(1260)}}$ | $0.78$
$\frac{\Gamma_{G_{3}(4200)\rightarrow\omega(782)\,f_{1}(1285)}}{\Gamma_{G_{3}(4200)\rightarrow\rho(770)\,a_{1}(1260)}}$ | $0.29$
$\frac{\Gamma_{G_{3}(4200)\rightarrow\omega(782)\,f_{1}(1420)}}{\Gamma_{G_{3}(4200)\rightarrow\rho(770)\,a_{1}(1260)}}$ | $0.0009$
$\frac{\Gamma_{G_{3}(4200)\rightarrow\phi(1020)\,f_{1}(1285)}}{\Gamma_{G_{3}(4200)\rightarrow\rho(770)\,a_{1}(1260)}}$ | $0.0011$
$\frac{\Gamma_{G_{3}(4200)\rightarrow\phi(1020)\,f_{1}(1420)}}{\Gamma_{G_{3}(4200)\rightarrow\rho(770)\,a_{1}(1260)}}$ | $0.16$

###  IV.5 Decay process $G_{3}\rightarrow B_{1}+P$

The second dominant decay of the $J^{PC}=3^{--}$ glueball is the one into a
pseudovector and a pseudoscalar meson. The interaction Lagrangian is given by

$\displaystyle\mathcal{L}_{g_{3}b_{1}p}=c_{g_{3}b_{1}p}\,G_{3}^{\mu\nu\rho}\,\mathrm{tr}\big{[}\big{\\{}B_{1,\mu},\,(\partial_{\nu}\partial_{\rho}P)\big{\\}}_{+}\big{]}\,,\vphantom{\bigg{(}\bigg{)}}$
(93)

and its extended version can be found in Eq. (136) in App. B. This implies
that the formula for the decay width is identical to Eq. (62),

$\displaystyle\Gamma_{G_{3}\rightarrow
B_{1}+P}(m_{g_{3}},m_{b_{1}},m_{p})=\vphantom{\Bigg{(}\Bigg{)}}$ (94)
$\displaystyle=\,$ $\displaystyle
c_{g_{3}b_{1}p}^{2}\,\frac{\big{|}\vec{k}_{b_{1},p}\big{|}^{5}}{420\uppi\,m_{g_{3}}^{2}}\,\Bigg{(}7+3\,\frac{\big{|}\vec{k}_{b_{1},p}\big{|}^{2}}{m_{b_{1}}^{2}}\Bigg{)}\times\vphantom{\Bigg{(}\Bigg{)}}$
$\displaystyle\times\kappa_{i}\,\Theta(m_{g_{3}}-m_{b_{1}}-m_{p})\,.\vphantom{\Bigg{(}\Bigg{)}}$

The factors $\kappa_{i}$ are listed in Table 28. Again, we study two possible
choices for the mixing angle ($\beta_{b_{1}}\approx 0^{\circ}$ and
$\beta_{b_{1}}\approx-40^{\circ})$ and upon using $m_{K_{1,B}}\approx
m_{K_{1}(1270)}$, we present the results in Table 17.

Table 17:  Predictions for the branching ratios of glueball $G_{3}(4200)\rightarrow B_{1}+P$.  branching ratio | theory for | theory for
---|---|---
$\beta_{b_{1}}=0^{\circ}$ | $\beta_{b_{1}}=-40^{\circ}$
$\frac{\Gamma_{G_{3}(4200)\rightarrow K_{1,B}\,K}}{\Gamma_{G_{3}(4200)\rightarrow b_{1}(1235)\,\pi}}$ | $1.15$ | $1.15$
$\frac{\Gamma_{G_{3}(4200)\rightarrow h_{1}(1170)\,\eta}}{\Gamma_{G_{3}(4200)\rightarrow b_{1}(1235)\,\pi}}$ | $0.17$ | $0.33$
$\frac{\Gamma_{G_{3}(4200)\rightarrow h_{1}(1170)\,\eta^{\prime}(958)}}{\Gamma_{G_{3}(4200)\rightarrow b_{1}(1235)\,\pi}}$ | $0.12$ | $0.001$
$\frac{\Gamma_{G_{3}(4200)\rightarrow h_{1}(1415)\,\eta}}{\Gamma_{G_{3}(4200)\rightarrow b_{1}(1235)\,\pi}}$ | $0.10$ | $0.001$
$\frac{\Gamma_{G_{3}(4200)\rightarrow h_{1}(1415)\,\eta^{\prime}(958)}}{\Gamma_{G_{3}(4200)\rightarrow b_{1}(1235)\,\pi}}$ | $0.08$ | $0.16$

###  IV.6 Decay process $G_{3}\rightarrow B_{1}+A_{1}$

Moreover, we present the results for the glueball into a pseudovector and an
axial-vector, $G_{3}\rightarrow B_{1}+A_{1}$ in Table 18 for
$\beta_{a_{1}}\approx 0^{\circ}$. Namely, the Lagrangian for this decay can be
obtained from Eq. (93) as

$\displaystyle\mathcal{L}_{g_{3}b_{1}a_{1}}=c_{g_{3}b_{1}a_{1}}\,G_{3}^{\mu\nu\rho}\,\mathrm{tr}\big{[}\big{\\{}B_{1,\mu},\,(\partial_{\nu}A_{1,\rho})\big{\\}}_{+}\big{]}\,.\vphantom{\bigg{(}\bigg{)}}$
(95)

The extended form is presented in Eq. (137) in App. B. The corresponding decay
formula is given by

$\displaystyle\Gamma_{G_{3}\rightarrow
B_{1}+A_{1}}(m_{g_{3}},m_{b_{1}},m_{a_{1}})=\vphantom{\Bigg{(}\Bigg{)}}$ (96)
$\displaystyle=\,$ $\displaystyle
c_{g_{3}b_{1}a_{1}}^{2}\,\frac{\big{|}\vec{k}_{b_{1},a_{1}}\big{|}^{3}}{840\uppi\,m_{b_{1}}^{2}\,m_{a_{1}}^{2}\,m_{g_{3}}^{2}}\,\Big{[}6\,|\vec{k}_{b_{1},a_{1}}\big{|}^{4}+\vphantom{\Bigg{(}\Bigg{)}}$
$\displaystyle+35\,m_{b_{1}}^{2}\,m_{a_{1}}^{2}+14\,\big{|}\vec{k}_{b_{1},a_{1}}\big{|}^{2}\,\big{(}m_{b_{1}}^{2}+m_{a_{1}}^{2}\big{)}\Big{]}\times\vphantom{\Bigg{(}\Bigg{)}}$
$\displaystyle\times\kappa_{i}\,\Theta(m_{g_{3}}-m_{b_{1}}-m_{a_{1}})\,,\vphantom{\Bigg{(}\Bigg{)}}$

where the $\kappa_{i}$’s are listed in Table 29.

Table 18:  Predictions for the branching ratios of glueball $G_{3}(4200)\rightarrow B_{1}+A_{1}$.  branching ratio | theory for | theory for
---|---|---
$\beta_{b_{1}}=0^{\circ}$ | $\beta_{b_{1}}=-40^{\circ}$
$\frac{\Gamma_{G_{3}(4200)\rightarrow K_{1,B}\,K_{1,A}}}{\Gamma_{G_{3}(4200)\rightarrow b_{1}(1235)\,a_{1}(1260)}}$ | $0.96$ | $0.96$
$\frac{\Gamma_{G_{3}(4200)\rightarrow h_{1}(1170)\,f_{1}(1285)}}{\Gamma_{G_{3}(4200)\rightarrow b_{1}(1235)\,a_{1}(1260)}}$ | $0.34$ | $0.20$
$\frac{\Gamma_{G_{3}(4200)\rightarrow h_{1}(1170)\,f_{1}(1420)}}{\Gamma_{G_{3}(4200)\rightarrow b_{1}(1235)\,a_{1}(1260)}}$ | $0$ | $0.11$
$\frac{\Gamma_{G_{3}(4200)\rightarrow h_{1}(1415)\,f_{1}(1285)}}{\Gamma_{G_{3}(4200)\rightarrow b_{1}(1235)\,a_{1}(1260)}}$ | $0$ | $0.092$
$\frac{\Gamma_{G_{3}(4200)\rightarrow h_{1}(1415)\,f_{1}(1420)}}{\Gamma_{G_{3}(4200)\rightarrow b_{1}(1235)\,a_{1}(1260)}}$ | $0.17$ | $0.10$

##  V Discussion of the construction of the model

In this section we discuss the interpretation of Eq. (22), the interaction
Lagrangian $\mathcal{L}_{W,\text{total}}$, which was constructed by keeping
only the lowest possible number of derivatives for a given interaction term.
We justify this choice by following two different lines of argumentation.
First, our effective model naturally emerges, if the Lagrangian
$\mathcal{L}_{W,\text{total}}$ is interpreted as being part of a more general
and complete chiral hadronic model in the vacuum, such as the (extended)
linear sigma model [57], which is aforementioned in the introduction. Second,
a (Functional) Renormalization Group perspective also indicates that terms
with the lowest derivatives might be retained as the dominant contributions
within our effective hadronic model.

The eLSM is built under the assumption of chiral symmetry, thus

$\displaystyle U_{\mathrm{L}}(3)\times
U_{\mathrm{R}}(3)=SU_{\mathrm{L}}(3)\times SU_{\mathrm{R}}(3)\times
U_{\mathrm{L}}(1)\times U_{\mathrm{R}}(1)\,,$

and the assumption of scale invariance. Besides the small explicit breaking of
both of these symmetries, because of the nonzero bare quark masses, which
derive from the electro-weak sector, the symmetries are also broken
dynamically. The $SU_{\mathrm{L}}(3)\times SU_{\mathrm{R}}(3)$ breaks down to
a $SU_{\mathrm{V}}(3)$ symmetry spontaneously, which was already discussed
before. Also the scale invariance breaks down via gluonic quantum fluctuations
and the emergence of a gluon condensate, which is also called the
conformal/trace anomaly [118, 10, 119]. Additionally, a second anomaly – the
chiral anomaly – breaks $U_{\mathrm{A}}(1)$ and explains the heavy mass of
$\eta^{\prime}(958)$ as well as the large pseudoscalar mixing angle [83]. As
discussed in Ref. [84] and commented in various part of this work, this
anomaly may affect also the masses and the decays of other nonets (those being
part of homochiral multiplets). In the chiral limit and ignoring the chiral
anomaly all the interaction terms of the eLSM are invariant under
scale/dilatation transformations. Consequently, they are parametrized by
dimensionless coupling constants.

The Lagrangian $\mathcal{L}_{w_{3},\text{total}}$ (22), that we use in this
work, contains various decay constants, see Table 3. With the exception of the
last entry, all of them are dimensionful. At a first sight, this feature seems
in disagreement with the requirement that should descend from a chiral model
such as the eLSM [53, 56]. Yet, this is not the case and a closer inspection
also shows why the lowest number of derivatives should be kept. First, let us
recall that the spontaneous symmetry breaking implies a peculiar scalar-axial-
vector mixing, that is removed by the a shift of the axial-vector nonet
$A_{1}$. In the simplified case with vanishing bare quark masses, where
$U_{\mathrm{V}}(3)$ is exact (this is enough for our illustrative purposes),
it takes the form [53, 120]:

$\displaystyle A_{1}^{\mu}\mapsto A_{1}^{\mu}+Z\,w\,\partial^{\mu}P\,,$ (97)

where $Z\approx 1.6$ and $w\approx\frac{g_{1}\phi_{N}}{m_{a_{1}}^{2}}$ where
$g_{1}\approx g_{\rho}\approx 5.5$, $m_{a_{1}}\simeq 1.4\,\,$GeV, and
$\phi_{N}\approx Zf_{\pi}$ ($f_{\pi}=92.4\,\,$MeV is the pion decay constant).

Now, the eLSM was studied in a variety of frameworks: Besides (pseudo)scalar
and (axial-)vector mesons [53] and the aforementioned calculation of glueballs
[56, 79, 117], it was also applied to study hybrid mesons [120] and excited
scalar mesons [121]. Even if various mesonic nonets were not included in the
eLSM yet, it is clear, which features such terms should have.

Within this framework, the $W_{3}PP$-Lagrangian,

$\displaystyle\mathcal{L}_{w_{3}pp}=g_{w_{3}pp}\,\mathrm{tr}\big{[}W^{\mu\nu\rho}\,\big{[}P,\,(\partial_{\mu}\partial_{\nu}\partial_{\rho}P)\big{]}_{-}\big{]}\,,$
(98)

where $g_{w_{3}pp}$ has dimension $E^{-2}$ and the $W_{3}A_{1}P$-Lagrangian,

$\displaystyle\mathcal{L}_{w_{3}a_{1}p}=g_{w_{3}a_{1}p}\,\mathrm{tr}\big{[}W^{\mu\nu\rho}\,\big{[}A_{1,\mu},\,(\partial_{\nu}\partial_{\rho}P)\big{]}_{-}\big{]}\,,$
(99)

where $g_{w_{3}a_{1}p}$ has dimension $E^{-1},$ can be seen as the result of
the interaction Lagrangian

$\displaystyle\mathcal{L}_{w_{3}a_{1}a_{1}}=g_{w_{3}a_{1}a_{1}}\,\mathrm{tr}\big{(}W^{\mu\nu\rho}\,\big{[}A_{1,\mu},\,\partial_{\nu}A_{1,\rho}\big{]}_{-}\big{)}\,,$
(100)

involving solely the dimensionless coupling $g_{w_{3}a_{1}a_{1}}$ via the
shift of Eq. (97) applied once/twice. In other words, only the Lagrangian
$\mathcal{L}_{w_{3}a_{1}a_{1}}$ is part of a generalized eLSM Lagrangian that
includes fields with $J=3$, in agreement with dilatation invariance. Then,
upon spontaneous symmetry breaking and mixing/shifts, the Lagrangians
$\mathcal{L}_{w_{3}a_{1}p}$ and $\mathcal{L}_{w_{3}pp}$ are a consequence of
$\mathcal{L}_{w_{3}a_{1}a_{1}}$, thus explaining how dimensional couplings
appear in the model, even if one starts solely with dimensionless ones.

While the decay $W_{3}\rightarrow A_{1}\,A_{1}$ is not kinematically allowed,
the corresponding term explains how the first and the fifth entries of Table 3
emerge. The simple relations

$\displaystyle g_{w_{3}pp}\approx Z^{2}\,w^{2}\,g_{w_{3}a_{1}a_{1}}$ (101)

and

$\displaystyle g_{w_{3}a_{1}p}\approx Z\,w\,g_{w_{3}a_{1}a_{1}}$ (102)

follow. Moreover, the ratio $g_{w_{3}a_{1}p}/g_{w_{3}pp}\approx(Z\,w)^{-1}$,
out of which we may estimate that

$\displaystyle g_{w_{3}a_{1}p}^{2}\approx 3\cdot 10^{-4}\,\,\text{MeV}^{-2}$
(103)

is obtained (it should be however stressed that the present heuristic
discussion cannot provide a precise determination of ratios of couplings, but
it solely offers a guide to understand the origin of the model). This value
leads to $\Gamma_{\rho_{3}(1690)\rightarrow a_{1}(1260)\,\pi}\approx
2\,\,$MeV, as previously mentioned.

Next, let us consider the $W_{3}B_{1}P$-Lagrangian,

$\displaystyle\mathcal{L}_{w_{3}b_{1}p}=g_{w_{3}b_{1}p}\,\mathrm{tr}\big{(}W^{\mu\nu\rho}\,\big{\\{}B_{1,\mu},\,(\partial_{\nu}\partial_{\rho}P)\big{\\}}_{+}\big{)}\,,$
(104)

with $g_{w_{3}b_{1}p}$, which can be seen as emerging from

$\displaystyle\mathcal{L}_{w_{3}b_{1}a_{1}}=g_{w_{3}b_{1}a_{1}}\,\mathrm{tr}\big{[}W^{\mu\nu\rho}\,\big{\\{}B_{1,\mu},\,(\partial_{\nu}A_{1,\rho})\big{\\}}_{+}\big{]}\,,$
(105)

with a dimensionless coupling $g_{w_{3}b_{1}a_{1}}$. The $A_{1}$-shift
introduced above implies that

$\displaystyle g_{w_{3}b_{1}p}\approx Z\,w\,g_{w_{3}b_{1}a_{1}}\,.$ (106)

Next, let us study the remaining terms. The $W_{3}V_{1}V_{1}$-Lagrangian is
already dilatation invariant, while the two other Lagrangians
$\mathcal{L}_{w_{3}v_{1}p}$ and $\mathcal{L}_{w_{3}a_{2}p}$ are not, since
they involve the coupling $g_{w_{3}v_{1}p}$ and $g_{w_{3}a_{2}p}$, having
dimensions $E^{-3}$ and $E^{-2}$. In this case, one could build analogous
Lagrangians $\mathcal{L}_{w_{3}v_{1}a_{1}}$ and
$\mathcal{L}_{w_{3}a_{2}a_{1}}$, but their coupling coupling constants
$g_{w_{3}v_{1}a_{1}}$ and $g_{w_{3}a_{2}a_{1}}$ still have dimension $E^{-2}$
and $E^{-1}$. This is however understandable, since these terms involve the
Levi-Civita pseudotensor $\varepsilon_{\mu\nu\rho\sigma}$ and hence are a
manifestation of the the chiral anomaly.

In conclusion, terms being part of the model are (i) dilatation invariant or
(ii) can be obtained from dilatation invariant terms of a more general
underlying chiral model via the shift of the axial-vector nonet or (iii) they
are linked to the chiral anomaly. Now, one also understand why only terms with
the lowest number of derivatives appear.

For the effective interaction terms of the $3^{--}$ glueball, the argument is
completely analogous.

Still, we also provide a brief alternative motivation for the construction of
the model from a renormalization group perspective. In this framework, we know
that the UV limit of all hadronic physics is QCD, where the degrees of freedom
are quarks and gluons. However, we want to describe low-energy QCD and the
effective hadronic degrees of freedom. By integrating out quantum fluctuations
of gluons and quarks via renormalization group transformations from the UV
scales to the IR scales including dynamical hadronization, one finally ends up
with an effective hadronic theory, were hadrons are the effective degrees of
freedom. While integrating out quantum fluctuations momentum shell after
momentum shell, all kinds of effective couplings/vertices, which are in
accordance with the UV symmetries of the theory – the symmetries of QCD – will
be generated. This process is effectively described as the RG flow of the
coupling constants of the theory, where effective couplings of hadrons are
initialy zero and only generated dynamically during the RG flow. The same
applies to the effective fermionic and hadronic quantum fields.

An efficient and modern framework to describe this process, which effectively
is an implementation of Wilsons idea of the RG [122, 123, 124], is the
Functional Renormalization Group (FRG) in its formulation via the Exact
Renormalization Group Equation [60, 61], see for example Refs. [58, 63, 59,
125, 126, 127] and Ref. [62] for an comprehensive up-to-date review. For our
purpose, it is solely important to note that the FRG is most efficiently
formulated on the level of the quantum effective IR action $\Gamma[\Phi]$,
which is the generating functional of all $1$PI-$n$-point-correlation
functions. The FRG allows to calculate $\Gamma[\Phi]$ – at least within
certain truncations – from the UV theory, which here is QCD. The interesting
part is that the quantum effective IR action contains all kinds of effective
vertices, that were generated during the RG flow and contain all information
about the higher energy scales. For the calculation of decays etc., however,
only these effective couplings are relevant and calculations are performed at
tree level, because all loop contributions are already contained in the
couplings and because we are working with the $1$PI-$n$-point-correlation
functions. Dynamical symmetry breaking is realized in this framework as a
nontrivial vacuum/ground-state, which realizes as a minimum of the effective
action $\Gamma[\Phi]$. All $1$PI-$n$-point functions must be extracted at this
physical point, which yields shifts like in Eq. (97).

In practice, it is almost impossible to really do this calculation from first
principles, especially concerning couplings for mesons with higher spin and
large masses. Nevertheless, we know from the previous argumentation that a low
IR-effective action $\Gamma[\Phi]$ must contain all possible interaction
terms, which are in accordance with the symmetry of the theory and if shifted
to the physical point, then it must contain all symmetries of the vacuum of
the theory. This also applies to the effective couplings of our spin-$3$
mesons. Thus, we can use the following argument: instead of calculating all
interaction terms and couplings, which are in accordance with the symmetries
of QCD, via an RG flow from first principles (which is anyhow impossible), we
just write down all possible effective interaction terms, that are invariant
under the residual IR vacuum-symmetries of QCD and simply fit the effective
coupling constants to experimental data. This approach additionally justifies
the use of tree level-calculations, because experimental measurements for
decay widths etc. already contain all information about the high energy scales
of QCD and so does an effective IR action $\Gamma[\Phi]$, from which
information is extracted at tree level. We therefore interpret all effective
Lagrangians $\mathcal{L}_{w_{3}\circ\circ}$ as being part of an effective IR
action $\Gamma[\Phi]$ for QCD.

Last, we argue, that as long as the momentum exchange in processes like three-
point-interactions (decays) is small, we can expand an effective IR action in
powers of momenta of the effective hadronic fields. In the context of the FRG,
this approach is denoted as a derivative expansion and turned out as a decent
approximation in a lot of low-energy models of QCD [64, 65, 66, 67, 68, 69,
70, 71, 72, 73, 74, 75].

All in all, both approaches complement each other and explain that we only
retain terms that are of the lowest orders in the derivatives of the fields,
that are of leading order in a large-$N_{\mathrm{c}}$ expansion and that are
flavor invariant.

##  VI Conclusions

In this paper, we have studied the decays of the lightest mesonic $\bar{q}q$
nonet with quantum numbers $J^{PC}=3^{--}$ using an effective QFT model based
on flavor symmetry. Our model retained only the dominant terms in an
large-$N_{\mathrm{c}}$ expansion and the lowest possible number of
derivatives. By comparing the theoretical results with the current status of
experimental data for decay width and some known branching ratios, which are
all reported by PDG [1], we conclude that the $\bar{q}q$ assignment works
quite well. Still, we remark that there are also other rather successful
approaches different from the $\bar{q}q$ picture towards a coherent
description of the nature of the $J^{PC}=3^{--}$ states, see e.g. via
multi-$\rho(770)$ resonances [128, 129]. However, also in our work some of the
decay channels deserve deeper theoretical and experimental investigation.
Additionally, we presented decay ratios of a putative and not yet detected
$J^{PC}=3^{--}$ glueball by considering decays into the vector-pseudoscalar
and pseudovector-pseudoscalar mesonic pairs. In summary, we provide many
qualitative predictions of strong and radiative decays of conventional
$J^{PC}=3^{--}$ mesons and the $J^{PC}=3^{--}$ glueball for future
experimental tests.

###### Acknowledgements.

S.J. acknowledges financial support through the project Development
Accelerator of the Jan Kochanowski University of Kielce, co-financed by the
European Union under the European Social Fund, with no. POWR.03.05. 00-00-Z212
/ 18. A.K. acknowledges the support of the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) through the collaborative research center
trans-regio CRC-TR 211 Strong-interaction matter under extreme conditions –
project number 315477589 – TRR 211. A.K. also acknowledges the support of the
Friedrich-Naumann-Foundation for Freedom, the Giersch Foundation and the
Helmholtz Graduate School for Hadron and Ion Research. F.G. acknowledges
support from the Polish National Science Centre (NCN) through the OPUS
projects 2019/33/B/ST2/00613 and no. 2018/29/B/ST2/02576. The authors thank J.
Braun, F. Divotgey, J. Eser, M. Piotrowska, D. H. Rischke, and M. J. Steil for
useful discussions.

##  Appendix A Masses and strange-nonstrange mixing angles

For a certain nonet, we denote as $\vec{a}$ the triplet of fields describing
the isospin $I=1$ elements $\bar{d}u$, $\bar{u}d$,
${\frac{1}{\sqrt{2}}\,(\bar{u}u-\bar{d}d)}$, with $K^{\pm}$ the complex kaonic
fields representing the doublet $\bar{s}u$ and $\bar{u}s$, and $\bar{K}^{0}$
the analogous fields $\bar{d}s$ and $\bar{s}d$ for the neutral kaonic
elements, and as $f_{8}$ and $f_{0}$ the octet and the singlet elements, whose
quark-antiquark configurations are
$\frac{1}{\sqrt{6}}\,(\bar{u}u+\bar{d}d-2\bar{s}s)$ and
$\frac{1}{\sqrt{3}}\,(\bar{u}u+\bar{d}d+\bar{s}s)$, respectively. The
corresponding $SU_{\mathrm{V}}(3)$-invariant mass term reads (omitting
possible Lorentz indices for simplicity),

$\displaystyle\tfrac{1}{2}\,m_{a}^{2}\,\vec{a}^{\,2}+m_{K}^{2}\,(K^{+}\,K^{-}+\bar{K}^{0}\,K^{0})+\vphantom{\bigg{(}\bigg{)}}$
(107)
$\displaystyle+\tfrac{1}{2}\,m_{8}^{2}\,f_{8}^{2}+\tfrac{1}{2}\,m_{0}^{2}\,f_{0}^{2}+m_{08}\,f_{0}\,f_{8}\,,\vphantom{\bigg{(}\bigg{)}}$

where also a singlet-octet mixing parameter $m_{08}$ has been introduced. The
masses of the fields $\vec{a}$, $\bar{K}^{0/\pm}$, and $f_{8}$ are expressed
as

$\displaystyle m_{a}^{2}=\,$ $\displaystyle
m_{\mathrm{nonet}}^{2}+2\,\delta_{n}^{2}\,,\vphantom{\bigg{(}\bigg{)}}$ (108)
$\displaystyle m_{K}^{2}=\,$ $\displaystyle
m_{\mathrm{nonet}}^{2}+\delta_{n}^{2}+\delta_{s}^{2}\,,\vphantom{\bigg{(}\bigg{)}}$
(109) $\displaystyle m_{8}^{2}=\,$ $\displaystyle
m_{\mathrm{nonet}}^{2}+\tfrac{2}{3}\,\delta_{n}^{2}+\tfrac{4}{3}\,\delta_{s}^{2}\,,\vphantom{\bigg{(}\bigg{)}}$
(110)

where $m_{\mathrm{nonet}}$ is the mass of the nonet in the chiral limit as a
consequence of spontaneous symmetry breaking (see e.g. Ref. [53]), and
$\delta_{n}^{2}$ represents the (almost negligible) contribution of the bare
nonstrange quarks $u$ and $d$ and $\delta_{s}^{2}$ the (on hadronic scales)
small but nonnegligible contribution of the bare quark $s$. (Besides the
pseudoscalar nonet, one simply approximates $\delta_{n}\propto m_{u}\simeq
m_{d}$ and $\delta_{s}\propto m_{s}$.) From the expressions above it follows
that

$\displaystyle m_{8}^{2}=\,$
$\displaystyle\tfrac{4}{3}\,m_{K}^{2}-\tfrac{1}{3}\,m_{a}^{2}=\vphantom{\bigg{(}\bigg{)}}$
(111) $\displaystyle=\,$ $\displaystyle
m_{\mathrm{nonet}}^{2}+\tfrac{2}{3}\,\delta_{n}^{2}+\tfrac{4}{3}\,\delta_{s}^{2}\,.\vphantom{\bigg{(}\bigg{)}}$

The mass of the singlet member of the nonet may contain an additional unknown
contribution (due to the anomaly and/or conversion to gluons), parametrized
by:

$\displaystyle m_{0}^{2}=\,$ $\displaystyle
m_{\mathrm{nonet}}^{2}+\tfrac{4}{3}\,\delta_{n}^{2}+\tfrac{2}{3}\,\delta_{s}^{2}+\alpha=\vphantom{\bigg{(}\bigg{)}}$
(112) $\displaystyle=\,$
$\displaystyle\tfrac{2}{3}\,m_{K}^{2}+\tfrac{1}{3}\,m_{a}^{2}+\alpha\,.\vphantom{\bigg{(}\bigg{)}}$

Next, we diagonalize the system in the isoscalar sector. We then introduce the
physical fields $f$ and $f^{\prime}$ as

$\displaystyle\begin{pmatrix}-f^{\prime}\\\
f\end{pmatrix}=\begin{pmatrix}\cos\beta_{\text{\tiny
PDG}}&-\sin\beta_{\text{\tiny PDG}}\\\ \sin\beta_{\text{\tiny
PDG}}&\cos\beta_{\text{\tiny PDG}}\end{pmatrix}\begin{pmatrix}f_{8}\\\
f_{0}\end{pmatrix}$ (113)

where $\beta_{\text{\tiny PDG}}$ is the corresponding mixing angle,

$\displaystyle\tan\big{(}2\beta_{\text{\tiny PDG}}\big{)}=\,$
$\displaystyle\frac{2m_{08}}{m_{0}^{2}-m_{8}^{2}}\,,$ (114)

which is obtained via diagonalization of the mass matrix together with the
relation,

$\displaystyle m_{8}^{2}=\,$ $\displaystyle
m_{f^{\prime}}^{2}\cos^{2}\beta_{\text{\tiny
PDG}}+m_{f}^{2}\sin^{2}\beta_{\text{\tiny PDG}}\,,\vphantom{\bigg{(}\bigg{)}}$
(115) $\displaystyle m_{0}^{2}=\,$ $\displaystyle
m_{f}^{2}\cos^{2}\beta_{\text{\tiny
PDG}}+m_{f^{\prime}}^{2}\sin^{2}\beta_{\text{\tiny
PDG}}\,.\vphantom{\bigg{(}\bigg{)}}$ (116)

In Eq. (113) the extra minus sign in front of $f^{\prime}$ is introduced for
later convenience. Using Eq. (115) one gets

$\displaystyle\frac{3m_{8}^{2}-3m_{f^{\prime}}^{2}}{-3m_{8}^{2}+3m_{f}^{2}}=\vphantom{\bigg{(}\bigg{)}}$
(117) $\displaystyle=\,$
$\displaystyle\frac{3m_{f^{\prime}}^{2}\cos^{2}\beta_{\text{\tiny
PDG}}+3m_{f}^{2}\sin^{2}\beta_{\text{\tiny
PDG}}-3m_{f^{\prime}}^{2}}{-3m_{f^{\prime}}^{2}\cos^{2}\beta_{\text{\tiny
PDG}}-3m_{f}^{2}\sin^{2}\beta_{\text{\tiny
PDG}}+3m_{f}^{2}}=\vphantom{\bigg{(}\bigg{)}}$ $\displaystyle=\,$
$\displaystyle\tan^{2}\beta_{\text{\tiny PDG}}\,,\vphantom{\bigg{(}\bigg{)}}$

but on the other hand using the above expressions for the masses the l.h.s.
can be written as

$\displaystyle\frac{3m_{8}^{2}-3m_{f^{\prime}}^{2}}{-3m_{8}^{2}+3m_{f}^{2}}=\frac{4m_{K}^{2}-m_{a}^{2}-3m_{f^{\prime}}^{2}}{-4m_{K}^{2}+m_{a}+3m_{f}^{2}}\,,$
(118)

leading to expressions used in the PDG [1] to calculate the mixing angle in
the singlet-octet basis:

$\displaystyle\tan^{2}\beta_{\text{\tiny
PDG}}=\frac{4m_{k}^{2}-m_{a}^{2}-3m_{f^{\prime}}^{2}}{-4m_{K}^{2}+m_{a}+3m_{f}^{2}}\,.$
(119)

In the last step we relate this to the (for our purposes) more convenient
nonstrange-strange basis $f_{N}$ and $f_{S}$ that corresponds to the
configurations $\frac{1}{\sqrt{2}}\,(\bar{u}u+\bar{d}d)$ and $\bar{s}s$. Via

$\displaystyle\begin{pmatrix}f_{8}\\\
f_{0}\end{pmatrix}=\tfrac{1}{\sqrt{3}}\begin{pmatrix}-\sqrt{2}&1\\\
1&\sqrt{2}\end{pmatrix}\begin{pmatrix}f_{S}\\\ f_{N}\end{pmatrix}\,,$ (120)

we introduce the strange-nonstrange mixing $\beta$ as:

$\displaystyle\begin{pmatrix}-f^{\prime}\\\ f\end{pmatrix}=$ (121)
$\displaystyle=\,$
$\displaystyle\tfrac{1}{\sqrt{3}}\begin{pmatrix}\cos\beta_{\text{\tiny
PDG}}&-\sin\beta_{\text{\tiny PDG}}\\\ \sin\beta_{\text{\tiny
PDG}}&\cos\beta_{\text{\tiny PDG}}\end{pmatrix}\begin{pmatrix}-\sqrt{2}&1\\\
1&\sqrt{2}\end{pmatrix}\begin{pmatrix}f_{S}\\\ f_{N}\end{pmatrix}=$
$\displaystyle=\,$ $\displaystyle\begin{pmatrix}-\cos\beta&\sin\beta\\\
\sin\beta&\cos\beta\end{pmatrix}\begin{pmatrix}f_{S}\\\ f_{N}\end{pmatrix}\,,$

or, as reported in the main text, as:

$\displaystyle\begin{pmatrix}f\\\
f^{\prime}\end{pmatrix}=\begin{pmatrix}\cos\beta&\sin\beta\\\
-\sin\beta&\cos\beta\end{pmatrix}\begin{pmatrix}f_{N}\\\ f_{S}\end{pmatrix}$
(122)

The strange-nonstrange mixing angle $\beta$ and the angle $\beta_{\text{\tiny
PDG}}$ can be obtained by Eq. (121) as:

$\displaystyle\sqrt{\tfrac{1}{3}}\cos\beta_{\text{\tiny
PDG}}-\sqrt{\tfrac{2}{3}}\sin\beta_{\text{\tiny
PDG}}=\vphantom{\bigg{(}\bigg{)}}$ (123) $\displaystyle=\,$
$\displaystyle\sin\beta\equiv\sin(-\beta_{\text{\tiny
PDG}}+\beta_{0})\,,\vphantom{\bigg{(}\bigg{)}}$

thus

$\displaystyle\beta=-\beta_{\text{\tiny PDG}}+\beta_{0}$ (124)

with

$\beta_{0}=\arccos\sqrt{\tfrac{2}{3}}\simeq 35.3^{\circ}\,.$ (125)

The mixing angle $\beta_{\text{\tiny PDG}}$ (and hence $\beta$) has been
calculated in the PDG for certain well-known mesonic nonets (pseudoscalar,
vector, as well as $J^{PC}=2^{++}$ and $J^{PC}=3^{--}$). While for the
pseudoscalar case we employ the result of Ref. [81] (where not only masses but
also decays are used to get $\beta$), in the remaining three cases we used the
value $\beta_{\text{\tiny PDG}}$ to determine the strange-nonstrange mixing
angle. With the exception of the peculiar pseudoscalar nonet, where the axial
anomaly is large, the other values of $\beta$ calculated from
$\beta_{\text{\tiny PDG}}$ are quite small – in agreement with the theoretical
expectations presented in Ref. [84]. As a consequence, the parameter $\alpha$
of Eq. (112) is negligible. In particular, for $J=3$ the numerical value
$\beta=\beta_{w_{3}}=3.5^{\circ}$ has been used throughout our calculations.
When future experimental data will be more accurate, one could also include
$\beta_{w_{3}}$ as a fit parameter in an improved model. In this respect,
better experimental data for radiative decays would be useful.

##  Appendix B Extended form of the Lagrangians of the model

In this appendix, we provide the explicit form of the interaction Lagrangians
in Table 3 and in Table 13.

$\displaystyle\mathcal{L}_{w_{3}pp}=\vphantom{\bigg{(}\bigg{)}}$ (126)
$\displaystyle=\,$ $\displaystyle
g_{w_{3}pp}\,\mathrm{tr}\big{(}W_{3}^{\mu\nu\rho}\big{[}P,\,(\partial_{\mu}\partial_{\nu}\partial_{\rho}P)\big{]}_{-}\big{)}=\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle=\,$
$\displaystyle\frac{g_{w_{3}pp}}{4}\,\Big{(}\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle+\rho^{0,\mu\nu\rho}_{3}\Big{\\{}+\bar{K}^{0}(\partial_{\mu}\partial_{\nu}\partial_{\rho}K^{0})-K^{0}(\partial_{\mu}\partial_{\nu}\partial_{\rho}\bar{K}^{0})+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle\quad+K^{+}(\partial_{\mu}\partial_{\nu}\partial_{\rho}K^{-})-K^{-}(\partial_{\mu}\partial_{\nu}\partial_{\rho}K^{+})+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle\quad+2\,\big{[}\pi^{+}(\partial_{\mu}\partial_{\nu}\partial_{\rho}\pi^{-})-\pi^{-}(\partial_{\mu}\partial_{\nu}\partial_{\rho}\pi^{+})\big{]}\Big{\\}}+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle+\rho^{+,\mu\nu\rho}_{3}\Big{\\{}\sqrt{2}\,\big{[}-K^{-}(\partial_{\mu}\partial_{\nu}\partial_{\rho}K^{0})+K^{0}(\partial_{\mu}\partial_{\nu}\partial_{\rho}K^{-})\big{]}+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle\quad+2\,\big{[}\pi^{-}(\partial_{\mu}\partial_{\nu}\partial_{\rho}\pi^{0})-\pi^{0}(\partial_{\mu}\partial_{\nu}\partial_{\rho}\pi^{-})\big{]}\Big{\\}}+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle+\rho^{-,\mu\nu\rho}_{3}\Big{\\{}\sqrt{2}\,\big{[}K^{+}(\partial_{\mu}\partial_{\nu}\partial_{\rho}\bar{K}^{0})-\bar{K}^{0}(\partial_{\mu}\partial_{\nu}\partial_{\rho}K^{+})\big{]}+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle\quad+2\,\big{[}-\pi^{+}(\partial_{\mu}\partial_{\nu}\partial_{\rho}\pi^{0})+\pi^{0}(\partial_{\mu}\partial_{\nu}\partial_{\rho}\pi^{+})\big{]}\Big{\\}}+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle+\bar{K}^{\ast
0,\mu\nu\rho}_{3}\Big{\\{}+K^{0}(\partial_{\mu}\partial_{\nu}\partial_{\rho}\pi^{0})-\pi^{0}(\partial_{\mu}\partial_{\nu}\partial_{\rho}K^{0})+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle\quad+\sqrt{2}\,\big{[}-K^{+}(\partial_{\mu}\partial_{\nu}\partial_{\rho}\pi^{-})+\pi^{-}(\partial_{\mu}\partial_{\nu}\partial_{\rho}K^{+})\big{]}+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle\quad+K^{0}\big{[}(\partial_{\mu}\partial_{\nu}\partial_{\rho}\eta)(-\cos\beta_{p}+\sqrt{2}\sin\beta_{p})+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle\qquad+(\partial_{\mu}\partial_{\nu}\partial_{\rho}\eta^{\prime})(\sin\beta_{p}+\sqrt{2}\cos\beta_{p})\big{]}-\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle\quad-\big{[}\eta(-\cos\beta_{p}+\sqrt{2}\sin\beta_{p})+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle\qquad+\eta^{\prime}(\sin\beta_{p}+\sqrt{2}\cos\beta_{p})\big{]}(\partial_{\mu}\partial_{\nu}\partial_{\rho}K^{0})\Big{\\}}+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle+K^{\ast
0,\mu\nu\rho}_{3}\Big{\\{}-\bar{K}^{0}(\partial_{\mu}\partial_{\nu}\partial_{\rho}\pi^{0})+\pi^{0}(\partial_{\mu}\partial_{\nu}\partial_{\rho}\bar{K}^{0})+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle\quad+\sqrt{2}\,\big{[}K^{-}(\partial_{\mu}\partial_{\nu}\partial_{\rho}\pi^{+}-\pi^{+}(\partial_{\mu}\partial_{\nu}\partial_{\rho}K^{-})\big{]}-\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle\quad-\bar{K}^{0}\big{[}(\partial_{\mu}\partial_{\nu}\partial_{\rho}\eta)(-\cos\beta_{p}+\sqrt{2}\sin\beta_{p})+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle\qquad+(\partial_{\mu}\partial_{\nu}\partial_{\rho}\eta^{\prime})(\sin\beta_{p}+\sqrt{2}\cos\beta_{p})\big{]}+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle\quad+\big{[}\eta(-\cos\beta_{p}+\sqrt{2}\sin\beta_{p})+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle\qquad+\eta^{\prime}(\sin\beta_{p}+\sqrt{2}\cos\beta_{p})\big{]}(\partial_{\mu}\partial_{\nu}\partial_{\rho}\bar{K}^{0})\Big{\\}}+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle+K^{\ast+,\mu\nu\rho}_{3}\Big{\\{}+K^{-}(\partial_{\mu}\partial_{\nu}\partial_{\rho}\pi^{0})-\pi^{0}(\partial_{\mu}\partial_{\nu}\partial_{\rho}K^{-})+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle\quad+\sqrt{2}\,\big{[}\bar{K}^{0}(\partial_{\mu}\partial_{\nu}\partial_{\rho}\pi^{-})-\pi^{-}(\partial_{\mu}\partial_{\nu}\partial_{\rho}\bar{K}^{0})\big{]}-\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle\quad-K^{-}\big{[}(\partial_{\mu}\partial_{\nu}\partial_{\rho}\eta)(-\cos\beta_{p}+\sqrt{2}\sin\beta_{p})+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle\qquad+(\partial_{\mu}\partial_{\nu}\partial_{\rho}\eta^{\prime})(\sin\beta_{p}+\sqrt{2}\cos\beta_{p})\big{]}+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle\quad+\big{[}\eta(-\cos\beta_{p}+\sqrt{2}\sin\beta_{p})+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle\qquad+\eta^{\prime}(\sin\beta_{p}+\sqrt{2}\cos\beta_{p})\big{]}(\partial_{\mu}\partial_{\nu}\partial_{\rho}K^{-})\Big{\\}}+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle+K^{\ast-,\mu\nu\rho}_{3}\Big{\\{}-K^{+}(\partial_{\mu}\partial_{\nu}\partial_{\rho}\pi^{0})+\pi^{0}(\partial_{\mu}\partial_{\nu}\partial_{\rho}K^{+})+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle\quad+\sqrt{2}\,\big{[}-K^{0}(\partial_{\mu}\partial_{\nu}\partial_{\rho}\pi^{+})+\pi^{+}(\partial_{\mu}\partial_{\nu}\partial_{\rho}K^{0})\big{]}+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle\quad+K^{+}\big{[}(\partial_{\mu}\partial_{\nu}\partial_{\rho}\eta)(-\cos\beta_{p}+\sqrt{2}\sin\beta_{p})+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle\qquad+(\partial_{\mu}\partial_{\nu}\partial_{\rho}\eta^{\prime})(\sin\beta_{p}+\sqrt{2}\cos\beta_{p})\big{]}-\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle\quad-\big{[}\eta(-\cos\beta_{p}+\sqrt{2}\sin\beta_{p})+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle\qquad+\eta^{\prime}(\sin\beta_{p}+\sqrt{2}\cos\beta_{p})\big{]}(\partial_{\mu}\partial_{\nu}\partial_{\rho}K^{+})\Big{\\}}+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle+\omega^{\mu\nu\rho}_{3}\Big{\\{}(-\cos\beta_{w_{3}}+\sqrt{2}\sin\beta_{w_{3}})\times\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle\quad\times\big{[}\bar{K}^{0}(\partial_{\mu}\partial_{\nu}\partial_{\rho}K^{0})-K^{0}(\partial_{\mu}\partial_{\nu}\partial_{\rho}\bar{K}^{0})-\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle\qquad-K^{+}(\partial_{\mu}\partial_{\nu}\partial_{\rho}K^{-})+K^{-}(\partial_{\mu}\partial_{\nu}\partial_{\rho}K^{+})\big{]}\Big{\\}}+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle+\phi^{\mu\nu\rho}_{3}\Big{\\{}(\sin\beta_{w_{3}}+\sqrt{2}\cos\beta_{w_{3}})\times\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle\quad\times\big{[}\bar{K}^{0}(\partial_{\mu}\partial_{\nu}\partial_{\rho}K^{0})-K^{0}(\partial_{\mu}\partial_{\nu}\partial_{\rho}\bar{K}^{0})-\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle\qquad-K^{+}(\partial_{\mu}\partial_{\nu}\partial_{\rho}K^{-})+K^{-}(\partial_{\mu}\partial_{\nu}\partial_{\rho}K^{+})\big{]}\Big{\\}}\Big{)}\,.\vphantom{\bigg{(}\bigg{)}}$

$\displaystyle\mathcal{L}_{w_{3}v_{1}p}=\vphantom{\bigg{(}\bigg{)}}$ (127)
$\displaystyle=\,$ $\displaystyle
g_{w_{3}v_{1}p}\,\varepsilon^{\mu\nu\rho\sigma}\,\mathrm{tr}\big{[}W_{3,\mu\alpha\beta}\big{\\{}(\partial_{\nu}V_{1,\rho}),\,(\partial^{\alpha}\partial^{\beta}\partial_{\sigma}P)\big{\\}}_{+}\big{]}=\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle=\,$
$\displaystyle\frac{g_{w_{3}v_{1}p}}{4}\,\varepsilon^{\mu\nu\rho\sigma}\,\Big{(}\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle+\rho^{0}_{3,\mu\alpha\beta}\Big{\\{}-(\partial_{\nu}\bar{K}^{\ast
0}_{\rho})(\partial^{\alpha}\partial^{\beta}\partial_{\sigma}K^{0})-\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle\quad-(\partial_{\nu}K^{\ast
0}_{\rho})(\partial^{\alpha}\partial^{\beta}\partial_{\sigma}\bar{K}^{0})+(\partial_{\nu}K^{\ast+}_{\rho})(\partial^{\alpha}\partial^{\beta}\partial_{\sigma}K^{-})+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle\quad+(\partial_{\nu}K^{\ast-}_{\rho})(\partial^{\alpha}\partial^{\beta}\partial_{\sigma}K^{+})+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle\quad+2\,\big{[}(\partial_{\nu}\omega_{\rho})\cos\beta_{v}-(\partial_{\nu}\phi_{\rho})\sin\beta_{v}\big{]}(\partial^{\alpha}\partial^{\beta}\partial_{\sigma}\pi^{0})+\vphantom{\bigg{(}\bigg{)}}$
$\displaystyle\quad+2\,(\partial_{\nu}\rho_{\rho}^{0})\big{[}(\partial^{\alpha}\partial^{\beta}\partial_{\sigma}\eta)\cos\beta_{p}-\vphantom{\bigg{(}\bigg{)}}$