From well-known tensor mesons to yet unknown axial-tensor mesons

Shahriyar Jafarzadea111<EMAIL_ADDRESS>Arthur Vereijkena222
<EMAIL_ADDRESS>Milena Piotrowskaa333<EMAIL_ADDRESS>and Francesco Giacosaa,b444<EMAIL_ADDRESS>
aInstitute of Physics, Jan Kochanowski University,
ul. Uniwersytecka 7, 25-406,, Kielce, Poland ,

b Institute for Theoretical Physics, J. W. Goethe University,
Max-von-Laue-Str. 1, 60438 Frankfurt am Main, Germany

Abstract

While the ground-state tensor ($J^{PC}=2^{++}$) mesons $a_{2}(1320)$,
$K_{2}^{\ast}(1430)$, $f_{2}(1270)$, and $f_{2}^{\prime}(1525)$ are well known
experimentally and form an almost ideal nonet of quark-antiquark states, their
chiral partners, the ground-states axial-tensor ($J^{PC}=2^{--}$) mesons are
poorly settled: only the kaonic member $K_{2}(1820)$ of the nonet has been
experimentally found, whereas the isovector state $\rho_{2}$ and two isoscalar
states $\omega_{2}$ and $\phi_{2}$ are still missing. Here, we study masses,
strong, and radiative decays of tensor and axial-tensor mesons within a chiral
model that links them: the established tensor mesons are used to test the
model and to determine its parameters, and subsequently various predictions
for their chiral partners, the axial-tensor mesons, are obtained. The results
are compared to current lattice QCD outcomes as well as to other theoretical
approaches and show that the ground-state axial-tensor mesons are expected to
be quite broad, the vector-pseudoscalar mode being the most prominent decay
mode followed by the tensor-pseudoscalar one. Nonetheless, their experimental
finding seems to be possible in ongoing and/or future experiments.

###### Contents

  1. 1 Introduction
  2. 2 Chiral model for spin-2 mesons
    1. 2.1 Mesonic nonets
    2. 2.2 Chiral invariant Lagrangian terms
  3. 3 Results
    1. 3.1 Masses of (axial-)tensor mesons
    2. 3.2 Decays of tensor mesons
      1. 3.2.1 $T\longrightarrow P^{(1)}+P^{(2)}$ decay mode
      2. 3.2.2 $T\longrightarrow V+P$ decay mode
    3. 3.3 Radiative decays
      1. 3.3.1 Vector-vector decay mode of tensor mesons
    4. 3.4 Decays of axial-tensor mesons
      1. 3.4.1 $A_{2}\longrightarrow V+P$ decay mode
      2. 3.4.2 $A_{2}\longrightarrow A_{1}+P$ decay mode
      3. 3.4.3 $A_{2}\longrightarrow T+P$ decay mode
  4. 4 Conclusions

##  1\. Introduction

The classification of the observed mesonic resonances into nonets of quark-
antiquark ($q\bar{q}$) states using flavor symmetry (as well as chiral
symmetry) is one of the greatest achievements of modern high energy physics.

In the non-relativistic spectroscopic notation, $q\bar{q}$ nonets are
specified by the expression $n^{2S+1}L_{J}$, with $n$ being the radial quantum
number, and $S$, $L$ and $J$ being the spin, spacial, and total angular
momenta of the quark-antiquark system, respectively. Upon restricting to the
first radial excitation $n=1$, the situation is summarized in Table 1, in
which also the relativistic notation $J^{PC}$ (with parity $P=(-1)^{L+1}$ and
charge conjugation $C=(-1)^{L+S}$) is reported.

Examples of very well-known $q\bar{q}$ mesonic nonets include the pseudoscalar
mesons with $J^{PC}=0^{-+}$ (spectroscopic notation: ${}^{1}S_{0}$), the
vector mesons with $J^{PC}=1^{--}$ (${}^{3}S_{1}$), and the axial-vector
mesons with $J^{PC}=1^{++}$ (${}^{3}P_{1}$) [1]. The above resonances are in
good agreement with the quark model of Ref. [2] and the Bethe-Salpeter
approach of Ref. [3]. The scalar mesons with $J^{PC}=0^{++}$ (${}^{1}S_{0}$)
are in the centre of a long-standing debate, e.g. Ref. [4, 5, 6, 7, 8, 9, 10,
11]: there is no lack of candidates, on the contrary there are more states
than expected from $q\bar{q}$ alone, in agreement with the intrusion of four-
quark states and the scalar glueball. Summarizing, upon including the nonet of
pseudovector mesons with $J^{PC}=1^{+-}$ (${}^{1}P_{1}$) and the orbitally
excited vector mesons with $J^{PC}=1^{--}$ (${}^{3}D_{1}$), for all nonets
with $n=1$ and $J<2$ there are corresponding resonances in the PDG (with the
interesting exception of the missing orbitally excited vector state $\phi(?)$,
with an expected mass at about 1.9 GeV, see Ref. [12]).

Moving to $J\geq 2$, one recognizes the well established nonet of tensor
mesons with $J^{PC}=2^{++}$ (${}^{3}P_{2}$), see e.g Refs. [13, 14, 6], and
the quite well-known nonet of pseudotensor mesons with $J^{PC}=2^{-+}$
(${}^{1}D_{2}$), even if, for the latter, some questions are still open [15,
16]. The nonet of mesons with $J^{PC}=3^{--}$ (${}^{3}D_{3}$) is also well
known, as the recent detailed report of Ref. [17] shows.

On the other hand, as it is visible from Table 1, the nonet of the so-called
axial-tensor mesons with $J^{PC}=2^{--}$ (${}^{3}D_{2}$) is extremely poorly
known. Only the kaonic member $K_{2}(1820)$ (with eventual mixing with
$K_{2}(1770)$) is listed. The isovector member $\rho_{2}$ and the isotensor
ones $\omega_{2}$ and $\phi_{2}$ were not yet discovered. One may actually
regard this nonet as a sort of “missing multiplet”. It is then evident that it
deserves further theoretical and experimental investigations.

One of the main goals of the present paper is a phenomenological study of
masses, strong, and radiative decays of tensor and, most importantly, of
axial-tensor mesons. In the latter case, predictions for the missing states
are delivered. For our purposes, we shall use a well defined chiral model of
QCD, called extended Linear Sigma Model (eLSM), that was previously
successfully applied to pseudoscalar mesons and vector mesons and their chiral
partners, the scalar and the axial-vector mesons [18] (see the first 4 entries
of Table 1). Within this context, the linear realization of chiral symmetry,
together with its explicit and spontaneous breaking patterns, allows to study
chiral partners (that is, $q\bar{q}$ mesons with the same $J$ related by
chiral transformations) on the same footing. Applications of the eLSM to the
excited (pseudo)scalar mesons [19, 20], to pseudovector and orbitally excited
vector mesons [21] and more recently to hybrid mesons [22] were undertaken.

It then seems quite natural to apply the eLSM to the tensor sector, thus
describing in a chiral framework both tensor and axial-tensor mesons. Since
masses and decays of the ground-state tensor mesons are well-established, the
tensor mesons allow for the determination of the (additional) parameters of
the model and for a test of its validity in the energy range between 1-1.6
GeV. As we shall see, the overall agreement of theory with experiments is
rather good and therefore confirms the predominant $q\bar{q}$ nature of the
ground-state tensor mesons. [For different interpretations of tensor mesons
and some other states listed in Table 1, see e.g. Refs. [23, 24, 25]].

Next, we move to the yet mostly unknown axial-tensor mesons. The estimate of
masses for the nonet members is possible thanks to the tensor states and to
the known axial-tensor kaonic state $K_{2}^{*}(1820)$. The decays of axial-
tensor mesons into a vector-pseudoscalar pair turns out to be quite large and
dominant. Also the decays into tensor-pseudoscalar and axial-vector-
pseudoscalar are sizable and represent interesting decay modes for the future
search of the missing states of this nonet.

$n^{2S+1}L_{J}$ | $J^{PC}$ |  | I=1
---
$u\overline{d}$, $d\overline{u}$
$\frac{d\overline{d}-u\overline{u}}{\sqrt{2}}$
| I=1/2
---
$u\overline{s}$, $d\overline{s}$
$s\overline{d}$, $s\overline{u}$
| I=0
---
$\approx\frac{u\overline{u}+d\overline{d}}{\sqrt{2}}$
| I=0
---
$\approx s\overline{s}$
| Meson names
---
Chiral Partners
$1^{1}S_{0}$ | $0^{-+}$ | $\pi$ | $K$ | $\eta(547)$ | $\eta^{\prime}(958)$ | Pseudoscalar |  $J=0$
$1^{3}P_{0}$ | $0^{++}$ | $a_{0}(1450)$ | $K_{0}^{\star}(1430)$ | $f_{0}(1370)$ | $f_{0}(1500)/f_{0}(1710)$ | Scalar
$1^{3}S_{1}$ | $1^{--}$ | $\rho(770)$ | $K^{\star}(892)$ | $\omega(782)$ | $\phi(1020)$ | Vector |  $J=1$
$1^{3}P_{1}$ | $1^{++}$ | $a_{1}(1260)$ | $K_{1A}$ | $f_{1}(1285)$ | $f_{1}^{\prime}(1420)$ | Axial-vector
$1^{1}P_{1}$ | $1^{+-}$ | $b_{1}(1235)$ | $K_{1B}$ | $h_{1}(1170)$ | $h_{1}(1415)$ | Pseudovector |  $J=1^{\star}$
$1^{3}D_{1}$ | $1^{--}$ | $\rho(1700)$ | $K^{\star}(1680)$ | $\omega(1650)$ | $\phi(???)$ | Excited-vector
$1^{3}P_{2}$ | $2^{++}$ | $a_{2}(1320)$ | $K_{2}^{\star}(1430)$ | $f_{2}(1270)$ | $f_{2}^{\prime}(1525)$ | Tensor |  $J=2$
$1^{3}D_{2}$ | $2^{--}$ | $\rho_{2}(???)$ | $K_{2}(1820)$ | $\omega_{2}(???)$ | $\phi_{2}(???)$ | Axial-tensor
$1^{1}D_{2}$ | $2^{-+}$ | $\pi_{2}(1670)$ | $K_{2}(1770)$ | $\eta_{2}(1645)$ | $\eta_{2}(1870)$ | Pseudotensor |
$1^{3}D_{3}$ | $3^{--}$ | $\rho_{3}(1690)$ | $K_{3}^{\star}(1780)$ | $\omega_{3}(1670)$ | $\phi_{3}(1850)$ |  $J=3$ \- Tensor |
Table 1. List of conventional $q\bar{q}$ mesons following the quark model
review of Ref. [1] up to $J=3$ and for the lowest radial excitation. The
columns refer to different values of the isospin $I$. The first eight nonets
are grouped into four chiral partners. The kaonic states $K_{1,A}$ and
$K_{1,B}$ mix and give rise to $K_{1}(1270)$ and $K_{1}(1400)$. Note, the
entries for which no assignment is proposed are the orbitally excited vector
$\phi(???)$ [12] as well as three axial-tensor states
$\rho_{2}(???)\equiv\rho_{2}$, $\omega_{2}(???)\equiv\omega_{2}$ and
$\phi_{2}(???)\equiv\phi_{2}$, which represent one of the main subject of this
work.

Light mesons (both conventional and exotic) are in the centre of various
experimental searches, such as the BESIII [26, 27], CLAS12 [28], COMPASS [29],
GlueX [30, 31, 32] experiments as well as the planned PANDA experiment [33].
In particular, photoproduction of tensor mesons represents an active
experimental [34] and phenomenological [35, 36, 37] topic. It is well possible
that the missing axial-tensor mesons discussed in this work can be discovered
in the future.

This work is organized as follows: in Sec. 2 we introduce the fields and the
Lagrangian terms describing masses and decays of (axial-)tensor states; in
Sec. 3 we show the corresponding results. Finally, in Sec. 4 our conclusions
are outlined.

##  2\. Chiral model for spin-2 mesons

In this Section, we first recall the form of flavour and chiral multiplets and
then we present the novel chiral Lagrangian terms that involve (axial-)tensor
mesons.

For conventional $q\bar{q}$ pairs, the spin $\vec{S}$ can take the value
$S\in\\{0\,,1\\}$, while the angular momentum $\vec{L}$ any value
$L=0,1,2,...$. As a consequence, the parity eigenvalue $P=(-1)^{L+1}$ and the
charge conjugation eigenvalue $C=(-1)^{L+S}$ follow. The spin and angular
momentum couple to the total angular momentum $\vec{J}=\vec{L}+\vec{S}$.

For $S=0$, the $q\bar{q}$ state reads $\ket{S=0}$=
$\frac{1}{\sqrt{2}}\ket{\uparrow_{q}\downarrow_{\overline{q}}-\downarrow_{q}\uparrow_{\overline{q}}}$,
while for $S=1$ is $\ket{S=1}$ = { $\ket{\uparrow_{q}\uparrow_{\overline{q}}}$
,
$\frac{1}{\sqrt{2}}\ket{\uparrow_{q}\downarrow_{\overline{q}}+\downarrow_{q}\uparrow_{\overline{q}}}$
, $\ket{\downarrow_{q}\downarrow_{\overline{q}}}$ }. Thus, for $S=0$ the
possible quantum numbers for conventional $q\bar{q}$ mesons are
$J^{PC}\in\Big{\\{}\textbf{even}^{-+},\textbf{odd}^{+-}\Big{\\}}$, while for
$S=1$ they are
$J^{PC}\in\Big{\\{}\\{0,1,2,...\\}^{++},\\{1,2,...\\}^{--}\Big{\\}}$. [Mesons
with $J^{PC}\in\Big{\\{}\textbf{even}^{+-},\textbf{odd}^{-+}\Big{\\}}$ as well
as $0^{--}$ are considered exotic.]

In this work, we concentrate on (pseudo)scalar, (axial-)vector, as well as
(axial-)tensor multiplets. These fields enter into appropriate chirally
invariant Lagrangian terms that describes the masses and the decays of
(axial-)tensor mesons.

###  2.1. Mesonic nonets

The ground-state pseudoscalar mesons with $J^{PC}=0^{-+}$ (${}^{1}S_{0}$)
comprise 3 pions, four kaons, the $\eta(547)$ and the $\eta^{\prime}(958)$.
They can be collected into the following nonet with light-quark elements
$P_{ij}\equiv 2^{-1/2}\bar{q}_{j}i\gamma^{5}q_{i}$:

$P=\frac{1}{\sqrt{2}}\begin{pmatrix}\frac{\eta_{N}+\pi^{0}}{\sqrt{2}}&\pi^{+}&K^{+}\\\
\pi^{-}&\frac{\eta_{N}-\pi^{0}}{\sqrt{2}}&K^{0}\\\
K^{-}&\bar{K}^{0}&\eta_{S}\end{pmatrix}\,\text{,}$ (2.1)

where $\eta_{N}\equiv\sqrt{1/2}\,(\bar{u}u+\bar{d}d)$ stands for the purely
non-strange state and $\eta_{S}\equiv\bar{s}s$ for the purely strange one. The
isoscalar physical fields emerge upon mixing reported in e.g. Ref. [38] as:

$\left(\begin{array}[c]{c}\eta(547)\\\
\eta^{\prime}\equiv\eta(958)\end{array}\right)=\left(\begin{array}[c]{cc}\cos\beta_{P}&\sin\beta_{P}\\\
-\sin\beta_{P}&\cos\beta_{P}\end{array}\right)\left(\begin{array}[c]{c}\eta_{N}\\\
\eta_{S}\end{array}\right)\text{ ,}$ (2.2)

with a large mixing angle $\beta_{P}=-43.4^{\circ}$. This sizable admixture is
a consequence of the so-called $U_{A}(1)$ axial anomaly [39, 40]. Namely,
pseudoscalar mesons belong to a so-called “heterochiral” multiplet [41], see
below.

Scalar mesons with $J^{PC}=0^{++}$ (following from $L=S=1$, thus
${}^{3}P_{0}$) are the chiral partners of the pseudoscalar ones. The
discussion about their assignment is still ongoing and there is not yet a
general agreement. Nevertheless, the set of resonances {$a_{0}(1450),$
$K_{0}^{\ast}(1430),$ $f_{0}(1370),$ $f_{0}(1500)/f_{0}(1710)$} reported in
Table 1 is favoured (here, also the scalar glueball is expected to enter with
$f_{0}(1710)$ being a good candidate). Having the quantum numbers of the
vacuum, the isoscalar members of this nonet may condense, see Sec. 2.2. By
introducing the currents $S_{ij}\equiv 2^{-1/2}\bar{q}_{j}q_{i}$, the nonet of
scalar states can be written as

$S=\frac{1}{\sqrt{2}}\begin{pmatrix}\frac{\sigma_{N}+a_{0}^{0}}{\sqrt{2}}&a_{0}^{+}&K_{0}^{\star+}\\\
a_{0}^{-}&\frac{\sigma_{N}-a_{0}^{0}}{\sqrt{2}}&K_{0}^{\star 0}\\\
K_{0}^{\star-}&\bar{K}_{0}^{\star 0}&\sigma_{S}\end{pmatrix}\,\text{.}$ (2.3)

The isoscalar sector cannot be described by a simple $2\times 2$ mixing
because (at least) the scalar glueball should be taken into account, thus
implying a more complicated $3\times 3$ mixing patter leading to the
resonances $f_{0}(1370)$, $f_{0}(1500)$, and $f_{0}(1710)$, e.g. Refs. [42,
20, 43, 44, 45, 46]. Yet, these fields do not appear as final states of decays
of (axial-)tensor mesons, therefore its detailed treatment of the scalar-
isoscalar mixing is not required in this work.

The third entry in Table 1 refers to the $J^{PC}=1^{--}$ (${}^{3}S_{1}$) nonet
with $L=0$ and $S=1$. These are the very well known vector states
{$\rho(770),$ $K^{\ast}(892),$ $\omega(782),$ $\phi(1020)$}. The matrix
$V^{\mu}$ with elements $V_{ij}^{\mu}=2^{-1/2}\bar{q}_{j}\gamma^{\mu}q_{i}$
has the form

$V^{\mu}=\frac{1}{\sqrt{2}}\begin{pmatrix}\frac{\omega_{N}^{\mu}+\rho^{0\mu}}{\sqrt{2}}&\rho^{+\mu}&K^{\ast+\mu}\\\
\rho^{-\mu}&\frac{\omega_{N}^{\mu}-\rho^{0\mu}}{\sqrt{2}}&K^{\ast 0\mu}\\\
K^{\ast-\mu}&\bar{K}^{\ast 0\mu}&\omega_{S}^{\mu}\end{pmatrix}\,,$ (2.4)

where $\omega_{N}$ and $\omega_{S}$ are purely non-strange and strange,
respectively. Similar to the pseudoscalar case, the physical fields arise upon
mixing

$\left(\begin{array}[c]{c}\omega(782)\\\
\phi(1020)\end{array}\right)=\left(\begin{array}[c]{cc}\cos\beta_{V}&\sin\beta_{V}\\\
-\sin\beta_{V}&\cos\beta_{V}\end{array}\right)\left(\begin{array}[c]{c}\omega_{N}\\\
\omega_{S}\end{array}\right)\text{ ,}$ (2.5)

where the small isoscalar-vector mixing angle $\beta_{V}=-3.9^{\circ}$ is
taken from the PDG [1]. Thus, the physical states $\omega$ and $\phi$ are
dominated by non-strange and strange components, respectively. This is in
agreement with the so-called “homochiral” nature of these states [41], see
also below.

The chiral partners of the vector mesons [47] are realized for the quantum
numbers $L=S=1$ (just as for the scalar mesons discussed previously) coupled
to $J^{PC}=1^{++}$ (${}^{3}P_{1}$): this is the so-called axial-vector meson
nonet $A_{1}$, that arises from the microscopic current $A_{1,ij}^{\mu}\equiv
2^{-1/2}\bar{q}_{j}\gamma^{5}\gamma^{\mu}q_{i}$:

$A_{1}^{\mu}=\frac{1}{\sqrt{2}}\begin{pmatrix}\frac{f_{1,N}^{\mu}+a_{1}^{0\mu}}{\sqrt{2}}&a_{1}^{+\mu}&K_{1A}^{+\mu}\\\
a_{1}^{-\mu}&\frac{f_{1,N}^{\mu}-a_{1}^{0\mu}}{\sqrt{2}}&K_{1A}^{0\mu}\\\
K_{1A}^{-\mu}&\bar{K}_{1A}^{0\mu}&f_{1,S}^{\mu}\end{pmatrix}\text{ }\,.$ (2.6)

The isoscalar sector reads

$\left(\begin{array}[c]{c}f_{1}(1285)\\\
f_{1}(1420)\end{array}\right)=\left(\begin{array}[c]{cc}\cos\beta_{A_{1}}&\sin\beta_{A_{1}}\\\
-\sin\beta_{A_{1}}&\cos\beta_{A_{1}}\end{array}\right)\left(\begin{array}[c]{c}f_{1,N}^{\mu}\\\
f_{1,S}^{\mu}\end{array}\right)\text{ ,}$ (2.7)

where the mixing angle $\beta_{A_{1}}$ is expected to be small because of the
homochiral nature of the multiplet [41], see also Ref. [48]. Here, we shall
set this angle to zero for simplicity.

An important digression is needed for the kaonic axial-vector mesons. Since
these states are not eigenstates of $C$, the state $K_{1,A}$ belonging to the
$J^{PC}=1^{++}$ (${}^{3}P_{1}$) axial-vector nonet and $K_{1,B}$ belonging to
the $J^{PC}=1^{+-}$ (${}^{1}P_{1}$) pseudovector nonet mix:

$\left(\begin{array}[c]{c}K_{1}(1270)\\\
K_{1}(1400)\end{array}\right)^{\mu}=\left(\begin{array}[c]{cc}\cos\varphi_{K}&-i\sin\varphi_{K}\\\
-i\sin\varphi_{K}&\cos\varphi_{K}\end{array}\right)\left(\begin{array}[c]{c}K_{1,A}\\\
K_{1,B}\end{array}\right)^{\mu}\text{ .}$ (2.8)

For a detailed study on the $K_{1}(1270)$/$K_{1}(1400)$ system using an
approach similar to the one used in this work, see Ref. [48]. The numerical
value $\varphi_{K}=\left(56.4\pm 4.3\right)^{\circ}$ [48, 47] implies that
$K_{1}(1270)$ is predominantly $K_{1,B}$ and $K_{1}(1400)$ is predominantly
$K_{1,A}$ (but the mixing is large). The study of this mixing angle via
semileptonic decays can be found in e.g. Ref. [49].

The next (fifth and sixth) entries of Table 1 are the pseudovector (from which
the $K_{1,B}$ above stems) and their chiral partners, the (orbitally) excited
vector mesons. One can build similar nonets (see Refs. [21, 22]), yet they are
not part of the decay products of the (axial-)tensor mesons studied in this
paper, therefore we omit them in the following.

Finally, we present the two tensor ($J=2$) meson nonets (seventh and eighth
rows of Table 1), which constitute the main subject of this work.

The well-known $J^{PC}=2^{++}$ (${}^{3}P_{2}$) tensor states with elements
$T_{ij}^{\mu\nu}=2^{-1/2}\bar{q}_{j}(i\gamma^{\mu}\partial^{\nu}+\cdots)q_{i}$
form an almost ideal nonet of quark-antiquark states:

$T^{\mu\nu}=\frac{1}{\sqrt{2}}\begin{pmatrix}\frac{f_{2,N}^{\mu\nu}+a_{2}^{0\mu\nu}}{\sqrt{2}}&a_{2}^{+\mu\nu}&K_{2}^{\ast+\mu\nu}\\\
a_{2}^{-\mu\nu}&\frac{f_{2,N}^{\mu\nu}-a_{2}^{0\mu\nu}}{\sqrt{2}}&K_{2}^{\ast
0\mu\nu}\\\ K_{2}^{\ast-\mu\nu}&\bar{K}_{2}^{\ast
0\mu\nu}&f_{2,S}^{\mu\nu}\end{pmatrix}\,.$ (2.9)

The physical isoscalar-tensor states are

$\left(\begin{array}[c]{c}f_{2}(1270)\\\
f_{2}^{\prime}(1525)\end{array}\right)=\left(\begin{array}[c]{cc}\cos\beta_{T}&\sin\beta_{T}\\\
-\sin\beta_{T}&\cos\beta_{T}\end{array}\right)\left(\begin{array}[c]{c}f_{2,N}\\\
f_{2,S}\end{array}\right)\text{ ,}$ (2.10)

where $\beta_{T}\simeq 5.7^{\circ}$ is the small mixing angle reported in the
PDG, in agreement with the fact that tensor mesons belong to a homochiral
multiplet, just as (axial-)vector states. Later on, we shall re-calculate the
mixing angle $\beta_{T}$ by a fit to data on decays. The decays of tensor
mesons were studied in detail in Refs. [14, 13] and refs. therein, and fit
well into the standard $q\bar{q}$ picture [2]. This point is corroborated by
the results of this work, see the next Section.

The axial-tensor nonet with $J^{PC}=2^{--}$ (${}^{3}D_{2}$) with currents
$2^{-1/2}\bar{q}_{j}\left(\gamma^{5}\gamma^{\mu}\partial^{\nu}+...\right)q_{i}$,
builds the chiral partners of the previously introduced tensor mesons. This
nonet is however poorly known: at present, only the kaonic states
$K_{2}(1820)$ and $K_{2}(1770)$ have been measured, the heavier one being
considered as predominately axial-tensor and the lighter mainly pseudotensor,
see Table 1 (as for the axial-vector and pseudovector kaons, mixing is
possible, but at present not calculable due to lack of experimental data). For
some recent theoretical works devoted to the vacuum properties and thermal
properties of these resonances, see Refs. [50, 51] and Refs. [52, 53]
respectively. The matrix for this nonet reads:

$\displaystyle A_{2}^{\mu\nu}$
$\displaystyle=\frac{1}{\sqrt{2}}\begin{pmatrix}\frac{\omega_{2,N}^{\mu\nu}+\rho_{2}^{0\mu\nu}}{\sqrt{2}}&\rho_{2}^{+\mu\nu}&K_{2}^{+\mu\nu}\\\
\rho_{2}^{-\mu\nu}&\frac{\omega_{2,N}^{\mu\nu}-\rho_{2}^{0\mu\nu}}{\sqrt{2}}&K_{2}^{0\mu\nu}\\\
K_{2}^{-\mu\nu}&\bar{K}_{2}^{0\mu\nu}&\omega_{2,S}^{\mu\nu}\end{pmatrix}\,.$
(2.11)

The isoscalar mixing is expected to be small in view of the homochiral nature
of the corresponding chiral multiplet in which this nonet is embedded [41]
(see also below). Thus, for the isoscalar members:
$\omega_{2,N}\simeq\omega_{2}$ and $\omega_{2,S}\simeq\phi_{2}$.

The transformation rules under parity, charge conjugation, and flavor
transformations for all the relevant nonets of this work
($P,S,V,A_{1},T,A_{2}$) are summarized in Table 2.

Nonet | Parity $(P)$ | Charge conjugation $(C)$ | Flavour $(U_{V}(3))$
---|---|---|---
$0^{-+}=P$ | $-P(t,-\vec{x})$ | $P^{t}$ | $UPU^{\dagger}$
$0^{++}=S$ | $S(t,-\vec{x})$ | $S^{t}$ | $USU^{\dagger}$
$1^{--}=V^{\mu}$ | $V_{\mu}(t,-\vec{x})$ | $-(V^{\mu})^{t}$ | $UV^{\mu}U^{\dagger}$
$1^{++}=A_{1}^{\mu}$ | $-A_{1\,\mu}(t,-\vec{x})$ | $(A_{1}^{\mu})^{t}$ | $UA_{1}^{\mu}U^{\dagger}$
$2^{++}=T^{\mu\nu}$ | $T_{\mu\nu}(t,-\vec{x})$ | $(T^{\mu\nu})^{t}$ | $UT^{\mu\nu}U^{\dagger}$
$2^{--}=A_{2}^{\mu\nu}$ | $-A_{2\,\mu\nu}(t,-\vec{x})$ | $-(A_{2}^{\mu\nu})^{t}$ | $UA_{2}^{\mu\nu}U^{\dagger}$
Table 2. Mesonic nonet transformations under P, C and $U_{V}(3)$.

Finally, we introduce the chiral multiplets which contain chiral partners and
have simple transformations under the chiral group $U_{R}(3)\times U_{L}(3)$.

The pseudoscalar and scalar nonets build the matrix

$\displaystyle\Phi=S+iP\text{ ,{}}$ (2.12)

whose “heterochiral” transformation can be found in Table 3. As a consequence,
the strange-nonstrange mixing angle can be large, as it is the case for the
pseudoscalar mesons (the scalar one is not easy to determine because one has a
three-state mixing problem) and possibly for the pseudotesnor mesons [41, 15].

Vector and axial-vector nonets and similarly tensor and axial-tensor nonets
build the following left-handed and right-handed objects:

$\displaystyle L^{\mu}:=V^{\mu}+A_{1}^{\mu}\ \text{ ,
}R^{\mu}:=V^{\mu}-A_{1}^{\mu}\text{ ,}$ (2.13)
$\displaystyle\mathbf{L}^{\mu\nu}:=T^{\mu\nu}+A_{2}^{\mu\nu}\text{ ,
}\mathbf{R}^{\mu\nu}:=T^{\mu\nu}-A_{2}^{\mu\nu}\text{ .}$ (2.14)

Their homochiral transformations are also listed in Table 3. One expects small
isoscalar mixing angles in these four nonets [41].

The fields listed in Table 3 are the needed ingredients to write down chiral
Lagrangian terms, as we shall show in the next subsection.

Nonet | Parity $(P)$ | Charge conjugation $(C)$ | $U_{R}(3)\times U_{L}(3)$
---|---|---|---
$\Phi(t,\vec{x})$ | $\Phi^{\dagger}(t,-\vec{x})$ | $\Phi^{t}(t,\vec{x})$ | $U_{L}\Phi U^{\dagger}_{R}$
$R^{\mu}(t,\vec{x})$ | $L_{\mu}(t,-\vec{x})$ | $-(L^{\mu}(t,\vec{x}))^{t}$ | $U_{R}R^{\mu}U_{R}^{\dagger}$
$L^{\mu}(t,\vec{x})$ | $R_{\mu}(t,-\vec{x})$ | $-(R^{\mu}(t,\vec{x}))^{t}$ | $U_{L}L^{\mu}U_{L}^{\dagger}$
$\mathbf{R}^{\mu\nu}(t,\vec{x})$ | $\mathbf{L}_{\mu\nu}(t,-\vec{x})$ | $(\mathbf{L}^{\mu\nu}(t,\vec{x}))^{t}$ | $U_{R}\mathbf{R}^{\mu\nu}U^{\dagger}_{R}$
$\mathbf{L}^{\mu\nu}(t,\vec{x})$ | $\mathbf{R}_{\mu\nu}(t,-\vec{x})$ | $(\mathbf{R}^{\mu\nu}(t,\vec{x}))^{t}$ | $U_{L}\mathbf{L}^{\mu\nu}U^{\dagger}_{L}$
Table 3. Transformations of the chiral multiplets under P, C, and
$U_{R}(3)\times U_{L}(3)$.

###  2.2. Chiral invariant Lagrangian terms

The standard eLSM Lagrangian is a function of (pseudo)scalar and
(axial-)vector mesons as well as a dilaton (or scalar glueball) field $G$, see
Refs. [18, 20, 22]:

$\displaystyle\mathcal{L}_{\text{eLSM}}=\mathcal{L}_{\text{dil}}+\text{Tr}\Big{[}\Big{(}D_{\mu}\Phi\Big{)}^{\dagger}\Big{(}D_{\mu}\Phi\Big{)}\Big{]}-m_{0}^{2}\Big{(}\frac{G}{G_{0}}\Big{)}^{2}\text{Tr}\Big{[}\Phi^{\dagger}\Phi\Big{]}-\frac{1}{4}\text{Tr}\Big{[}\Big{(}L_{\mu\nu}^{2}+R_{\mu\nu}^{2}\Big{)}\Big{]}$
$\displaystyle+\text{Tr}\Big{[}\Big{(}\frac{m_{\text{vec}}^{2}G^{2}}{2G_{0}^{2}}+\Delta\Big{)}\Big{(}L_{\mu}^{2}+R_{\mu}^{2}\Big{)}\Big{]}+\cdots\text{
,}$ (2.15)

where $\mathcal{L}_{\text{dil}}$ is the dilaton Lagrangian for the field $G$
[54, 55] and $G_{0}$ is the corresponding vacuum expectation value (v.e.v.).
[Note, the dilaton field $G$ will not be relevant but is formally important
for dilatation invariance, but will not appear as a decay product.] The
covariant derivative entering in the previous equation reads
$D_{\mu}\Phi:=\partial_{\mu}\Phi-ig_{1}\big{(}L_{\mu}\Phi-\Phi
R_{\mu}\big{)}$. The matrix $\Delta$ breaks flavour symmetry by including a
different mass for mesons carrying the strange quark:

$\Delta=\begin{pmatrix}\delta_{N}&0&0\\\ 0&\delta_{N}&0\\\
0&0&\delta_{S}\end{pmatrix}\;\;\text{where}\;\;\delta_{N}\thicksim m_{u}^{2}\
\text{ ,}\quad\delta_{S}\thicksim m_{s}^{2}\text{ .}$ (2.16)

The eLSM is based on the proper treatment of dilatation and chiral invariances
and their explicit and spontaneous breaking patterns. [For other chiral models
that make use of dilatation invariance, see e.g. Refs. [56, 57] and refs.
therein]. The results for masses and decays of (pseudo)scalar and
(axial-)vector mesons can be found in the afore mentioned Refs. [18, 20]. For
our present work, we need to recall some of its basic features.

(i) The scalar-isoscalar fields of the model acquire nonzero v.e.v. The
dilaton field $G$ condenses to a certain value $G_{0}$ (whose numerical value
is not needed here), which is related to the breaking of dilatation
invariance. Similarly, $q\bar{q}$ scalar-isoscalar fields develop a v.e.v. as
a consequence of spontaneous symmetry breaking (SSB), which in simple terms is
due to the Mexican hat form of the effective potential when $m_{0}^{2}<0$. As
a consequence, shifts are necessary:

$\displaystyle G\rightarrow G+G_{0}\text{ , }S\rightarrow
S+\Phi_{0}\qquad\text{with}\qquad\Phi_{0}:=\frac{1}{\sqrt{2}}\begin{pmatrix}\frac{\phi_{N}}{\sqrt{2}}&0&0\\\
0&\frac{\phi_{N}}{\sqrt{2}}&0\\\ 0&0&\phi_{S}\end{pmatrix}\text{ .}$ (2.17)

where the numerical values are reported in Table 4.

(ii) The axial-vector matrix must be shifted because SSB induces a mixing of
scalar and axial-vector fields, which is proportional to the parameter
$g_{1}=\frac{m_{a_{1}}}{Z_{\pi}f_{\pi}}\sqrt{1-\frac{1}{Z_{\pi}^{2}}}\simeq
5.8$ [18, 19]. This operation amounts to:

$\displaystyle A_{1\,\mu}\rightarrow
A_{1\,\mu}+\partial_{\mu}\mathcal{P},\,\,\mathcal{P}:=\frac{1}{\sqrt{2}}\begin{pmatrix}\frac{Z_{\pi}w_{\pi}(\eta_{N}+\pi^{0})}{\sqrt{2}}&Z_{\pi}w_{\pi}\pi^{+}&Z_{K}w_{K}K^{+}\\\
Z_{\pi}w_{\pi}\pi^{-}&\frac{Z_{\pi}w_{\pi}(\eta_{N}-\pi^{0})}{\sqrt{2}}&Z_{K}w_{K}K^{0}\\\
Z_{K}w_{K}K^{-}&Z_{K}w_{K}\bar{K}^{0}&Z_{\eta_{S}}w_{\eta_{S}}\eta_{S}\end{pmatrix}\text{
,}$ (2.18)

where the constant terms are also given in Table 4. The matrix $\mathcal{P}$
contains the pseudoscalar fields, just as the matrix $P$ of Eq. (2.1), but
proper renormalization factors have been included. The shifts in Eqs. (2.17)
and (2.18) shall also be applied to the novel Lagrangian terms introduced
below.

Parameters | Expressions |  Numerical values  [18]
---|---|---
$\,\;Z_{\pi}=Z_{\eta_{N}}$ | $\frac{m_{a_{1}}}{\sqrt{m_{a_{1}}^{2}-g_{1}^{2}\phi_{S}^{2}}}$ | $1.709$
$\,\;w_{\pi}=w_{\eta_{N}}$ | $\frac{g_{1}\phi_{N}}{m_{a_{1}}^{2}}$ | $0.683\,\text{GeV}^{-1}$
$\,\;Z_{K}$ | $\frac{2m_{K_{1A}}}{\sqrt{4m_{K_{1A}}^{2}-g_{1}^{2}(\phi_{N}+\sqrt{2}\phi_{S})^{2}}}$ | $1.604$
$\,\;w_{K}$ | $\frac{g_{1}(\phi_{N}+\sqrt{2}\phi_{S})}{2m_{K_{1,A}}^{2}}$ | $0.611\,\text{GeV}^{-1}$
$\,\;Z_{\eta_{S}}$ | $\frac{m_{f_{1,S}}}{\sqrt{m_{f_{1,S}}^{2}-2g_{1}^{2}\phi_{S}^{2}}}$ | $1.539$
$\,\;w_{\eta_{S}}$ | $\frac{\sqrt{2}g_{1}\phi_{S}}{m_{f_{1,S}}^{2}}$ | $0.554\,\text{GeV}^{-1}$
$\,\;\phi_{N}$ | $Z_{\pi}f_{\pi}$ | $0.158\,\text{GeV}$
$\,\;\phi_{S}$ | $\frac{2Z_{K}f_{K}-\phi_{N}}{\sqrt{2}}$ | $0.138\,\text{GeV}$
$\,\;g_{1}$ | $\frac{m_{a_{1}}}{Z_{\pi}f_{\pi}}\sqrt{1-\frac{1}{Z_{\pi}^{2}}}$ | $5.8$
Table 4. Numerical values of (scalar and axial-vector) shift-related
quantities needed in this work.

Next, we turn to the chirally invariant Lagrangian terms that contain
(axial-)tensor mesons. For convenience, we shall split them into four distinct
ones.

First, the chiral invariant Lagrangian that generates, among other
interactions, the masses of the spin-2 mesons reads

$\displaystyle\mathcal{L}_{\text{mass}}=\text{Tr}\Big{[}\Big{(}\frac{m_{\text{ten}}^{2}\,G^{2}}{2\,G_{0}^{2}}+\Delta^{\text{ten}}\Big{)}\Big{(}\mathbf{L}_{\mu\nu}^{2}+\mathbf{R}_{\mu\nu}^{2}\Big{)}\Big{]}+\frac{h_{1}^{\text{ten}}}{2}\text{Tr}\Big{[}\Phi^{\dagger}\Phi\Big{]}\text{Tr}\Big{[}\mathbf{L}^{\mu\nu}\mathbf{L}_{\mu\nu}+\mathbf{R}^{\mu\nu}\mathbf{R}_{\mu\nu}\Big{]}$
(2.19)
$\displaystyle+h_{2}^{\text{ten}}\text{Tr}\Big{[}\Phi^{\dagger}\mathbf{L}^{\mu\nu}\mathbf{L}_{\mu\nu}\Phi+\Phi\mathbf{R}^{\mu\nu}\mathbf{R}_{\mu\nu}\Phi^{\dagger}\Big{]}+2h_{3}^{\text{ten}}\text{Tr}\Big{[}\Phi\mathbf{R}^{\mu\nu}\Phi^{\dagger}\mathbf{L}_{\mu\nu}\Big{]}\text{,}$

where
$\Delta^{\text{ten}}=\text{diag}\\{\delta_{N}^{\text{ten}}\,,\delta_{N}^{\text{ten}}\,,\delta_{S}^{\text{ten}}\\}$;
one may set the (anyhow small) light-quark contribution
$\delta_{N}^{\text{ten}}=0$, since a term proportional to the identity matrix
can be reabsorbed into the term proportional to $m_{\text{ten}}$; the element
$\delta_{S}^{\text{ten}}$ is then obtained from the mass relation
$\delta_{S}^{\text{ten}}=m_{K_{2}}^{2}-m_{\textbf{a}_{2}}^{2}\simeq
0.3\,\text{GeV}^{2}$ using PDG values for the tensor mesons (details in Sec.
2.18). Besides the term proportional to $\Delta^{\text{ten}}$, all the other
terms involve dimensionless coupling constants.

According to the large-$N_{c}$ limit, the coupling constants
$h_{2}^{\text{ten}}$ and $h_{3}^{\text{ten}}$ scale as $N_{c}^{-1}$, while
$h_{1}^{\text{ten}}$ as $N_{c}^{-3}$, thus being suppressed. The mass term
$m_{\text{ten}}$ is $N_{c}$-independent (it is a mass term that sets the mass
scale of (axial-)tensor mesons), while the condensate $G_{0}$ scales as
$N_{c}$. As a consequence, the dimensionless coupling
$m_{\text{ten}}^{2}/G_{0}^{2}$ scales as $N_{c}^{-2}$.

We then turn to other interaction terms that generate the decays of
(axial-)tensor mesons. The first of such terms involves solely left- and
right-handed chiral fields:

$\displaystyle\mathcal{L}_{g_{2}^{\text{ten}}}=\frac{g_{2}^{\text{ten}}}{2}\Big{(}\text{Tr}\Big{[}\mathbf{L}_{\mu\nu}\\{L^{\mu},L^{\nu}\\}\Big{]}+\text{Tr}\Big{[}\mathbf{R}_{\mu\nu}\\{R^{\mu},R^{\nu}\\}\Big{]}\Big{)}+\frac{g_{2}^{\prime\,\text{ten}}}{6}\text{Tr}\Big{[}\mathbf{L}_{\mu\nu}+\mathbf{R}_{\mu\nu}\Big{]}\text{Tr}\Big{[}\\{L^{\mu},L^{\nu}\\}+\\{R^{\mu},R^{\nu}\\}\Big{]}\
\text{ ,}$ (2.20)

whose coupling constants scales as $g_{2}^{\text{ten}}\propto N_{c}^{-1/2}$
(dominant) and $g_{2}^{\prime\text{ten}}\propto N_{c}^{-3/2}$ (suppressed).
Further suppressed terms are possible, but are not considered here. The
coupling constants have dimension [Energy], thus the Lagrangian can be made
dilatation invariant by multiplying by $G/G_{0}$. (Upon condensation of $G$,
the term above is reobtained; this is omitted here, since no relevant
additional interactions emerge). The Lagrangian
$\mathcal{L}_{g_{2}^{\text{ten}}}$ enables us to compute the following two
types of decays:

  1. (1)

$2^{++}\longrightarrow 0^{-+}+0^{-+}$ ;

  2. (2)

$2^{--}\longrightarrow 0^{-+}+1^{--}$ .

Both decays are extremely relevant in this work, since the pseudoscalar-
pseudoscalar mode is the dominant one for the tensor mesons, while the
pseudoscalar-vector one is the dominant channel for the axial-tensor states.
Eq. (2.20) shows that they are related by chiral transformations, and hence
depend on the same coupling constant(s).

The next Lagrangian that couples left-handed and right-handed fields involves
derivatives:

$\displaystyle\mathcal{L}_{a^{\text{ten}}}=\frac{a^{\text{ten}}}{2}\text{Tr}\Big{[}\mathbf{L}_{\mu\nu}\\{L^{\mu}_{\beta},L^{\nu\beta}\\}+\mathbf{R}_{\mu\nu}\\{R^{\mu}_{\beta},R^{\nu\beta}\\}\Big{]}+\frac{a^{\prime\,\text{ten}}}{6}\text{Tr}\Big{[}\mathbf{L}_{\mu\nu}+\mathbf{R}_{\mu\nu}\big{]}\text{Tr}\Big{[}\\{L^{\mu}_{\beta},L^{\nu\beta}\\}+\\{R^{\mu}_{\beta},R^{\nu\beta}\\}\Big{]}\,,$
(2.21)

where $L^{\mu\nu}:=\partial^{\mu}L^{\nu}-\partial^{\nu}L^{\mu}$ and
analogously for $R^{\mu\nu}$. The large-$N_{c}$ scaling behaviors are
$a^{\text{ten}}\propto N_{c}^{-1/2}$ (dominant) and
$a^{\prime\text{ten}}\propto N_{c}^{-3/2}$ (subdominant). This Lagrangian
cannot be made dilatation invariant555Note, formally one could multiply the
Lagrangian of Eq. (2.21) by $G_{0}/G$, but that implies a non-analytic point
of the interaction potential for $G\rightarrow 0$; this is why only terms with
coupling constants with zero or positive dimension can be made dilatation
invariant. because the coupling constants have dimension [Energy-1], thus it
is expected to deliver suppressed decays into mesons (as it turns out to be
the case). Yet, it is relevant because, after an appropriate inclusion of the
photon field via vector meson dominance (VMD), it describes the decay of the
tensor mesons to two photons, one photon and one vector mesons, as well as
giving a subleading but important contribution to the decay into two vector
mesons.

The next chiral interaction term reads

$\displaystyle\mathcal{L}_{c^{\text{ten}}}=c_{1}^{\text{ten}}\,\text{Tr}\Big{[}\partial^{\mu}\textbf{L}^{\nu\alpha}\tilde{L}_{\mu\nu}\,\partial_{\alpha}\Phi\,\Phi^{\dagger}-\partial^{\mu}\textbf{R}^{\nu\alpha}\Phi^{\dagger}\,\partial_{\alpha}\Phi\tilde{R}_{\mu\nu}-\partial^{\mu}\textbf{R}^{\nu\alpha}\tilde{R}_{\mu\nu}\partial_{\alpha}\Phi^{\dagger}\Phi+\partial^{\mu}\textbf{L}^{\nu\alpha}\Phi\partial_{\alpha}\Phi^{\dagger}\tilde{L}_{\mu\nu}\Big{]}$
$\displaystyle+c_{2}^{\text{ten}}\,\text{Tr}\Big{[}\partial^{\mu}\textbf{L}^{\nu\alpha}\partial_{\alpha}\Phi\tilde{R}_{\mu\nu}\,\,\Phi^{\dagger}-\partial^{\mu}\textbf{R}^{\nu\alpha}\Phi^{\dagger}\,\tilde{L}_{\mu\nu}\partial_{\alpha}\Phi-\partial^{\mu}\textbf{R}^{\nu\alpha}\partial_{\alpha}\Phi^{\dagger}\tilde{L}_{\mu\nu}\Phi+\partial^{\mu}\textbf{L}^{\nu\alpha}\Phi\tilde{R}_{\mu\nu}\partial_{\alpha}\Phi^{\dagger}\Big{]}\text{
,}$ (2.22)

where
$\tilde{L}_{\mu\nu}:=\frac{\varepsilon_{\mu\nu\rho\sigma}}{2}(\partial^{\rho}L^{\sigma}-\partial^{\sigma}L^{\rho})$
and similarly for $\tilde{R}_{\mu\nu}$. This Lagrangian breaks also dilatation
invariance as it is typical for terms involving the Levi-Civita tensor. Both
constants $c_{1}^{\text{ten}}$ and $c_{2}^{\text{ten}}$ scale as
$N_{c}^{-1/2}$. Large-$N_{c}$ suppressed terms are possible but are omitted
here since present data do not allow to constrain them. The relevant decay
terms are:

  1. (1)

$2^{++}\longrightarrow 0^{-+}+1^{--}$;

  2. (2)

$2^{--}\longrightarrow 0^{-+}+1^{++}$.

These decay channels are related by chiral symmetry and for tensor mesons they
are quite relevant. Note, the pseudoscalar meson as a decay product is a
consequence of the shift of Eq. (2.18).

The decay of a generic spin-2 state has the form:

$\Gamma_{T\rightarrow
A+B}^{tl}=\frac{|\vec{k}_{a,b}|}{40\,\pi\,m_{t}^{2}}\times|\mathcal{M}|^{2}\times\kappa_{i}\times\Theta(m_{t}-m_{a}-m_{b})\,,$
(2.23)

where $m_{t}$ is the mass of the decaying (axial-)tensor particle, $m_{a}$ and
$m_{b}$ are the masses of the decay products, $\Theta(x)$ denotes the
Heaviside step-function, and the modulus of the outgoing particles momentum
has the following analytic expression:

$|\vec{k}_{a,b}|:=\frac{1}{2\,m_{t}}\sqrt{(m_{t}^{2}-m_{a}^{2}-m_{b}^{2})^{2}-4m_{a}^{2}m_{b}^{2}}\
\text{ .}$ (2.24)

We obtain the (eventually dimensionful) factors $\kappa_{i}$ entering in Eq.
(2.23) for the $i$-th decay channel from the explicit forms of the Lagrangian
(the index $i$ runs over the flavor channels, see the following). The decay
amplitudes $|\mathcal{M}|^{2}$, derived via Feynman rules under the use of the
polarization vectors (tensors) and their corresponding completeness relations,
are listed in Table 5.

Decay Mode | $\frac{1}{5}\times|\mathcal{M}|^{2}$
---|---
$2^{++}\longrightarrow 0^{-+}+0^{-+}$ | $g_{2}^{\text{ten}\,2}\times\frac{2|\vec{k}_{p^{(1)},p^{(2)}}|^{4}}{15}$
$2^{--}\longrightarrow 0^{-+}+1^{--}$ | $g_{2}^{\text{ten}\,2}\times\frac{|\vec{k}_{v,p}|^{2}}{15}\Big{(}5+\frac{2|\vec{k}_{v,p}|^{2}}{m_{v}^{2}}\Big{)}$
$2^{--}\longrightarrow 0^{-+}+2^{++}$ | $\frac{4h_{3}^{\text{ten}\,2}}{45}\Big{(}45+\frac{4|\vec{k}_{t,p}|^{4}}{m_{t}^{4}}+\frac{30|\vec{k}_{t,p}|^{2}}{m_{t}^{2}}\Big{)}$
$2^{++}\longrightarrow 0^{-+}+1^{--}$ | $\frac{(c_{1}^{\text{ten}}+c_{2}^{\text{ten}})^{2}\,m_{t}^{2}|\vec{k}_{v,p}|^{4}}{5}$
$2^{--}\longrightarrow 0^{-+}+1^{++}$ | $\frac{(c_{1}^{\text{ten}}-c_{2}^{\text{ten}})^{2}\,m_{a_{2}}^{2}|\vec{k}_{a_{1},p}|^{4}}{5}$
Table 5. Decay amplitudes for different decay modes.

To this end, we recall that one has to average over all incoming spin-
polarizations, sum up all possible outgoing ones, as well as considering the
following completeness relations (for further details see e.g. [58]):

$\displaystyle\sum_{\lambda=1}^{3}\epsilon_{\mu}(\lambda,\,\vec{k})\,\epsilon_{\nu}(\lambda,\,\vec{k})$
$\displaystyle=-G_{\mu\nu}\,,$ (2.25)
$\displaystyle\sum_{\lambda=1}^{5}\epsilon_{\mu\nu}(\lambda,\,\vec{k})\,\epsilon_{\alpha\beta}(\lambda,\,\vec{k})$
$\displaystyle=-\frac{G_{\mu\nu}G_{\alpha\beta}}{3}+\frac{G_{\mu\alpha}G_{\nu\beta}+G_{\mu\beta}G_{\nu\alpha}}{2}\,,$
(2.26)

where

$\displaystyle G_{\mu\nu}$
$\displaystyle\equiv\eta_{\mu\nu}-\frac{k_{\mu}k_{\nu}}{m^{2}}\,,\,\,\,\,\,\eta_{\mu\nu}\equiv\begin{pmatrix}1&0&0&0\\\
0&-1&0&0\\\ 0&0&-1&0\\\ 0&0&0&-1\end{pmatrix}\text{ .}$ (2.27)

##  3\. Results

In this section we present the results for the (axial-)tensor mesons. First,
we discuss the masses of the well established tensor mesons and of their
poorly known chiral partners, the axial-tensor ones. Then, we turn to the
decays, first those of tensor mesons (that are well measured and serve as a
test of the approach and for parameter determination), and then to the decays
of the ground-state axial-tensor mesons (that are mostly experimentally
unknown). For the reader’s convenience, in Table 6 we summarize the parameters
of the eLSM related to the (axial-)tensor sector that will be obtained via
various fit procedures later on.

Parameters | Numerical Values | Fitting Data
---|---|---
$h_{3}^{\text{ten}}$ | $-41$ | Table 7
$\delta_{S}^{\text{ten}}$ | $0.3\,\text{GeV}^{2}$ | Table 7
$g_{2}^{\text{ten}}$ | $(1.392\pm 0.024)\cdot 10^{4}(\text{MeV})$ | Table 9
$g_{2}^{\prime\,\text{ten}}$ | $(0.024\pm 0.041)\cdot 10^{4}(\text{MeV})$ | Table 9
$g^{\text{ten}}_{2\,\text{lat}}$ | $(0.7\pm 0.4)\cdot 10^{4}(\text{MeV})$ | Table 16
$c^{\text{ten}}$ | $(4.8\pm 0.9)\cdot 10^{-7}(\text{MeV})^{-3}$ | Table 10
$a^{\text{ten}}$ | $(-2.09\pm 0.06)\cdot 10^{-2}(\text{MeV})^{-1}$ | Table 12
$a^{\prime\,\text{ten}}$ | $(3.5\pm 0.4)\cdot 10^{-3}(\text{MeV})^{-1}$ | Table 12
$\beta_{T}$ | $(3.16\pm 0.81)^{\circ}$ | Table 9
Table 6. Parameters of the eLSM connected to the (axial-)tensor mesons as
determined by fits to known experimental data (see the following for details).
Note, the suffix “lat” in $g^{\text{ten}}_{2\,\text{lat}}$ indicates that it
was obtained via a comparison to lattice results.

###  3.1. Masses of (axial-)tensor mesons

The expressions for the masses of tensor mesons are obtained by expanding Eq.
(2.19) and taking into account the v.e.v.s and shifts of Eqs. (2.17) and
(2.18):

$\displaystyle
m_{\mathbb{a}_{2}}^{2}=\big{(}m^{2}_{\text{ten}}+2\delta_{N}^{\text{ten}}\big{)}+\frac{h_{3}^{\text{ten}}\phi_{N}^{2}}{2}+\frac{h_{2}^{\text{ten}}\phi_{N}^{2}}{2}+\frac{h_{1}^{\text{ten}}}{2}(\phi_{N}^{2}+\phi_{S}^{2})\,,$
(3.1) $\displaystyle
m_{K_{2}}^{2}=\big{(}m^{2}_{\text{ten}}+\delta_{N}^{\text{ten}}+\delta_{S}^{\text{ten}}\big{)}+\frac{h_{1}^{\text{ten}}}{2}(\phi_{N}^{2}+\phi_{S}^{2})+\frac{1}{\sqrt{2}}h_{3}^{\text{ten}}\phi_{N}\phi_{S}+\frac{h_{2}^{\text{ten}}}{4}(\phi_{N}^{2}+2\phi_{S}^{2})\,,$
(3.2) $\displaystyle
m_{f_{2,N}}^{2}=\big{(}m^{2}_{\text{ten}}+2\delta_{N}^{\text{ten}}\big{)}+\frac{h_{3}^{\text{ten}}\phi_{N}^{2}}{2}+\frac{h_{1}^{\text{ten}}}{2}(\phi_{N}^{2}+\phi_{S}^{2})+\frac{h_{2}^{\text{ten}}\phi_{N}^{2}}{2}\,,$
(3.3) $\displaystyle
m_{f_{2,S}}^{2}=\big{(}m^{2}_{\text{ten}}+2\delta_{S}^{\text{ten}}\big{)}+\frac{h_{1}^{\text{ten}}}{2}(\phi_{N}^{2}+\phi_{S}^{2})+h_{3}^{\text{ten}}\phi_{S}^{2}+h_{2}^{\text{ten}}\phi_{S}^{2}\,.$
(3.4)

The analogous expressions for the axial-tensor mesons read:

$\displaystyle
m_{\rho_{2}}^{2}=\big{(}m^{2}_{\text{ten}}+2\delta_{N}^{\text{ten}}\big{)}-\frac{h_{3}^{\text{ten}}\phi_{N}^{2}}{2}+\frac{h_{2}^{\text{ten}}\phi_{N}^{2}}{2}+\frac{h_{1}^{\text{ten}}}{2}(\phi_{N}^{2}+\phi_{S}^{2})\,,$
(3.5) $\displaystyle
m_{K_{2A}}^{2}=\big{(}m^{2}_{\text{ten}}+\delta_{N}^{\text{ten}}+\delta_{S}^{\text{ten}}\big{)}+\frac{h_{1}^{\text{ten}}}{2}(\phi_{N}^{2}+\phi_{S}^{2})-\frac{1}{\sqrt{2}}h_{3}^{\text{ten}}\phi_{N}\phi_{S}+\frac{h_{2}^{\text{ten}}}{4}(\phi_{N}^{2}+2\phi_{S}^{2})\,,$
(3.6) $\displaystyle
m_{\omega_{2,N}}^{2}=\big{(}m^{2}_{\text{ten}}+2\delta_{N}^{\text{ten}}\big{)}-\frac{h_{3}^{\text{ten}}\phi_{N}^{2}}{2}+\frac{h_{2}^{\text{ten}}\phi_{N}^{2}}{2}+\frac{h_{1}^{\text{ten}}}{2}(\phi_{N}^{2}+\phi_{S}^{2})\,,$
(3.7) $\displaystyle
m_{\omega_{2,S}}^{2}=\big{(}m^{2}_{\text{ten}}+2\delta_{S}^{\text{ten}}\big{)}-h_{3}^{\text{ten}}\phi_{S}^{2}+h_{2}^{\text{ten}}\phi_{S}^{2}+\frac{h_{1}^{\text{ten}}}{2}(\phi_{N}^{2}+\phi_{S}^{2})\,.$
(3.8)

We observe that in each nonet the masses of the isovector and the nonstrange
isoscalar members are identical:

$m_{\rho_{2}}^{2}=m_{\omega_{2,N}}^{2}\,\,\,\text{and}\,\,\,m_{\mathbb{a}_{2}}^{2}=m_{f_{2,N}}^{2}\text{
.}$ (3.9)

This degeneracy is broken by the (small) isoscalar mixing angle.

The following three equations relate the masses of chiral partners:

$\displaystyle m_{\rho_{2}}^{2}$
$\displaystyle=m_{\mathbb{a}_{2}}^{2}-h_{3}^{\text{ten}}\phi_{N}^{2},$ (3.10)
$\displaystyle m_{K_{2A}}^{2}$
$\displaystyle=m_{K_{2}}^{2}-\sqrt{2}h_{3}^{\text{ten}}\phi_{N}\phi_{S},$
(3.11) $\displaystyle m_{\omega_{2,S}}^{2}$
$\displaystyle=m_{f_{2,S}}^{2}-2h_{3}^{\text{ten}}\phi_{S}^{2}\text{ ,}$
(3.12)

out of which one obtains the following numerical estimate (using $K_{2}(1820)$
and $K_{2}^{*}(1430)$ as inputs):

$h_{3}^{\text{ten}}=\frac{m_{K_{2A}}^{2}-m_{K_{2}}^{2}}{\sqrt{2}\phi_{N}\phi_{S}}\simeq-41\;\text{
.}$ (3.13)

Next, by using the PDG masses of the two tensor mesons and considering
$\delta_{N}^{\text{ten}}\simeq 0$ and $\phi_{N}^{2}\simeq 2\phi_{S}^{2}$, one
gets:

$\delta_{S}^{\text{ten}}=m_{K_{2}}^{2}-m_{\mathbb{a}_{2}}^{2}\simeq
0.3\,\text{GeV}^{2}\text{ .}$ (3.14)

The following considerations are in order:

  1. (1)

The mass of the purely strange resonance $\omega_{2,S}\simeq\phi_{2}$ is
predicted to be:

$m_{\phi_{2}}^{2}\simeq
m_{\omega_{2,S}}^{2}=m_{\mathbb{a}_{2}}^{2}+2\delta_{S}^{\text{ten}}-\frac{3}{2}h_{3}^{\text{ten}}\phi_{N}^{2}\,,$
(3.15)

which implies the numerical value for the missing resonance
$\phi_{2}\simeq\omega_{2,S}$ as:

$m_{\phi_{2}}\approx 1971\,\text{MeV .}$ (3.16)

In other words, the missing state $\phi_{2}(???)$ of Table 1 could appear as a
novel resonance $\phi_{2}(1971)$.

  2. (2)

Using the following relation between the masses of chiral partners

$\displaystyle
m_{\rho_{2}}^{2}=m_{\mathbb{a}_{2}}^{2}-h_{3}^{\text{ten}}\phi_{N}^{2}\text{
,}$

we get the following prediction for the not-yet discovered resonance
$\rho_{2}$:

$m_{\rho_{2}}=1663\,\text{MeV .}$ (3.17)

Since we neglect the isoscalar mixing, we also have $m_{\rho_{2}}\simeq 1663$
MeV. Thus, the new resonances $\rho_{2}(1663)$ and $\omega_{2}(1663)$ could
possibly appear in the mesonic spectrum.

  3. (3)

As a consistency check, the eLSM predicts the mass of the purely strange
isoscalar tensor meson as

$m_{f_{2,S}}^{2}=m_{\omega_{2,S}}^{2}+2h_{3}^{\text{ten}}\phi_{S}^{2}\text{
,}$ (3.18)

out of which $m_{f_{2,S}}\simeq 1520$ MeV, slightly lighter than the physical
value of the resonance $f_{2}^{\prime}(1525)$ that amounts to
$1538\,\text{MeV}$. According to the relation
$m_{\mathbb{a}_{2}}^{2}=m_{f_{2,N}}^{2}$, one has
$m_{f_{2,N}}=1317\,\text{MeV}$, to be compared with the mass of $f_{2}(1270)$
of about $1297$ MeV.

The small departures of the two isoscalar-tensor states can be easily
explained by introducing a small isoscalar mixing angle. In our case, we shall
study this mixing angle in the context of two-pseudoscalar and two-photon
decay processes. When including the isoscalar mixing in the tensor sector,
Table 7 summarizes our results for the (axial-)tensor masses.

Resonances | Masses (in MeV) | Resonances | Masses (in MeV)
---|---|---|---
$\,\;a_{2}(1320)$ | 1317 | $\,\;\rho_{2}(?)$ | $1663$
$\,\;K_{2}^{\ast}(1430)$ | 1427 | $\,\;K_{2}(1820)$ | 1819
$\,\;f_{2}(1270)$ | $1297$ | $\,\;\omega_{2,N}$ | $1663$
$\,\;f_{2}^{\prime}(1525)$ | $1538$ | $\,\;\omega_{2,S}$ | $1971$
Table 7. Masses of the mesons predicted by the eLSM. Bold entries are the PDG
values. The (small) mixing for the tensor-isoscalar mesons has been taken into
account.

###  3.2. Decays of tensor mesons

In this subsection, we compare the predictions for the strong and radiative
decay of tensor mesons with available PDG data.

####  3.2.1. $T\longrightarrow P^{(1)}+P^{(2)}$ decay mode

The interaction Lagrangian describing the decay of spin-$2$ tensor mesons to
two pseudoscalar mesons follows from the chiral Lagrangian of Eq. (2.20) by
applying the shift of Eq. (2.18):

$\mathcal{L}_{tpp}=g_{2}^{\text{ten}}\,\mathrm{Tr}\Big{[}T^{\mu\nu}\big{\\{}(\partial_{\mu}\mathcal{P})\,,(\partial_{\nu}\mathcal{P})\big{\\}}\Big{]}+\frac{g_{2}^{\prime\,\text{ten}}}{3}\mathrm{Tr}\Big{[}T^{\mu\nu}\Big{]}\mathrm{Tr}\Big{[}\big{\\{}(\partial_{\mu}\mathcal{P})\,,(\partial_{\nu}\mathcal{P})\big{\\}}\Big{]}\,\text{
.}$ (3.19)

The factor $1/3$ in front of the second term is chosen in such a way that for
the singlet state the same normalization occurs. Namely, by using the
decomposition
$\sqrt{2}T_{2,\mu\nu}=\sum_{a=0}^{8}T_{\mu\nu}^{a}\frac{\lambda^{a}}{\sqrt{2}}$
together with $\lambda^{0}=\sqrt{\frac{2}{3}}\mathbf{1,}$ it follows
$T_{\mu\nu}=\frac{1}{\sqrt{6}}T_{\mu\nu}^{0}\mathbf{1+...}$ . For the singlet
state the Lagrangian reduces to
$\mathcal{L}_{tpp}=(g_{2}^{\text{ten}}+g_{2}^{\prime\text{ten}})T^{0\,\mu\nu}\,\mathrm{Tr}\Big{[}\big{\\{}(\partial_{\mu}\mathcal{P})\,,(\partial_{\nu}\mathcal{P})\big{\\}}\Big{]}+...$
. This convention (which of course does not affect the results) is used in
other terms as well.

The tree-level decay rate is

$\Gamma_{T\longrightarrow
P^{(1)}+P^{(2)}}^{tl}(m_{t},m_{p^{(1)}},m_{p^{(2)}})=\kappa_{tpp\,,i}\,\frac{|\vec{k}_{p^{(1)},p^{(2)}}|^{5}}{60\,\pi\,m_{t}^{2}}\Theta(m_{t}-m_{p^{(1)}}-m_{p^{(2)}})\,\text{
,}$ (3.20)

where the coefficients $\kappa_{tpp\,,i}$ (with dimension [Energy-2]) are
listed in Table 8.

Decay process | $\kappa_{tpp\,,i}$
---|---
$a_{2}(1320)\longrightarrow\bar{K}\,K$ | $2\Big{(}\frac{g_{2}^{\text{ten}}Z_{k}^{2}w_{k}^{2}}{2}\Big{)}^{2}$
$a_{2}(1320)\longrightarrow\pi\,\eta$ | $\Big{(}g_{2}^{\text{ten}}Z_{\pi}Z_{\eta_{N}}w_{\pi}w_{\eta_{N}}\cos\beta_{P}\Big{)}^{2}$
$a_{2}(1320)\longrightarrow\pi\,\eta^{\prime}$ | $\Big{(}-g_{2}^{\text{ten}}Z_{\pi}Z_{\eta_{N}}w_{\pi}w_{\eta_{N}}\sin\beta_{P}\Big{)}^{2}$
$K_{2}^{\ast}(1430)\longrightarrow\pi\,\bar{K}$ | $3\Big{(}\frac{g_{2}^{\text{ten}}Z_{k}w_{k}Z_{\pi}w_{\pi}}{2}\Big{)}^{2}$
$f_{2}(1270)\longrightarrow\bar{K}\,K$ | $2\Big{(}\frac{Z_{k}^{2}w_{k}^{2}\big{(}(4g_{2}^{\prime\,\text{ten}}+3g_{2}^{\text{ten}})\cos\beta_{T}+\sqrt{2}(2g_{2}^{\prime\,\text{ten}}+3g_{2}^{\text{ten}})\sin\beta_{T}\big{)}}{6}\Big{)}^{2}$
$f_{2}(1270)\longrightarrow\pi\,\pi$ | $6\Big{(}\frac{Z_{\pi}^{2}w_{\pi}^{2}\big{(}(2g_{2}^{\prime\,\text{ten}}+3g_{2}^{\text{ten}})\cos\beta_{T}+\sqrt{2}g_{2}^{\prime\,\text{ten}}\sin\beta_{T}\big{)}}{6}\Big{)}^{2}$
$f_{2}(1270)\longrightarrow\eta\,\eta$ | $2\Big{(}\frac{Z_{\eta_{N}}^{2}w_{\eta_{N}}^{2}\cos\beta_{P}^{2}\big{(}(2g_{2}^{\prime\,\text{ten}}+3g_{2}^{\text{ten}})\cos\beta_{T}+\sqrt{2}g_{2}^{\prime\,\text{ten}}\sin\beta_{T}\big{)}+Z_{\eta_{S}}^{2}w_{\eta_{S}}^{2}\sin\beta_{P}^{2}\big{(}\sqrt{2}(g_{2}^{\prime\,\text{ten}}+3g_{2}^{\text{ten}})\sin\beta_{T}+2g_{2}^{\prime\,\text{ten}}\cos\beta_{T}\big{)}}{6}\Big{)}^{2}$
$f_{2}^{\prime}(1525)\longrightarrow\bar{K}\,K$ | $2\Big{(}\frac{Z_{k}^{2}w_{k}^{2}\big{(}-(4g_{2}^{\prime\,\text{ten}}+3g_{2}^{\text{ten}})\sin\beta_{T}+\sqrt{2}(2g_{2}^{\prime\,\text{ten}}+3g_{2}^{\text{ten}})\cos\beta_{T}\big{)}}{6}\Big{)}^{2}$
$f_{2}^{\prime}(1525)\longrightarrow\pi\,\pi$ | $6\Big{(}\frac{Z_{\pi}^{2}w_{\pi}^{2}\big{(}-(2g_{2}^{\prime\,\text{ten}}+3g_{2}^{\text{ten}})\sin\beta_{T}+\sqrt{2}g_{2}^{\prime\,\text{ten}}\cos\beta_{T}\big{)}}{6}\Big{)}^{2}$
$f_{2}^{\prime}(1525)\longrightarrow\eta\,\eta$ | $2\Big{(}\frac{Z_{\eta_{N}}^{2}w_{\eta_{N}}^{2}\cos\beta_{P}^{2}\big{(}-(2g_{2}^{\prime\,\text{ten}}+3g_{2}^{\text{ten}})\sin\beta_{T}+\sqrt{2}g_{2}^{\prime\,\text{ten}}\cos\beta_{T}\big{)}+Z_{\eta_{S}}^{2}w_{\eta_{S}}^{2}\sin\beta_{P}^{2}\big{(}\sqrt{2}(g_{2}^{\prime\,\text{ten}}+3g_{2}^{\text{ten}})\cos\beta_{T}-2g_{2}^{\prime\,\text{ten}}\sin\beta_{T}\big{)}}{6}\Big{)}^{2}$
Table 8. Coefficients for the decay channel $T\longrightarrow
P^{(1)}+P^{(2)}$.

We perform a fit to the PDG data of Table 9 with three parameters: two
coupling constants, the large-$N_{c}$ dominant $g_{2}^{\text{ten}}$ and the
subdominant $g_{2}^{\prime\,\text{ten}}$, as well as the mixing anlge
$\beta_{T}$. The fit gives:

$g_{2}^{\text{ten}}=(1.392\pm 0.024)\cdot 10^{4}(\text{MeV})\,,\qquad
g_{2}^{\prime\,\text{ten}}=(0.024\pm 0.041)\cdot
10^{4}(\text{MeV})\,,\qquad\beta_{T}=(3.16\pm 0.81)^{\circ}\text{ .}$ (3.21)

Decay process (in model) | eLSM (MeV) | PDG (MeV)
---|---|---
$\,\;a_{2}(1320)\longrightarrow\bar{K}\,K$ | $4.06\pm 0.14$ | $7.0^{+2.0}_{-1.5}\leftrightarrow(4.9\pm 0.8)\%$
$\,\;a_{2}(1320)\longrightarrow\pi\,\eta$ | $25.37\pm 0.87$ | $18.5\pm 3.0\leftrightarrow(14.5\pm 1.2)\%$
$\,\;a_{2}(1320)\longrightarrow\pi\,\eta^{\prime}(958)$ | $1.01\pm 0.03$ | $0.58\pm 0.10\leftrightarrow(0.55\pm 0.09)\%$
$\,\;K_{2}^{\ast}(1430)\longrightarrow\pi\,\bar{K}$ | $44.82\pm 1.54$ | $49.9\pm 1.9\leftrightarrow(49.9\pm 0.6)\%$
$\,\;f_{2}(1270)\longrightarrow\bar{K}\,K$ | $3.54\pm 0.29$ | $8.5\pm 0.8\leftrightarrow(4.6^{+0.5}_{-0.4})\%$
$\,\;f_{2}(1270)\longrightarrow\pi\,\pi$ | $168.82\pm 3.89$ | $157.2^{+4.0}_{-1.1}\leftrightarrow(84.2^{+2.9}_{-0.9})\%$
$\,\;f_{2}(1270)\longrightarrow\eta\,\eta$ | $0.67\pm 0.03$ | $0.75\pm 0.14\leftrightarrow(0.4\pm 0.08)\%$
$\,\;f_{2}^{\prime}(1525)\longrightarrow\bar{K}\,K$ | $23.72\pm 0.60$ | $75\pm 4\leftrightarrow(87.6\pm 2.2)\%$
$\,\;f_{2}^{\prime}(1525)\longrightarrow\pi\,\pi$ | $0.67\pm 0.14$ | $0.71\pm 0.14\leftrightarrow(0.83\pm 0.16)\%$
$\,\;f_{2}^{\prime}(1525)\longrightarrow\eta\,\eta$ | $1.81\pm 0.05$ | $9.9\pm 1.9\leftrightarrow(11.6\pm 2.2)\%$
Table 9. Decay rates for $T\longrightarrow P^{(1)}+P^{(2)}$.

As predicted by large-$N_{c}$ arguments, the results confirm that the coupling
constant $g_{2}^{\prime\,\text{ten}}$ is much smaller than
$g_{2}^{\text{ten}}$. Due to the quite large error, it is also compatible with
zero. Yet, the phenomenology of large-$N_{c}$ suppressed decay (as e.g. the
decays of the $J/\psi$ [59]) shows that such terms, even if small, do not
vanish. Moreover, the following prediction is obtained

$\text{Th}:\;\Gamma\Big{[}K_{2}^{\ast}(1430)\longrightarrow\eta\,\bar{K}\Big{]}=(1.13\pm
0.03)\cdot
10^{-3}\;\text{MeV},\;\;\text{PDG}:\;\;0.15^{+0.34}_{-0.10}\;\text{MeV.}$
(3.22)

which due to large uncertainties was excluded from the fitting procedure.

Note, the isospin mixing angle $\beta_{T}=3.16^{\circ}$ is close (albeit not
equal) to the value $\beta_{T}=5.7^{\circ}$ of PDG calculated via the
following formula

$\displaystyle\beta_{T}=35.3^{\circ}-\arctan{\Bigg{(}\sqrt{\frac{4m_{K_{2}}^{2}-m_{a_{2}}^{2}-3m_{f_{2}^{\prime}}^{2}}{-4m_{K_{2}}^{2}+m_{a_{2}}^{2}+3m_{f_{2}}^{2}}}\Bigg{)}}\text{
.}$ (3.23)

In general, the results of Table 9 show a good qualitative agreement. Yet,
some entries are at odds with experimental ones, but one should recall that
the experimental data are quite precise and only three parameters were used.
The results could be improved by adding further large-$N_{c}$ suppressed terms
as well as flavor-symmetry breaking contributions. In this way a more accurate
description of the tensor decays could be possible. Such a precision study of
tensor states could be interesting on its own but is not the main goal of this
paper, thus it is left for the future.

####  3.2.2. $T\longrightarrow V+P$ decay mode

The next experimentally well-known decay channel is the one into a vector-
pseudoscalar pair. Upon considering the shift of Eq. (2.17), the chiral
Lagrangian of Eq. (2.2) contains the term:

$\displaystyle\mathcal{L}_{tvp}=(c_{1}^{\text{ten}}+c_{2}^{\text{ten}})\,\varepsilon_{\mu\nu\rho\sigma}\,\text{Tr}\Big{[}\,\partial^{\mu}T^{\nu\alpha}\,\Big{(}\partial^{\rho}V^{\sigma}(\partial_{\alpha}P)\Phi_{0}-\Phi_{0}(\partial_{\alpha}P\partial^{\rho}V^{\sigma})\Big{)}\Big{]}\text{
.}$ (3.24)

Defining $c^{\text{ten}}:=c_{1}^{\text{ten}}+c_{2}^{\text{ten}}$, the tree-
level decay rate formula reads

$\Gamma_{T\longrightarrow
V+P}^{tl}(m_{t},m_{v},m_{p})=\frac{c^{\text{ten}\,2}\,|\vec{k}_{v,p}|^{5}}{40\,\pi}\,\kappa_{tvp\,,i}\,\Theta(m_{t}-m_{v}-m_{p})\,,$
(3.25)

where the factors $\kappa_{tvp\,,i}$ of mass dimension 2 are reported in Table
10.

We obtain the value for the coupling $c^{\text{ten}}$ by performing a
$\chi^{2}$-fit to the data of Table 10 (for a brief overview of this procedure
including the determination of the error of the obtained parameter(s), see the
Appendix of Ref. [12]):

$c^{\text{ten}}=(4.8\pm 0.9)\cdot 10^{-7}(\text{MeV})^{-3}\text{ .}$ (3.26)

In the very same table the theoretical results are also presented.

Decay process (in model) | eLSM (MeV) | PDG-2020 (MeV) | $\kappa_{tvp\,,i}$
---|---|---|---
$\,\;a_{2}(1320)\longrightarrow\rho(770)\,\pi$ | $71.0\pm 2.6$ | $73.61\pm 3.35\leftrightarrow(70.1\pm 2.7)\%$ | $2\Big{(}\frac{\phi_{N}}{4}\Big{)}^{2}$
$\,\;K_{2}^{\ast}(1430)\longrightarrow\bar{K}^{\ast}(892)\,\pi$ | $27.9\pm 1.0$ | $26.92\pm 2.14\leftrightarrow(24.7\pm 1.6)\%$ | $3\Big{(}\frac{\phi_{N}}{8}\Big{)}^{2}$
$\,\;K_{2}^{\ast}(1430)\longrightarrow\rho(770)\,K$ | $10.3\pm 0.4$ | $9.48\pm 0.97\leftrightarrow(8.7\pm 0.8)\%$ | $3\Big{(}\frac{\sqrt{2}\phi_{S}}{8}\Big{)}^{2}$
$\,\;K_{2}^{\ast}(1430)\longrightarrow\omega(782)\,\bar{K}$ | $3.5\pm 0.1$ | $3.16\pm 0.88\leftrightarrow(2.9\pm 0.8)\%$ | $\Big{(}\frac{-\phi_{S}\cos{\beta_{V}}+\phi_{N}\sin{\beta_{V}}}{4\sqrt{2}})\Big{)}^{2}$
$\,\;f_{2}^{\prime}(1525)\longrightarrow\bar{K}^{\ast}(892)\,K+\mathrm{c}.\mathrm{c}.$ | $19.89\pm 0.73$ |  | $4\Big{(}\frac{2\phi_{S}\cos{\beta_{T}}+\phi_{N}\sin{\beta_{T}}}{8})\Big{)}^{2}$
Table 10.  Decay rates and coefficients for the decay channel
$T\longrightarrow P+V$.

The model results are in good agreement with the experimental data. Moreover,
a noteworthy prediction concerning
$f_{2}^{\prime}(1525)\rightarrow\bar{K}^{\ast}(892)\,K+\mathrm{c}.\mathrm{c}.$
is obtained.

###  3.3. Radiative decays

Radiative decays can be obtained via vector meson dominance [60] by applying
the additional shift

$V_{\mu\nu}\rightarrow V_{\mu\nu}+\frac{e}{g_{\rho}}F_{\mu\nu}Q\text{ ,}$
(3.27)

where $V_{\mu\nu}\equiv\partial_{\mu}V_{\nu}-\partial_{\nu}V_{\mu}$,
$Q=\text{diag}\\{2/3,-1/3,-1/3\\}$ is the quark charge matrix,
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the
electromagnetic field tensor, $e=\sqrt{4\pi\alpha}$ is the electric coupling
constant, and $g_{\rho}\simeq 5.5\pm 0.5$ parametrizes the photon-vector-meson
transition. As a first application, we consider $T\longrightarrow\gamma P$ by
applying the VMD shift of Eq. (3.27) into Eq. (3.24) that leads to the
Lagrangian

$\displaystyle\mathcal{L}_{t\gamma
p}=\frac{e}{g_{\rho}}c^{\text{ten}}\,\text{Tr}\Big{[}\,\partial^{\mu}T^{\nu\alpha}\,\Big{(}F_{\mu\nu}Q(\partial_{\alpha}P)\Phi_{0}-\Phi_{0}(\partial_{\alpha}PF_{\mu\nu}Q\Big{)}\Big{]}\text{
,}$ (3.28)

and to the corresponding decay rate

$\Gamma_{T\longrightarrow\gamma+P}^{tl}(m_{t},m_{p})=\frac{c^{\text{ten}\,2}\,|\vec{k}_{\gamma,p}|^{5}}{40\,\pi}\,\kappa_{t\gamma
p\,,i}\,\Theta(m_{t}-m_{p})\,.$ (3.29)

Decay process (in model) | eLSM | PDG-2020 | $\kappa_{t\gamma p\,,i}$
---|---|---|---
$\,\;K_{2}^{\pm}(1430)\longrightarrow\gamma\,K^{\pm}$ | $1.12\pm 0.47\,\,\text{MeV}$ | $0.24\pm 0.05\,\,\text{MeV}$ | $\frac{e^{2}}{g_{\rho}^{2}}\Big{(}\frac{\phi_{N}+2\sqrt{2}\phi_{S}}{12}\Big{)}^{2}$
$\,\;K_{2}^{0}(1430)\longrightarrow\gamma\,K^{0}$ | $5.1\pm 2.2\,\,\text{keV}$ | $<5.4\,\,\text{keV}$ | $\frac{e^{2}}{g_{\rho}^{2}}\Big{(}\frac{\phi_{N}-\sqrt{2}\phi_{S}}{12}\Big{)}^{2}$
$\,\;a_{2}^{\pm}(1320)\longrightarrow\gamma\,\pi^{\pm}$ | $1.01\pm 0.43\,\,\text{MeV}$ | $0.31\pm 0.03\,\,\text{MeV}$ | $\frac{e^{2}}{g_{\rho}^{2}}\Big{(}\frac{\phi_{N}}{4}\Big{)}^{2}$
Table 11.  Decay rates and coefficients for the decay channel
$T\longrightarrow\gamma+P$.

Taking into account $c^{\text{ten}}$ in Eq. (3.26) as well as $g_{\rho}=5.5\pm
0.5$, we use the following equation to calculate the theoretical errors in
Table 11:

$\delta\Gamma_{T\rightarrow\gamma
P}(m_{t},m_{p})=\sqrt{\Big{(}\frac{\partial\Gamma_{T\rightarrow\gamma
P}(m_{t},m_{p})}{\partial c^{\text{ten}}}\delta
c^{\text{ten}}\Big{)}^{2}+\Big{(}\frac{\partial\Gamma_{T\rightarrow\gamma
P}(m_{t},m_{p})}{\partial g_{\rho}}\delta g_{\rho}\Big{)}^{2}}\text{ .}$
(3.30)

Our results for the photon-vector decays are reported in Table 11: they are
somewhat larger but, in consideration of the large errors, in qualitative
agreement with the PDG results.

Next, we describe the two-photon decays. We note that the chiral Lagrangian of
Eq. (2.21) delivers the following TVV interaction term:

$\mathcal{L}_{tvv}=a^{\text{ten}}\text{Tr}\Big{[}T_{\mu\nu}\\{V^{\mu}_{\beta},V^{\nu\beta}\\}\Big{]}+\frac{a^{\prime\,\text{ten}}}{3}\text{Tr}\Big{[}T_{\mu\nu}\Big{]}\text{Tr}\big{[}\\{V^{\mu}_{\beta},V^{\nu\beta}\\}\Big{]}\,.$
(3.31)

Then, through the VMD shift (3.27) for both vector fields, we get the
Lagrangian

$\mathcal{L}_{t\gamma\gamma}=a^{\text{ten}}\frac{e^{2}}{g_{\rho}^{2}}\text{Tr}\Big{[}T_{\mu\nu}\\{QF^{\mu}_{\beta}\,,QF^{\nu\beta}\\}\Big{]}+\frac{a^{\prime\,\text{ten}}}{3}\frac{e^{2}}{g_{\rho}^{2}}\text{Tr}\Big{[}T_{\mu\nu}\Big{]}\text{Tr}\Big{[}\\{QF^{\mu}_{\beta}\,,QF^{\nu\beta}\\}\Big{]}\text{
,}$ (3.32)

and the decay rate of the type

$\Gamma_{T\rightarrow\gamma\gamma}^{tl}=\frac{e^{4}}{g_{\rho}^{4}}\frac{2\,|\vec{k}|^{5}}{5\,\pi\,m_{t}^{2}}\kappa_{t\gamma\gamma\,,i}=\frac{e^{4}}{g_{\rho}^{4}}\frac{m_{t}^{3}}{80\pi}\kappa_{t\gamma\gamma\,,i}\text{
,}$ (3.33)

where $m_{t}=2|\vec{k}|$ was used.

By using the experimental values of Table 12, a standard $\chi^{2}$ approach
with two free parameters (the coupling constants $a^{\text{ten}}$ and
$a^{\prime\,\text{ten}}$, $\beta_{T}=3.16^{\circ}$ kept fixed) and three
experimental data-points, gives:

$a^{\text{ten}}=(-2.09\pm 0.06)\cdot 10^{-2}(\text{MeV})^{-1}\,,\qquad
a^{\prime\,\text{ten}}=(3.5\pm 0.4)\cdot 10^{-3}(\text{MeV})^{-1}\text{ }$
(3.34)

The theoretical results are also listed in Table 12. Thus, in this case the
coupling constants are directly fitted to the decay rates. The fact that all 3
data points are well described confirm that the value the mixing angle
$\beta_{T}=3.16^{\circ}$ is consistent with data. Moreover, the parameter
$a^{\prime\,\text{ten}}$ turns out to be much smaller than $a^{\text{ten}}$ in
agreement with the large-$N_{c}$ expectations. [Note: in the fit, we fixed
$g_{\rho}=5.5$ and determined the coupling constant $a^{\text{ten}}$;
actually, the combination $a^{\text{ten}}/g_{\rho}^{2}$ enters into the
expressions. Yet, allowing for a variation of $g_{\rho}$ would not lead to any
significant changes here, since it could be reabsorbed by a change of
$a^{\text{ten}}$.]

These radiative decay channels also show that the coupling with photons is
sizable, thus the production of these not-yet discovered states $\rho_{2}$,
$\omega_{2}$, and $\phi_{2}$ in photoproduction experiments is well upheld.

Decay process (in model) | eLSM (keV) | PDG-2020 (keV) | $\kappa_{t\gamma\gamma\,,i}$
---|---|---|---
$\,\;a_{2}(1320)\longrightarrow\gamma\gamma$ | $1.01\pm 0.06$ | $1.00\pm 0.06$ | $\frac{a^{\text{ten}\,2}}{36}$
$\,\;f_{2}(1270)\longrightarrow\gamma\gamma$ | $1.95\pm 0.10$ | $2.6\pm 0.5$ | $\Big{(}\frac{(5a^{\text{ten}}+4a^{\prime\,\text{ten}})\cos{\beta_{T}}+\sqrt{2}(a^{\text{ten}}+2a^{\prime\,\text{ten}})\sin{\beta_{T}}}{18}\Big{)}^{2}$
$\,\;f_{2}^{\prime}(1525)\longrightarrow\gamma\gamma$ | $0.083\pm 0.009$ | $0.082\pm 0.009$ | $\Big{(}\frac{-(5a^{\text{ten}}+4a^{\prime\,\text{ten}})\sin{\beta_{T}}+\sqrt{2}(a^{\text{ten}}+2a^{\prime\,\text{ten}})\cos{\beta_{T}}}{18}\Big{)}^{2}$
Table 12. Decay rates and coefficients for the decay channel
$T\longrightarrow\gamma\gamma$ .

It is possible to use the two parameters presented in Eq. (3.34) for the
predictions of numerous photon-vector decay mode by shifting only one vector
field of the Lagrangian of Equation (3.31)

$\mathcal{L}_{t\gamma
V}=2a^{\text{ten}}\frac{e}{g_{\rho}}\text{Tr}\Big{[}T_{\mu\nu}\\{QF^{\mu}_{\beta}\,,V^{\nu\beta}\\}\Big{]}+\frac{2a^{\prime\,\text{ten}}}{3}\frac{e}{g_{\rho}}\text{Tr}\Big{[}T_{\mu\nu}\Big{]}\text{Tr}\Big{[}\\{QF^{\mu}_{\beta}\,,V^{\nu\beta}\\}\Big{]}\text{
.}$ (3.35)

The vector-photon decay rates take the form

$\displaystyle\Gamma_{T\longrightarrow\gamma+V}^{tl}(m_{t},m_{v})=\frac{\kappa_{t\gamma
v\,,i}\,e^{2}}{15\,\pi\,m_{t}^{2}m_{v}^{2}g_{\rho}^{2}}\Big{(}3|\vec{k}_{v,\gamma}|^{7}$
$\displaystyle+5|\vec{k}_{v,\gamma}|^{3}m_{v}^{2}+5|\vec{k}_{v,\gamma}|^{4}m_{v}^{2}\sqrt{|\vec{k}_{v,\gamma}|^{2}+m_{v}^{2}}$
$\displaystyle\qquad+|\vec{k}_{v,\gamma}|^{5}\big{(}10m_{v}^{2}-3(|\vec{k}_{v,\gamma}|^{2}+m_{v}^{2})\big{)}\Big{)}\,\text{
.}$ (3.36)

The corresponding coefficients are reported in Table 13 and the numerical
predictions for the not-yet measured decay rates in Table 14. The
corresponding errors are obtained by considering Eq. (3.34) together with
$g_{\rho}=5.5\pm 0.5$. [Note, the variation for $g_{\rho}=5.5\pm 0.5$ must be
taken into account for the evaluation of the errors since the different ratio
$a^{\text{ten}}/g_{\rho}$ enters into the expressions.] Additional sources of
errors, such as the uncertainties of masses, were neglected for simplicity.
Thus, the actual errors can be somewhat larger but of the same order of the
ones reported in Table 14.

Decay process (in model) | $\kappa_{t\gamma v\,,i}$
---|---
$\,\;a_{2}(1320)\longrightarrow\rho(770)\,\gamma$ | $\Big{(}\frac{a^{\text{ten}}}{6}\Big{)}^{2}$
$\,\;a_{2}^{0}(1320)\longrightarrow\omega(782)\,\gamma$ | $\Big{(}\frac{a^{\text{ten}}\cos{\beta_{V}}}{2}\Big{)}^{2}$
$\,\;a_{2}(1320)\longrightarrow\phi(1020)\,\gamma$ | $\Big{(}\frac{a^{\text{ten}}\sin{\beta_{V}}}{2}\Big{)}^{2}$
$\,\;K_{2}^{\ast\pm}(1430)\longrightarrow\bar{K}^{\ast\pm}(892)\,\gamma$ | $\Big{(}\frac{a^{\text{ten}}}{6}\Big{)}^{2}$
$\,\;K_{2}^{\ast\;0}(1430)\longrightarrow\bar{K}^{\ast\;0}(892)\,\gamma$ | $\Big{(}\frac{a^{\text{ten}}}{3}\Big{)}^{2}$
$\,\;f_{2}(1270)\longrightarrow\rho(770)\,\gamma$ | $\Big{(}\frac{(a^{\text{ten}}+2a^{\prime\,\text{ten}})\cos{\beta_{T}}+\sqrt{2}a^{\prime\,\text{ten}}\sin{\beta_{T}}}{2}\Big{)}^{2}$
$\,\;f_{2}(1270)\longrightarrow\omega(782)\,\gamma$ | $\Big{(}\frac{\cos{\beta_{V}}\big{(}(a^{\text{ten}}+2a^{\prime\,\text{ten}})\cos{\beta_{T}}+\sqrt{2}a^{\prime\,\text{ten}}\sin{\beta_{T}}\big{)}-2\sin{\beta_{V}}\big{(}(a^{\text{ten}}+a^{\prime\,\text{ten}})\sin{\beta_{T}}+\sqrt{2}a^{\prime\,\text{ten}}\cos{\beta_{T}}\big{)}}{6}\Big{)}^{2}$
$\,\;f_{2}(1270)\longrightarrow\phi(1020)\,\gamma$ | $\Big{(}\frac{\sin{\beta_{V}}\big{(}(a^{\text{ten}}+2a^{\prime\,\text{ten}})\cos{\beta_{T}}+\sqrt{2}a^{\prime\,\text{ten}}\sin{\beta_{T}}\big{)}+2\cos{\beta_{V}}\big{(}(a^{\text{ten}}+a^{\prime\,\text{ten}})\sin{\beta_{T}}+\sqrt{2}a^{\prime\,\text{ten}}\cos{\beta_{T}}\big{)}}{6}\Big{)}^{2}$
$\,\;f_{2}^{\prime}(1525)\longrightarrow\rho(770)\,\gamma$ | $\Big{(}\frac{-(a^{\text{ten}}+2a^{\prime\,\text{ten}})\sin{\beta_{T}}+\sqrt{2}a^{\prime\,\text{ten}}\cos{\beta_{T}}}{2}\Big{)}^{2}$
$\,\;f_{2}^{\prime}(1525)\longrightarrow\omega(782)\,\gamma$ | $\Big{(}\frac{\sin{\beta_{T}}\big{(}-(a^{\text{ten}}+2a^{\prime\,\text{ten}})\cos{\beta_{V}}+2\sqrt{2}a^{\prime\,\text{ten}}\sin{\beta_{V}}\big{)}+\cos{\beta_{T}}\big{(}\sqrt{2}a^{\prime\,\text{ten}}\cos{\beta_{V}}-2(a^{\text{ten}}+a^{\prime\,\text{ten}})\sin{\beta_{V}}\big{)}}{6}\Big{)}^{2}$
$\,\;f_{2}^{\prime}(1525)\longrightarrow\phi(1020)\,\gamma$ | $\Big{(}\frac{\cos{\beta_{T}}\big{(}-2(a^{\text{ten}}+a^{\prime\,\text{ten}})\cos{\beta_{V}}-\sqrt{2}a^{\prime\,\text{ten}}\sin{\beta_{V}}\big{)}+\sin{\beta_{T}}\big{(}2\sqrt{2}a^{\prime\,\text{ten}}\cos{\beta_{V}}+(a^{\text{ten}}+2a^{\prime\,\text{ten}})\sin{\beta_{V}}\big{)}}{6}\Big{)}^{2}$
Table 13.  Coefficients for the decay channel $T\longrightarrow\gamma+V$ decays.  Decay process (in model) | Decay Width (MeV)
---|---
$\,\;a_{2}(1320)\longrightarrow\rho(770)\,\gamma$ | $0.22\pm 0.04$
$\,\;a_{2}^{0}(1320)\longrightarrow\omega(782)\,\gamma$ | $1.94\pm 0.04$
$\,\;a_{2}(1320)\longrightarrow\phi(1020)\,\gamma$ | $0.0024\pm 0.0005$
$\,\;K_{2}^{\ast\pm}(1430)\longrightarrow\bar{K}^{\ast\pm}(892)\,\gamma$ | $0.23\pm 0.04$
$\,\;K_{2}^{\ast\;0}(1430)\longrightarrow\bar{K}^{\ast\;0}(892)\,\gamma$ | $0.94\pm 0.18$
$\,\;f_{2}(1270)\longrightarrow\rho(770)\,\gamma$ | $0.70\pm 0.17$
$\,\;f_{2}(1270)\longrightarrow\omega(782)\,\gamma$ | $0.068\pm 0.017$
$\,\;f_{2}(1270)\longrightarrow\phi(1020)\,\gamma$ | $0.007\pm 0.002$
$\,\;f_{2}^{\prime}(1525)\longrightarrow\rho(770)\,\gamma$ | $0.32\pm 0.08$
$\,\;f_{2}^{\prime}(1525)\longrightarrow\omega(782)\,\gamma$ | $0.012\pm 0.005$
$\,\;f_{2}^{\prime}(1525)\longrightarrow\phi(1020)\,\gamma$ | $0.61\pm 0.12$
Table 14.  Predictions for $T\longrightarrow\gamma+V$ .

####  3.3.1. Vector-vector decay mode of tensor mesons

We conclude the present subsection by considering the following PDG data:

$\displaystyle\text{PDG}:$
$\displaystyle\,\,\,\Gamma[f_{2}(1270)\longrightarrow\pi^{+}\pi^{-}2\pi^{0}]\approx
19.5\,\text{MeV;}$ $\displaystyle\text{PDG}:$
$\displaystyle\,\,\,\Gamma[a_{2}(1320)\longrightarrow\omega(782)\pi\pi]\approx
11.3\,\text{MeV;}$ $\displaystyle\text{PDG}:$
$\displaystyle\,\,\,\Gamma[K_{2}^{\star}(1430)\longrightarrow
K^{\star}(892)\pi\pi]=13.5\pm 2.3\,\text{MeV.}$

It is natural to interpret these decays as vector-vector decay modes
($\rho(770)\rho(770)$, $\omega(782)\rho(770)$, and $K^{*}(892)\rho(770)$), in
which the $\rho(770)$ meson further decays into two pions. In order to
describe them, we introduce the properly normalized “Sill” spectral function
discussed in Ref. [61] for the $\rho$-meson as

$\displaystyle
d_{\rho}(y)=\frac{2y}{\pi}\frac{\sqrt{y^{2}-4m_{\pi}^{2}}\,\tilde{\Gamma}_{\rho}}{(y^{2}-m_{\rho}^{2})^{2}+(\sqrt{y^{2}-4m_{\pi}^{2}}\,\tilde{\Gamma}_{\rho})^{2}}\,\Theta(x-2m_{\pi}),\,\,\int_{0}^{\infty}dy\,d_{\rho}(y)=1\text{
,}$ (3.37)

where $\tilde{\Gamma}_{\rho}$ is defined as

$\displaystyle\tilde{\Gamma}_{\rho}\equiv\frac{\Gamma_{\rho\rightarrow
2\pi}\,m_{\rho}}{\sqrt{m_{\rho}^{2}-4m_{\pi}^{2}}}\;\text{ ,}$ (3.38)

in which $\Gamma_{\rho\rightarrow 2\pi}=149.1$ MeV and $m_{\rho}=775.11$ MeV
are taken from the PDG (small errors omitted).

The Sill spectral function is useful because the normalization is guaranteed
also for broad states and because the contribution of virtual particles (the
pion-pion loop for the $\rho$ meson) has a vanishing real part, thus only the
imaginary part describing the decay width is sufficient (in other words, no
modification of the $\rho(770)$-mass part of the denominator appears), see
details in Ref. [61].

There are two chiral invariant Lagrangians, Eq. (2.20) and Eq. (2.21), that
contribute to the coupling of tensor mesons to two vector mesons. We present
the following approximate results by considering destructive interference
between the corresponding amplitudes (the constructive case delivers too large
decay rates):

$\displaystyle\Gamma_{f_{2}\rightarrow\rho\rho\rightarrow
4\pi}\simeq\int_{0}^{\infty}dy_{1}\,d_{\rho}(y_{1})\,\int_{0}^{\infty}dy_{2}\,d_{\rho}(y_{2})\,\Gamma_{f_{2}\rightarrow\rho\rho}(m_{f_{2}},y_{1},y_{2})\approx
29.9\,\text{MeV ;}$ (3.39)
$\displaystyle\Gamma_{a_{2}\rightarrow\omega\rho\rightarrow\omega
2\pi}\simeq\int_{0}^{\infty}dy_{1}\,d_{\rho}(y_{1})\,\Gamma_{a_{2}\rightarrow\rho\omega}(m_{\mathbb{a}_{2}},m_{\omega_{1}},y_{1})\approx
11.1\,\text{MeV ;}$ (3.40) $\displaystyle\Gamma_{K_{2}\rightarrow
K^{\star}\rho\rightarrow
K2\pi}\simeq\int_{0}^{\infty}dy_{1}\,d_{\rho}(y_{1})\,\Gamma_{K_{2}\rightarrow\rho\omega}(m_{K_{2}},m_{K_{1}},y_{1})\approx
6.6\,\text{MeV .}$ (3.41)

The results are in qualitative agreement with the experimental ones: this is
quite remarkable since no additional parameter is requested. Moreover, this is
a nontrivial process that involves two distinct interaction terms (with
destructive interference) as well as the integral over the spectral function
of the $\rho$-meson.

###  3.4. Decays of axial-tensor mesons

The ground-state mesons with $J^{PC}=2^{--}$ are elusive. In this section we
use the eLSM to predict their decay rates and compare with the available
lattice QCD (LQCD) and other theoretical results.

####  3.4.1. $A_{2}\longrightarrow V+P$ decay mode

The chiral Lagrangian (2.20) contains the decay of axial-vector states into a
vector and pseudoscalar pair:

$\mathcal{L}_{a_{2}vp}=g_{2}^{\text{ten}}\,\mathrm{Tr}\Big{[}A_{2}^{\mu\nu}\big{\\{}(\partial_{\mu}\mathcal{P})\,,V_{\nu}\big{\\}}\Big{]}\,\text{
.}$ (3.42)

The tree-level decay rate takes the form:

$\Gamma_{A_{2}\longrightarrow
V+P}^{tl}(m_{a_{2}},m_{v},m_{p})=\frac{g_{2}^{\text{ten}\,2}\,|\vec{k}_{v,p}|^{3}}{120\,\pi\,m_{a_{2}}^{2}}\Big{(}5+\frac{2\,|\vec{k}_{v,p}|^{2}}{m_{v}^{2}}\Big{)}\,\kappa_{a_{2}vp\,,i}\,\Theta(m_{a_{2}}-m_{v}-m_{p})\,,$
(3.43)

where the $\kappa_{a_{2}vp\,,i}$ can be found in Table 15 together with the
decay widths. We observe quite large decay rates for this decay mode. Yet, our
results can be considered qualitative for various reasons:

(i) the interaction strength is determined solely by the knowledge on tensor
mesons: while this is in agreement with chiral symmetry, no corrections to it
have been taken into account, but chiral symmetry breaking effects are
expected to be relevant between 1-2 GeV;

(ii) the result depends on quantities of the type $(Zw)^{2}\propto g_{1}^{4}$
(see Tables 15 and 4), thus an even small uncertainty in this parameter of the
original eLSM generates a quite large change of the results of the decays of
axial-tensor states;

(iii) the parameter $g_{1}$ (as well as others, such as $g_{2}^{\text{ten}}$)
is taken as a constant over a large range of energies, yet it is reasonable to
assume that it has a (soft) energy dependence. One might expect that it
becomes smaller when the mass of the decaying particle increases, thus
embedding the effect of form factors (and eventually corrections due to
quantum fluctuations). Also in this case, even a small change would have a
non-negligible consequence leading to the left side of the quoted ranges in
table 15.

Summarizing, the used parameters (in particular $g_{1}$) imply a quite large
and difficult to determine source of uncertainty of our decays of axial-tensor
states. In order to be on the safe side, we thus estimate a quite large error
of about 50%, for the reported numbers in Table 15.

Experimental data is missing for the $\rho_{2}$,
$\omega_{2,N}\simeq\omega_{2}$, and $\omega_{2,S}\simeq\phi_{2}$. In the
kaonic sectors, the resonances $K_{2}(1820)$ and $K_{2}(1770)$ are expected to
emerge from the bare axial-tensor and pseudotensor mesons which can mix in a
way that resembles the mixing of the axial-vector and pseudovector mesons
described in Eq. (2.8).

According to the assignment of Table 1, $K_{2A}$ is closer to $K_{2}(1820)$,
(but this is not yet settled). The decay width for the $K_{2A}$ state
described in our work is approximately twice of the experimentally measured
decay of $K_{2}(1820)$. This fact suggests that the left sides of the range of
Table 15 for the $K_{2A}$ state is favoured.

This interpretation can be extended to the other decay rates by a comparison
with the lattice QCD results reported in Table 16. Namely, our theoretical
results for decays are of the same order of magnitude but about 2-3 times
larger than the LQCD ones. Even though both approaches have large
uncertainties, this comparison (as well as the previous arguments) signalizes
that the low side of the range of each decay mode of the eLSM should be
preferred.

For completeness we also perform a parameter determination by directly using
the LQCD results, that leads to $g^{\text{ten}}_{2\,\text{lat}}\simeq(0.7\pm
0.4)\cdot 10^{4}(\text{MeV})$ (assuming 50 percent error in [62])666Note that,
in the $SU(3)$ flavor symmetry based LQCD calculation, the lightest
pseudoscalar states have a mass of about 700 MeV. . This value is within two
sigma of the previous determination in Eq. (3.21). In Table 16, we present the
corresponding results. Quite interestingly, the various entries are consistent
with each other, thus showing that any ratio of decays constructed with
lattice results is in agreement with the corresponding eLSM one.

In summary, the outcome of our theoretical study is pretty clear: although the
uncertainties are still large, the axial-tensor states are expected to be
broad even when taking the low side of our results. This fact could eventually
explain the missing experimental observation of the putative states
$\rho_{2}$, $\omega_{2}$, and $\phi_{2}$ up to now.

Decay process (in model) | eLSM (MeV) | $\kappa_{a_{2}vp\,,i}$
---|---|---
$\,\;\rho_{2}(?)\longrightarrow\rho(770)\,\eta$ | $\approx 99\pm 50$ | $\Big{(}-w_{\eta_{N}}Z_{\eta_{N}}\cos{\beta_{P}}\Big{)}^{2}$
$\,\;\rho_{2}(?)\longrightarrow\bar{K}^{\ast}(892)\,K+\mathrm{c}.\mathrm{c}.$ | $\approx 85\pm 43$ | $4\Big{(}\frac{Z_{K}w_{K}}{2}\Big{)}^{2}$
$\,\;\rho_{2}(?)\longrightarrow\omega(782)\,\pi$ | $\approx 419\pm 210$ | $\Big{(}-w_{\pi}Z_{\pi}\cos{\beta_{V}}\Big{)}^{2}$
$\,\;\rho_{2}(?)\longrightarrow\phi(1020)\,\pi$ | $\approx 0.8$ | $\Big{(}w_{\pi}Z_{\pi}\sin{\beta_{V}}\Big{)}^{2}$
$\,\;K_{2,A}\longrightarrow\rho(770)\,K$ | $\approx 195\pm 98$ | $3\Big{(}\frac{Z_{K}w_{K}}{2}\Big{)}^{2}$
$\,\;K_{2,A}\longrightarrow\bar{K}^{\ast}(892)\,\pi$ | $\approx 316\pm 158$ | $3\Big{(}\frac{Z_{\pi}w_{\pi}}{2}\Big{)}^{2}$
$\,\;K_{2,A}\longrightarrow\bar{K}^{\ast}(892)\,\eta$ | $\approx 0.01$ | $\Big{(}-\frac{1}{2}(\sqrt{2}Z_{\eta_{S}}w_{\eta_{S}}\sin{\beta_{P}}+Z_{\eta_{N}}w_{\eta_{N}}\cos{\beta_{P}})\Big{)}^{2}$
$\,\;K_{2,A}\longrightarrow\omega(782)\,\bar{K}$ | $\approx 51\pm 26$ | $\Big{(}-\frac{w_{K}Z_{K}}{2}(\sqrt{2}\sin{\beta_{V}}+\cos{\beta_{V}})\Big{)}^{2}$
$\,\;K_{2,A}\longrightarrow\phi(1020)\,\bar{K}$ | $\approx 50\pm 25$ | $\Big{(}\frac{w_{K}Z_{K}}{2}(-\sqrt{2}\cos{\beta_{V}}+\sin{\beta_{V}})\Big{)}^{2}$
$\,\;\omega_{2,N}\longrightarrow\rho(770)\,\pi$ | $\approx 1314\pm 657$ | $3\Big{(}-Z_{\pi}w_{\pi}\Big{)}^{2}$
$\,\;\omega_{2,N}\longrightarrow\bar{K}^{\ast}(892)\,K+\mathrm{c}.\mathrm{c}.$ | $\approx 85\pm 43$ | $4\Big{(}-\frac{Z_{K}w_{K}}{2}\Big{)}^{2}$
$\,\;\omega_{2,N}\longrightarrow\omega(782)\,\eta$ | $\approx 93\pm 47$ | $\Big{(}Z_{\eta_{N}}w_{\eta_{N}}\cos{\beta_{P}}\cos{\beta_{V}}\Big{)}^{2}$
$\,\;\omega_{2,N}\longrightarrow\phi(1020)\,\eta$ | $\approx 0.06$ | $\Big{(}Z_{\eta_{N}}w_{\eta_{N}}\cos{\beta_{P}}\sin{\beta_{V}}\Big{)}^{2}$
$\,\;\omega_{2,S}\longrightarrow\bar{K}^{\ast}(892)\,K+\mathrm{c}.\mathrm{c}.$ | $\approx 510\pm 255$ | $4\Big{(}-\frac{Z_{K}w_{K}}{\sqrt{2}}\Big{)}^{2}$
$\,\;\omega_{2,S}\longrightarrow\omega(782)\,\eta$ | $\approx 1.0\pm 0.5$ | $\Big{(}-\sqrt{2}Z_{\eta_{S}}w_{\eta_{S}}\sin{\beta_{P}}\sin{\beta_{V}}\Big{)}^{2}$
$\,\;\omega_{2,S}\longrightarrow\omega(782)\,\eta^{\prime}(958)$ | $\approx 0.3$ | $\Big{(}-\sqrt{2}Z_{\eta_{S}}w_{\eta_{S}}\cos{\beta_{P}}\sin{\beta_{V}}\Big{)}^{2}$
$\,\;\omega_{2,S}\longrightarrow\phi(1020)\,\eta$ | $\approx 101\pm 51$ | $\Big{(}-\sqrt{2}Z_{\eta_{S}}w_{\eta_{S}}\cos{\beta_{V}}\sin{\beta_{P}}\Big{)}^{2}$
Table 15. Predictions and coefficients for the decay channel $A_{2}\longrightarrow V+P$. Decay process (in model) | eLSM (MeV) | LQCD (MeV) [62]
---|---|---
$\,\;\rho_{2}(?)\longrightarrow\rho(770)\,\eta$ | $\approx 30$ |
$\,\;\rho_{2}(?)\longrightarrow\bar{K}^{\ast}(892)\,K+\mathrm{c}.\mathrm{c}.$ | $\approx 27$ | $36$
$\,\;\rho_{2}(?)\longrightarrow\omega(782)\,\pi$ | $\approx 122$ | $125$
$\,\;\rho_{2}(?)\longrightarrow\phi(1020)\,\pi$ | $\approx 0.3$ |
$\,\;K_{2,A}\longrightarrow\rho(770)\,K$ | $\approx 53$ |
$\,\;K_{2,A}\longrightarrow\bar{K}^{\ast}(892)\,\pi$ | $\approx 87$ |
$\,\;K_{2,A}\longrightarrow\bar{K}^{\ast}(892)\,\eta$ | $\approx 0.004$ |
$\,\;K_{2,A}\longrightarrow\omega(782)\,\bar{K}$ | $\approx 13.8$ |
$\,\;K_{2,A}\longrightarrow\phi(1020)\,\bar{K}$ | $\approx 13.7$ |
$\,\;\omega_{2,N}\longrightarrow\rho(770)\,\pi$ | $\approx 363$ | $365$
$\,\;\omega_{2,N}\longrightarrow\bar{K}^{\ast}(892)\,K+\mathrm{c}.\mathrm{c}.$ | $\approx 25$ | $36$
$\,\;\omega_{2,N}\longrightarrow\omega(782)\,\eta$ | $\approx 27$ | $17$
$\,\;\omega_{2,N}\longrightarrow\phi(1020)\,\eta$ | $\approx 0.02$ |
$\,\;\omega_{2,S}\longrightarrow\bar{K}^{\ast}(892)\,K+\mathrm{c}.\mathrm{c}.$ | $\approx 100$ | $148$
$\,\;\omega_{2,S}\longrightarrow\omega(782)\,\eta$ | $\approx 0.2$ |
$\,\;\omega_{2,S}\longrightarrow\omega(782)\,\eta^{\prime}(958)$ | $\approx 0.02$ |
$\,\;\omega_{2,S}\longrightarrow\phi(1020)\,\eta$ | $\approx 17$ | $44$
Table 16.  Predictions for the $A_{2}\longrightarrow V+P$ based on LQCD
results.

####  3.4.2. $A_{2}\longrightarrow A_{1}+P$ decay mode

The chiral Lagrangian of Eq. (2.2) contains the decay into an axial-tensor and
pseudoscalar pair:

$\displaystyle\mathcal{L}_{a_{2}a_{1}p}=c^{\prime\,\text{ten}}\,\varepsilon_{\mu\nu\rho\sigma}\,\text{Tr}\Big{[}\,\partial^{\mu}A_{2}^{\nu\alpha}\,\Big{(}(\partial^{\rho}A_{1}^{\sigma})(\partial_{\alpha}P)\Phi_{0}-\Phi_{0}(\partial_{\alpha}P)(\partial^{\rho}A_{1}^{\sigma})\Big{)}\Big{]}\text{
,}$ (3.44)

where $c^{\prime\,\text{ten}}:=c_{1}^{\text{ten}}-c_{2}^{\text{ten}}$. The
corresponding decay is:

$\Gamma_{A_{2}\longrightarrow
A_{1}+P}^{tl}(m_{a_{2}},m_{a_{1}},m_{p})=\frac{c^{\prime\,\text{ten}\,2}\,|\vec{k}_{a_{1},p}|^{5}}{40\,\pi}\,\kappa_{a_{2}ap\,,i}\Theta(m_{a_{2}}-m_{a_{1}}-m_{p})\,\text{
.}$ (3.45)

The theoretical predictions (as well as the $\kappa_{a_{2}ap\,,i}$ values) for
each channel proportional to the factor
$\frac{c^{\prime\,\text{ten}}}{c^{\text{ten}}}$ are presented in Table 17 (we
assume here, for simplicity, that $K_{1A}\approx K_{1}(1400)$).

Thus, in this case the decay of the tensor mesons does not fix the strength
for the axial-tensor since two equivalent terms are present in the original
Lagrangian of Eq. (2.2), and their sum/difference appears for tensors and
axial-tensors, respectively. Nevertheless, it is reasonable to expect that
$c^{\prime\,\text{ten}}$ is of the same order of $c^{\text{ten}}$. In general,
rather small decay widths are therefore expected. Within this context, it is
worth to mention that our prediction for the $\rho_{2}\rightarrow
a_{1}(1260)\pi$ is quite similar to the prediction in Ref. [50] for the case
of $\frac{c^{\prime\,\text{ten}}}{c^{\text{ten}}}\simeq 1$ .

Decay process (in model) | eLSM (MeV) | $\kappa_{a_{2}ap\,,i}$
---|---|---
$\,\;\rho_{2}(?)\longrightarrow a_{1}(1260)\,\pi$ | $\simeq 13\Big{(}\frac{c^{\prime\,\text{ten}}}{c^{\text{ten}}}\Big{)}^{2}$ | $2\Big{(}\frac{\phi_{N}}{4}\Big{)}^{2}$
$\,\;K_{2,A}\longrightarrow a_{1}(1260)\,K$ | $\simeq 0.1\Big{(}\frac{c^{\prime\,\text{ten}}}{c^{\text{ten}}}\Big{)}^{2}$ | $3\Big{(}\frac{\sqrt{2}\phi_{S}}{8}\Big{)}^{2}$
$\,\;K_{2,A}\longrightarrow\bar{K}_{1,A}\,\pi$ | $\simeq 11\Big{(}\frac{c^{\prime\,\text{ten}}}{c^{\text{ten}}}\Big{)}^{2}$ | $3\Big{(}\frac{\phi_{N}}{8}\Big{)}^{2}$
$\,\;K_{2,A}\longrightarrow f_{1}(1285)\,\bar{K}$ | $\simeq 0.2\Big{(}\frac{c^{\prime\,\text{ten}}}{c^{\text{ten}}}\Big{)}^{2}$ | $\Big{(}\frac{\sqrt{2}}{8}(\phi_{N}\sin{\beta_{A_{1}}}-\phi_{S}\cos{\beta_{A_{1}}})\Big{)}^{2}$
$\,\;\omega_{2,S}\longrightarrow\bar{K}_{1,A}\,K+\mathrm{c}.\mathrm{c}.$ | $\simeq 6\Big{(}\frac{c^{\prime\,\text{ten}}}{c^{\text{ten}}}\Big{)}^{2}$ | $4\Big{(}\frac{\phi_{S}}{4}\Big{)}^{2}$
Table 17. Predictions and coefficients for the decay channel
$A_{2}\longrightarrow A_{1}+P$.

####  3.4.3. $A_{2}\longrightarrow T+P$ decay mode

The last term of the Lagrangian (2.19) enables us to make some predictions for
the decay of the axial-tensor mesons into a tensor and pseudoscalar pair.
Namely, that Lagrangian reduces to the following interaction

$\displaystyle\mathcal{L}_{a_{2}tp}=2ih_{3}^{\text{ten}}\text{Tr}\Big{[}A_{2}^{\mu\nu}(PT_{\mu\nu}\Phi_{0}-\Phi_{0}T_{\mu\nu}P)\Big{]}\,,$
(3.46)

which leads to the corresponding decay

$\displaystyle\Gamma_{A_{2}\longrightarrow
T+P}^{tl}(m_{a_{2}},m_{t},m_{p})=\frac{(h_{3}^{\text{ten}})^{2}|\vec{k}_{t,p}|}{2\,m_{a_{2}}^{2}\pi}\Big{(}1+\frac{4|\vec{k}_{t,p}|^{4}}{45m_{t}^{4}}+\frac{2|\vec{k}_{t,p}|^{2}}{3m_{t}^{2}}\Big{)}\,\kappa_{a_{2}tp\,,i}\,\Theta(m_{a_{2}}-m_{t}-m_{p})\,.$
(3.47)

Decay process (in model) | eLSM (MeV) | $\kappa_{a_{2}tp\,,i}$
---|---|---
$\,\;\rho_{2}(?)\longrightarrow a_{2}(1320)\,\pi$ | $\approx 88$ | $2\Big{(}\frac{\phi_{N}}{4}\Big{)}^{2}$
$\,\;K_{2,A}\longrightarrow K_{2}^{\star}(1430)\,\pi$ | $\approx 49$ | $3\Big{(}\frac{\sqrt{2}\phi_{S}}{8}\Big{)}^{2}$
$\,\;K_{2,A}\longrightarrow a_{2}(1320)\,K$ | $\approx 84$ | $3\Big{(}\frac{\sqrt{2}\phi_{N}}{8}\Big{)}^{2}$
$\,\;K_{2,A}\longrightarrow f_{2}(1270)\,K$ | $\approx 4$ | $\Big{(}\frac{\phi_{N}\cos{\beta_{T}}-2\phi_{S}\sin{\beta_{T}}}{8}\Big{)}^{2}$
$\,\;\omega_{2,S}\longrightarrow K_{2}^{\star}(1430)\,K+\mathrm{c}.\mathrm{c}.$ | $\approx 15$ | $4\Big{(}\frac{\phi_{S}}{8}\Big{)}^{2}$
Table 18. Predictions and coefficients for the decay channel $A_{2}\rightarrow
T+P$.

The theoretical results are reported in Table 18. As it is visible, quite
sizable decays are obtained in some channels, which are then promising for
future investigations of the axial-tensor states. Note, in [50] the decay rate
of $\rho_{2}(?)\longrightarrow a_{2}(1320)\,\pi$ is calculated to be about
$200\,\text{MeV}$.

In the end, it should be stressed that the sum of the various decay rates for
the axial-tensor mesons generates quite large decay widths for these states,
even by taking the lowest side of the given ranges. Nevertheless, the
discovery of the missing axial-tensor states seems possible, especially by
looking into the decay channels described in this subsection.

##  4\. Conclusions

In this work we have studied the well known lightest tensor mesons and their
(still poorly known) chiral partners, the axial-tensor mesons, in the
framework of a chiral model for low-energy QCD, the eLSM.

The masses as well as the strong and radiative decays of the ground-state
tensor nonet fit well into the quark-antiquark picture, thus confirming the
assignment of the quark model review of the PDG. Thanks to chiral symmetry,
the parameters determined in the tensor sector allow to make predictions for
masses and decays for the basically unknown ground-state axial-tensor
resonances (the states $\rho_{2}$, $\omega_{2,N}\simeq\omega_{2}$, and
$\omega_{2,S}\simeq\phi_{2}$ were not yet seen in experiments; only the kaonic
state $K_{2}(1820)$ is listed in the quark model review of the PDG). Even
though the uncertainties are large -the widths are determined solely by the
chiral link of tensors to axial-tensors and small changes in some key
parameters generate quite sizable variations- the emerging picture is quite
clear: the axial-tensor mesons are broad, the largest decay channel is the one
into vector-pseudoscalar, followed by tensor-pseudoscalar and axial-vector-
pseudoscalar pairs, respectively. Large decay widths into vector-pseudoscalar
pairs are also obtained by LQCD and the corresponding decay ratios from
lattice fit well with the ones determined by our model calculations.

We may conclude that the states $\rho_{2}$, $\omega_{2}$, and $\phi_{2}$ can
be discovered in ongoing and future experiments (see e.g. Refs. [26, 27, 28,
29, 30, 31, 32, 33, 34, 35, 36, 37] and refs. therein), even if that shall not
be an easy task, due to the expected large decay widths.

The study of partial wave analysis [16, 63] of the decays of (axial-)tensor
mesons represents an interesting direction to further constrain models in the
future. Moreover, a detailed investigation of the tensor glueball (whose LQCD
predicted mass is about 2.2 GeV [64]) in a chiral framework is a promising
outlook, since the topic of glueballs is a relevant one in low-energy QCD and
various tensor mesons in the energy region at about 2 GeV exist.

## Acknowledgement

We are thankful to Adrian Königstein for fruitful discussions on the tensor
mesons. 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. V. and F. G. acknowledge support from the Polish National Science
Centre (NCN) through the OPUS project no 2019/33/B/ST2/00613. F. G. and S. J.
acknowledge also support through the NCN OPUS no. 2018/29/B/ST2/02576.

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