$\Gamma\equiv\Gamma_{{}^{3}F_{2}}^{\mu\nu}=\,\overleftrightarrow{D}^{\mu}\overleftrightarrow{D}^{\nu}-\left(\frac{2}{3}g^{\mu\nu}-\frac{k^{\mu}k^{\nu}}{k^{2}}\right)\overleftrightarrow{D}_{\alpha}\,\overleftrightarrow{D}^{\alpha}$
(2.156)

leading to the elements

$T_{2F,ij}^{\mu\nu}=\bar{q}_{j}\,\Gamma_{{}^{3}F_{2}}^{\mu\nu}q_{i}\,.$
(2.157)

The excited tensor matrix reads:

$T_{2F}^{\mu\nu}=\frac{1}{\sqrt{2}}\begin{pmatrix}\frac{f_{2F,N}+a_{2F}^{0}}{\sqrt{2}}&a_{2F}^{+}&K_{2F}^{\ast+}\\\
a_{2F}^{-}&\frac{f_{2F,N}-a_{2F}^{0}}{\sqrt{2}}&K_{2F}^{\ast 0}\\\
K_{2F}^{\ast-}&\bar{K}_{2F}^{\ast 0}&f_{2F,S}\end{pmatrix}^{\mu\nu}\,.$
(2.158)

Under $\mathcal{P}$ and $\mathcal{C}$ , one has
$T_{2F}^{\mu\nu}(t,\mathbf{x})\xrightarrow{\mathcal{P}}T_{2F,\mu\nu}(t,-\mathbf{x})$
($\mathcal{P}=1$) and
$T_{2F}^{\mu\nu}\xrightarrow{\mathcal{C}}T_{2F}^{\mu\nu,T}$ ($\mathcal{C}=1$).

A possible candidate member for this nonet is the isoscalar state
$f_{2}(2150)$ [128] (it could be the predominantly non-strange isoscalar
member) but the mixing between the isoscalar members is unknown.

Hybrid meson nonet I: $J^{\mathcal{P}\mathcal{C}}=1^{-+}$

We present the lightest nonet of hybrid states, which are a bound states of a
quark-antiquark pair and one gluon (schematically, $\bar{q}qg$). Even if the
experimental status is not complete yet, some candidates exist. First, let us
discuss the structure. Formally, the currents can be obtained by choosing

$\Gamma=\Gamma_{\text{hyb},1^{-+}}^{\mu}=G^{\mu\nu}\gamma_{\nu}$ (2.159)

leading to

$P_{ij}^{\text{hyb},\mu}=\frac{1}{\sqrt{2}}\bar{q}_{j}\Gamma_{\text{hyb}}^{\mu}q_{i}\,=\frac{1}{\sqrt{2}}\bar{q}_{j}G^{\mu\nu}\gamma_{\nu}q_{i}\,,$
(2.160)

which contain at least one gluon in the wave function, look at
$\left(P_{ij}^{\text{hyb},\mu}\right)^{\dagger}\left|0_{\text{QCD}}\right\rangle.$
Note that since the $\bar{q}q$ spectroscopic notation ${}^{2S+1}L_{J}$ does
not apply to this case (it is a 3-body system), it is omitted.

Intuitively, the vector current $\bar{q}_{j}\gamma_{\nu}q_{i}$ is complemented
by the gluon field tensor $G^{\mu\nu}$, which flips the charge-conjugation but
preserves parity. In matrix form:

$P^{\text{hyb},\mu}=\frac{1}{\sqrt{2}}\left(\begin{array}[c]{ccc}\frac{\eta_{1,N}^{\text{hyb}}+\pi_{1}^{0}}{\sqrt{2}}&\pi_{1}^{\text{hyb}+}&K_{1}^{\text{hyb}+}\\\
\pi_{1}^{\text{hyb}-}&\frac{\eta_{1,N}^{\text{hyb}}-\pi_{1}^{0}}{\sqrt{2}}&K_{1}^{\text{hyb}0}\\\
K_{1}^{\text{hyb}-}&\bar{K}_{1}^{\text{hyb}0}&\eta_{1,S}^{\text{hyb}}\end{array}\right)^{\mu}\;\text{
.}$ (2.161)

Under $\mathcal{P}$ and $\mathcal{C}$ , one has
$P^{\text{hyb}\,,\mu}(t,\mathbf{x})\xrightarrow{\mathcal{P}}P^{\text{hyb}}_{\mu}(t,-\mathbf{x})$
($\mathcal{P}=-1$) and
$P^{\text{hyb}}_{\mu}\xrightarrow{\mathcal{C}}{P^{\text{hyb}}}^{T}_{\mu}$
($\mathcal{C}=1$). The quantum numbers $J^{\mathcal{P}\mathcal{C}}=1^{-+}$ are
not allowed for a purely $\bar{q}q$ system, thus such hybrid object displays
exotic quantum numbers (this is not the case for the kaonic members, since
$\mathcal{C}$ is not defined for them).

The experimental status of the hybrid mesons is as follows: up to the PDG 2022
(and 2023 update) two isovector states with
$J^{\mathcal{P}\mathcal{C}}=1^{-+}$ were reported, the state $\pi_{1}(1400)$
found by the E852 collaboration at Brookhaven [210, 211], and $\pi_{1}(1600)$
observed also by E852 [212, 213, 214, 215] and by the Compass collaboration
[216, 217].

Yet, according to lattice calculations, only one hybrid state close to 1.5 GeV
should appear [218, 219, 106]. This is also the case of models, as e.g.
holographic QCD approaches [220, 221] that would favor a $\pi_{1}(1400)$
hybrid state. The analysis of Ref. [222] has shown a way out of this
conundrum: the resonances $\pi_{1}(1400)$ and $\pi_{1}(1600)$ correspond to
the same pole, hence to the same physical state, whose mass is closer to
$\pi_{1}(1600)$. This view is confirmed by a simple observation (PDG 2022
[13]): $\pi_{1}(1400)$ was observed only in $\eta\pi,$ while $\pi_{1}(1600)$
in all other channels. In the latest version of the PDG 2024 the entry
$\pi_{1}(1400)$ has been removed form the summary table [14].

Quite recently, the BESIII experiment measured an isoscalar hybrid state,
denoted as $\eta_{1}(1855)$ [223]. It is natural to accommodate this state
within a nonet of hybrid states. Due to its larger mass, one may consider it
as a predominantly $\bar{s}sg$ state. In Refs. [224, 225], a detailed study of
the phenomenology of the whole nonet that (with strong and radiative decays)
includes also a kaonic hybrid state with a mass of $1.75$ GeV and a light
$\eta_{1}$ (mostly nonstrange) resonance with a mass about $1.66$ GeV is
presented.

Hybrid meson nonet II: $J^{\mathcal{P}\mathcal{C}}=1^{+-}$

The next nonet of hybrid state is built by inserting a $\gamma^{5}$ to the
currents, leading to the chiral partners of the $1^{-+}$-hybrid nonet:

$\Gamma_{\text{hyb},1^{+-}}^{\mu}=G^{\mu\nu}\gamma^{5}\gamma_{\nu}\text{ ,{}}$
(2.162)

hence:

$B_{ij}^{\text{hyb},\mu}=\frac{1}{\sqrt{2}}\bar{q}_{j}\Gamma_{\text{hyb},1^{+-}}^{\mu}q_{i}=\frac{1}{\sqrt{2}}\bar{q}_{j}G^{\mu\nu}\gamma^{5}\gamma_{\nu}q_{i}\text{
.}$ (2.163)

These states carry the quantum numbers $J^{\mathcal{P}\mathcal{C}}=1^{+-}$
just as the pseudovector mesons, thus making them crypto-exotics. In matrix
form:

$B^{\text{hyb},\mu}=\frac{1}{\sqrt{2}}\left(\begin{array}[c]{ccc}\frac{h_{1N,B}^{\text{hyb}}+b_{1}^{\text{hyb},0}}{\sqrt{2}}&b_{1}^{\text{hyb},+}&K_{1,B}^{\text{hyb}+}\\\
b_{1}^{\text{hyb},+}&\frac{h_{1N,B}^{\text{hyb}}-b_{1}^{\text{hyb},0}}{\sqrt{2}}&K_{1,B}^{\text{hyb}0}\\\
K_{1,B}^{\text{hyb}-}&\bar{K}_{1,B}^{\text{hyb}0}&h_{1S,B}^{\text{hyb}}\end{array}\right)^{\mu}\text{
.}$ (2.164)

Under $\mathcal{P}$ and $\mathcal{C}$ , one has
$B^{\text{hyb}\,,\mu}(t,\mathbf{x})\xrightarrow{\mathcal{P}}-B^{\text{hyb}}_{\mu}(t,-\mathbf{x})$
($\mathcal{P}=1$) and
$B^{\text{hyb},\mu}\xrightarrow{\mathcal{C}}-B^{\text{hyb},\mu T}$
($\mathcal{C}=-1$). According to lattice QCD results, their masses should be
above 2 GeV [218, 219, 106], but at present no experimental candidates can be
listed.

The most important properties of the different nonets are summarized in Tables
2.3, 2.4 and 2.5.

$J^{\mathcal{P}\mathcal{C}}$ | Resonances | Mixing Relation | Mixing Angle
---|---|---|---
$0^{++}$ |  | $\eta$
---
$\eta^{\prime}(958)$
| $\eta=\eta_{N}\,\cos{\beta_{ps}}+\eta_{S}\,\sin{\beta_{ps}}$
---
$\eta^{\prime}=-\eta_{N}\,\sin{\beta_{ps}}+\eta_{S}\cos{\beta_{ps}}$
$\beta_{ps}\simeq-43.4^{\circ}$ Ref. [226]
$1^{--}$ |  | $\omega(782)$
---
$\phi(1020)$
| $\omega=\omega_{1,N}\,\cos{\beta_{v}}+\omega_{1,S}\,\sin{\beta_{v}}$
---
$\phi=-\omega_{1,N}\,\sin{\beta_{v}}+\omega_{1,S}\cos{\beta_{v}}$
$\beta_{v}\simeq-3.9^{\circ}$ Ref. [13]
$1^{++}$ |  | $f_{1}(1285)$
---
$f_{1}(1420)$
| $f_{1}=f_{1,N}\,\cos{\beta_{av}}+f_{1,S}\,\sin{\beta_{av}}$
---
$f_{1}^{\prime}=-f_{1,N}\,\sin{\beta_{av}}+f_{1,S}\cos{\beta_{av}}$
$\beta_{av}^{\dagger}\simeq 24^{\circ\dagger}$ Ref. [189]
$1^{+-}$ |  | $h_{1}(1170)$
---
$h_{1}(1415)$
| $h_{1}=h_{1,N}\,\cos{\beta_{pv}}+h_{1,S}\,\sin{\beta_{pv}}$
---
$h_{1}^{\prime}=-h_{1,N}\,\sin{\beta_{pv}}+h_{1,S}\cos{\beta_{pv}}$
$\beta_{pv}\simeq 1^{\circ\dagger}$ Ref. [194]
$2^{++}$ |  | $f_{2}(1270)$
---
$f_{2}(1525)$
| $f_{2}=f_{2,N}\,\cos{\beta_{t}}+f_{2,S}\,\sin{\beta_{t}}$
---
$f_{2}^{\prime}=-f_{2,N}\,\sin{\beta_{t}}+f_{2,S}\cos{\beta_{t}}$
$\beta_{t}\simeq 3.16^{\circ}$ Ref. [13]
$2^{-+}$ |  | $\eta_{2}(1645)$
---
$\eta_{2}^{\prime}(1870)$
| $\eta_{2}=\eta_{2,N}\,\cos{\beta_{pt}}+\eta_{2,S}\,\sin{\beta_{pt}}$
---
$\eta_{2}^{\prime}=-\eta_{2,N}\,\sin{\beta_{pt}}+\eta_{2,S}\cos{\beta_{pt}}$
$\beta_{pt}\simeq-42.3^{\circ\dagger}$ Ref. [203, 190]
Table 2.3: Mixing angles between the isoscalar mesons of the nonets. Numerical values indicated by $\dagger$ are under debate. $n^{2S+1}L_{J}$ | $J^{\mathcal{P}\mathcal{C}}$ |  | 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
---
$\approx s\overline{s}$
| Nonet Name
---
PDG inspired
| Nonet Name
---
Chirally inspired
| Chiral
---
Multiplet
$1^{1}S_{0}$ | $0^{-+}$ | $\pi$ | $K$ | $\eta(547)$ | $\eta^{\prime}(958)$ | $P$ | $P$ |  $\Phi=S+iP$
$1^{3}P_{0}$ | $0^{++}$ | $a_{0}(1450)$ | $K_{0}^{\star}(1430)$ | $f_{0}(1370)$ |  | $f_{0}(1500)$
---
$f_{0}(1710)$
$S$ | $S$
$1^{3}S_{1}$ | $1^{--}$ | $\rho(770)$ | $K^{\star}(892)$ | $\omega(782)$ | $\phi(1020)$ | $V^{\mu}$ | $V^{\mu}$ |  $L^{\mu}=V^{\mu}+A^{\mu}$ $R^{\mu}=V^{\mu}-A^{\mu}$
$1^{3}P_{1}$ | $1^{++}$ | $a_{1}(1260)$ | $K_{1A}$ | $f_{1}(1285)$ | $f_{1}^{\prime}(1420)$ | $A_{1}^{\mu}$ | $A^{\mu}$
$1^{1}P_{1}$ | $1^{+-}$ | $b_{1}(1235)$ | $K_{1B}$ | $h_{1}(1170)$ | $h_{1}(1415)$ | $B^{\mu}$ | $P^{\mu}$ |  $\Phi^{\mu}=S^{\mu}+iP^{\mu}$
$1^{3}D_{1}$ | $1^{--}$ | $\rho(1700)$ | $K^{\star}(1680)$ | $\omega(1650)$ | $\phi(???)$ | $V_{D}^{\mu}$ | $S^{\mu}$
$1^{3}P_{2}$ | $2^{++}$ | $a_{2}(1320)$ | $K_{2}^{\star}(1430)$ | $f_{2}(1270)$ | $f_{2}^{\prime}(1525)$ | $T^{\mu\nu}$ | $V^{\mu\nu}$ |  $\mathbf{L}^{\mu\nu}=V^{\mu\nu}+A^{\mu\nu}$ $\mathbf{R}^{\mu\nu}=V^{\mu\nu}-A^{\mu\nu}$
$1^{3}D_{2}$ | $2^{--}$ | $\rho_{2}(???)$ | $K_{2}(1820)$ | $\omega_{2}(???)$ | $\phi_{2}(???)$ | $A_{2}^{\mu\nu}$ | $A^{\mu\nu}$
$1^{1}D_{2}$ | $2^{-+}$ | $\pi_{2}(1670)$ | $K_{2}(1770)$ | $\eta_{2}(1645)$ | $\eta_{2}(1870)$ | $P_{2}^{\mu\nu}$ | $P^{\mu\nu}$ |  $\Phi^{\mu\nu}=S^{\mu\nu}{}+iP^{\mu\nu}$
$1^{3}F_{2}$ | $2^{++}$ | $a_{2}(???)$ | $K_{2}^{\ast}(???)$ | $f_{2}(???)$ | $f_{2}(???)$ | $T_{2F}^{\mu\nu}$ | $S^{\mu\nu}$
Table 2.4: List of conventional $q\bar{q}$ mesonic nonets, grouped in pair of chiral partners, together with different naming conventions. On the last column, the chiral multiplets described in Section 2.4 are displayed. Note, in this work we use the PDG-inspired name, but the chirally inspired name, which makes use solely of the letters $S$, $P$, for the heterochiral nonet $\Phi$ and $V$ and $A$ for the homochiral multiplets $L,R$, is also reported for completeness.  Nonet | $J^{\mathcal{P}\mathcal{C}}$ | Current | States | $\mathcal{P}$ | $\mathcal{C}$
---|---|---|---|---|---
$P$ | $0^{-+}$ | $P_{ij}=i\,\overline{q}_{j}\gamma_{5}q_{i}/\sqrt{2}$ |  |  $\pi$, $K$,
---
$\eta(547)$, $\eta^{\prime}(958)$
$-P(t,-\vec{x})$ | $P^{T}$
$S$ | $0^{++}$ | $S_{ij}=\,\overline{q}_{j}q_{i}/\sqrt{2}$ |  |  $a_{0}(1450)$, $K_{0}^{\ast}(1430)$,
---
$f_{0}(1370)$, $f_{0}(1500)/f_{0}(1710)$
$S(t,-\vec{x})$ | $S^{T}$
$V^{\mu}$ | $1^{--}$ | $V^{\mu}_{ij}=\,\overline{q}_{j}\gamma^{\mu}q_{i}/\sqrt{2}$ |  |  $\rho(770)$, $K^{\ast}(892)$,
---
$\omega(782)$, $\phi(1020)$
$V_{\mu}(t,-\vec{x})$ | $-V^{\mu\,,T}$
$A_{1}^{\mu}$ | $1^{++}$ | $A^{\mu}_{1\,,ij}=\,\overline{q}_{j}\gamma^{5}\gamma^{\mu}q_{i}/\sqrt{2}$ |  |  $a_{1}(1260)$, $K_{1}(1270)/K_{1}(1400)$,
---
$f_{1}(1420)$, $f_{1}(1285)$
$-A_{1\mu}(t,-\vec{x})$ | $A_{1}^{\mu\,,T}$
$B^{\mu}$ | $1^{+-}$ | $B^{\mu}_{ij}=\,\overline{q}_{j}\gamma^{5}\overleftrightarrow{D}^{\mu}q_{i}/\sqrt{2}$ |  |  $b_{1}(1235)$, $K_{1}(1270)/K_{1}(1400)$,
---
$h_{1}(1415)$, $h_{1}(1170)$
$-B_{\mu}(t,-\vec{x})$ | $-B^{\mu\,,T}$
$V_{D}^{\mu}$ | $1^{--}$ | $V^{\mu}_{D\,,ij}=i\,\overline{q}_{j}\overleftrightarrow{D}^{\mu}q_{i}/\sqrt{2}$ |  |  $\rho(1700)$, $K^{\ast}(1680)$,
---
$\omega(1650)$, $\phi(?)$
$V_{D\,,\mu}(t,-\vec{x})$ | $-V_{D}^{\mu\,,T}$
$T^{\mu\nu}$ | $2^{++}$ |  | $T_{ij}^{\mu\nu}=i\bar{q}_{j}\,\Big{[}\gamma^{\mu}\,\overleftrightarrow{D}^{\nu}+\gamma^{\nu}\,\overleftrightarrow{D}^{\mu}$
---
$-\frac{2}{3}\Big{(}g^{\mu\nu}-\frac{k^{\mu}k^{\nu}}{k^{2}}\Big{)}\overleftrightarrow{D}_{\alpha}\gamma^{\alpha}\Big{]}\,q_{i}$
|  $a_{2}(1320)$, $K_{2}^{\ast}(1430)$,
---
$f_{2}(1270),f_{2}(1525)$
$T_{\mu\nu}(t,-\vec{x})$ | $\Big{(}T^{\mu\nu}\Big{)}^{T}$
$A_{2}^{\mu\nu}$ | $2^{--}$ |  | $A_{2,ij}^{\mu\nu}=i\bar{q}_{j}\,\Big{[}\gamma^{5}\gamma^{\mu}\,\overleftrightarrow{D}^{\nu}+\gamma^{5}\gamma^{\nu}\,\overleftrightarrow{D}^{\mu}$
---
$-\frac{2}{3}\Big{(}g^{\mu\nu}-\frac{k^{\mu}k^{\nu}}{k^{2}}\Big{)}\overleftrightarrow{D}_{\alpha}\gamma^{5}\gamma^{\alpha}\Big{]}\,q_{i}$
|  $\rho_{2}(?)$, $K_{2}(1770)/K_{2}(1820)$,
---
$\omega_{2}(?)$, $\phi_{2}(?)$
$-A_{2,\mu\nu}(t,-\vec{x})$ | $-\Big{(}A_{2}^{\mu\nu}\Big{)}^{T}$
$P_{2}^{\mu\nu}$ | $2^{-+}$ |  | $P_{2,ij}^{\mu\nu}=i\bar{q}_{j}\,\Big{[}\gamma^{5}\,\overleftrightarrow{D}^{\mu}\overleftrightarrow{D}^{\nu}-$
---
$-\frac{2}{3}\Big{(}g^{\mu\nu}-\frac{k^{\mu}k^{\nu}}{k^{2}}\Big{)}\overleftrightarrow{D}_{\alpha}\,\gamma^{5}\,\overleftrightarrow{D}^{\alpha}\Big{]}\,q_{i}$
|  $\pi_{2}(1670)$, $K_{2}(1770)/K_{2}(1820)$,
---
$\eta_{2}(1870)$, $\eta_{2}(1645)$
$-P_{2,\mu\nu}(t,-\vec{x})$ | $\Big{(}P_{2}^{\mu\nu}\Big{)}^{T}$
$T_{2F}^{\mu\nu}$ | $2^{++}$ |  | $T_{2F,ij}^{\mu\nu}=\bar{q}_{j}\,\Big{[}\,\overleftrightarrow{D}^{\mu}\overleftrightarrow{D}^{\nu}-$
---
$-\frac{2}{3}\Big{(}g^{\mu\nu}-\frac{k^{\mu}k^{\nu}}{k^{2}}\Big{)}\overleftrightarrow{D}_{\alpha}\,\overleftrightarrow{D}^{\alpha}\Big{]}\,q_{i}$
|  $a_{2}(?)$, $K_{2}(?)$,
---
$f_{2}(?)$, $f_{2}^{\prime}(?)$
$T_{2F\,,\mu\nu}(t,-\vec{x})$ | $\Big{(}T_{2F}^{\mu\nu}\Big{)}^{T}$
$P^{\mu,\text{hyb}}$ | $1^{-+}$ | $P^{\mu,\text{hyb}}_{ij}=\,\overline{q}_{j}G^{\mu\nu}\gamma_{\nu}q_{i}/\sqrt{2}$ |  |  $\pi_{1}(1600)$, $K_{1}(?)$,
---
$\eta_{1}(?)$, $\eta_{1}(??)$
$P_{\mu}^{\text{hyb}}(t,-\vec{x})$ | $\Big{(}P^{\mu\,,\text{hyb}}\Big{)}^{T}$
$B^{\mu,\text{hyb}}$ | $1^{+-}$ | $B^{\mu,\text{hyb}}_{ij}=\,\overline{q}_{j}G^{\mu\nu}\gamma_{\nu}\gamma^{5}q_{i}/\sqrt{2}$ |  |  $b_{1}(1200?)$, $K_{1}(?)$,
---
$h_{1}(?)$, $h_{1}(??)$
$-B_{\mu}^{\text{hyb}}(t,-\vec{x})$ | $-\Big{(}B^{\mu\,,\text{hyb}}\Big{)}^{T}$
Table 2.5: Particle content and transformation properties under $\mathcal{P}$
parity and $\mathcal{C}$ charge conjugation of the different nonets. Under
flavor transformation, each nonet transforms as $M_{\Gamma}\rightarrow
U_{V}M_{\Gamma}U_{V}^{\dagger}$.

### 2.4 Chiral multiplets

#### 2.4.1 General ideas and hetero/homochirality

Let us consider a certain nonet $M_{1}\equiv 1/\sqrt{2}\,\bar{q}_{j}\Gamma
q_{i}$ for a given choice of $\Gamma.$ Then, the nonet of chiral partners is
obtained (up to a phase) by inserting a matrix $\gamma^{5}$, giving
$M_{2}\equiv 1/\sqrt{2}\,\bar{q}_{j}\Gamma\gamma^{5}q_{i}$, hence:

$M_{1}\equiv\frac{1}{\sqrt{2}}\,\bar{q}_{j}\Gamma
q_{i}\xLeftrightarrow{\text{chiral
partners}}\frac{1}{\sqrt{2}}\,\bar{q}_{j}\Gamma\gamma^{5}q_{i}=M_{2}\;.$
(2.165)

This is so because a chiral transformation $SU(3)_{L}\times SU(3)_{R}$ mixes
$M_{1}$ with $M_{2}.$ Indeed, the specific chiral transformation denoted as
‘axial’ with $q\rightarrow
e^{{i\sum_{k=1}^{8}\theta_{A}^{(k)}}t^{k}\gamma^{5}}q$ clearly does the job
due to the explicit presence of $\gamma^{5}.$ It is evident that $M_{1}$ with
$M_{2}$ carry the same total angular momentum $J$ but have opposite parity:
namely, $\gamma^{5}$ does not introduce any Lorentz index but switches
$\mathcal{P}$ due to $\\{\gamma^{0},\gamma^{5}\\}=0.$ The charge-conjugation
$\mathcal{C}$is more subtle: in some cases (homochiral multiplet) it switches,
in other (heterochiral partners) it does not.

We also recall that, in the $N_{f}=2$ case, chiral transformation link states
with the same $J,$ but opposite parity $\mathcal{P}$ and $G$-parity, in
particular [124, 227]:

$\displaystyle\sigma_{N}$ $\displaystyle\Longleftrightarrow\mathbf{\pi}\text{
, }\eta_{N}\Longleftrightarrow\mathbf{a}_{0}\text{ ,}$ (2.166)
$\displaystyle\mathbf{\rho}$
$\displaystyle\Longleftrightarrow\mathbf{a}_{1}\text{ ,
}\omega_{N}\Longleftrightarrow f_{1,N}\text{ .}$ (2.167)

In fact, for the (pseudo)scalars: $\sigma_{N}$ has
$\mathcal{G}=\mathcal{C}=1,$ but the pion triplet $\mathbf{\pi}$ has
$\mathcal{G}=-\mathcal{C}=-1,$ and $\eta_{N}$ has $\mathcal{G}=\mathcal{C}=1$
but $\mathbf{a}_{0}$ has $\mathcal{G}=-\mathcal{C}=-1$. For the (axial-)vector
sector: $\mathbf{\rho}$ has $\mathcal{G}=-\mathcal{C}=1$ and $\mathbf{a}_{1}$
has $\mathcal{G}=-\mathcal{C}=-1,$ while $\omega_{N}$ has
$\mathcal{G}=\mathcal{C}=-1$ and $f_{1,N}$ has $\mathcal{G}=\mathcal{C}=1.$
For the general $N_{f}=3$ case, kaons are involved for which
$\mathcal{G}$-parity is not a good quantum number, but we shall recover the
above results as a special case.

Next, we group the nonets $M_{1}$ and $M_{2}$ into appropriate chiral
multiplets, for which two possibilities exist: heterochiral multiplets and
homochiral multiplets [48], see also the summarizing tables 2.5 and 2.6
listing the main multiplets and their properties.

(1) Heterochiral multiplets

The first possibility refers to the case

$\left[\Gamma,\gamma^{5}\right]=0\text{ , }$ (2.168)

implying that

$\overline{q}_{j,R}\,\Gamma q_{i,R}=\overline{q}_{j,L}\,\Gamma q_{i,L}=0\text{
,}$ (2.169)

and

$\overline{q}_{j,R}\,\Gamma q_{i,L}=\overline{q}_{j}\,P_{L}\Gamma
P_{L}q_{i}=\overline{q}_{j}\,P_{L}^{2}\Gamma
q_{i}=\overline{q}_{j}\,P_{L}\Gamma q_{i}=\frac{1}{2}\overline{q}_{j}\,\Gamma
q_{i}-\frac{1}{2}\overline{q}_{j}\,\gamma^{5}\Gamma q_{i}\neq 0\text{ .}$
(2.170)

The corresponding chiral multiplets are defined as:

$\Phi_{\Gamma,ij}=\sqrt{2}\overline{q}_{j,R}\,\Gamma q_{i,L}\text{ .}$ (2.171)

It is then clear that the object $\Phi_{\Gamma}$ contains both chiral partners
$M_{1}$ and $M_{2}$ in a linear combination. In general one has

$\Phi_{\Gamma}=M_{1}+iM_{2}$ (2.172)

where the phase $i$ takes into account that the currents must be Hermitian.

The transformation of $\Phi_{\Gamma}$ under chiral transformations
$SU(3)_{L}\times SU(3)_{R}$ is easily obtained form Eq. (2.171) as:

$\Phi_{\Gamma}\rightarrow U_{L}\Phi U_{R}^{\dagger}\text{ .}$ (2.173)

This is also why this chiral multiplet is called heterochiral: both $U_{L}$
and $U_{R}$ appear in its chirally transformed expression. For heterochiral
multiplets, the chiral partners $M_{1}$ and $M_{2}$ have the same
$\mathcal{C}$, see below for examples.

(2) Homochiral multiplets

The second possibility refers to the case

$\left\\{\Gamma,\gamma^{5}\right\\}=0\text{ ,}$ (2.174)

implying that

$\overline{q}_{j,R}\,\Gamma q_{i,L}=0$ (2.175)

and

$\overline{q}_{j,L}\,\Gamma q_{i,L}\neq 0\text{ , }\overline{q}_{j,R}\,\Gamma
q_{i,R}\neq 0\text{ .}$ (2.176)

In this case, one introduces the left-handed and the right-handed currents

$L_{\Gamma,ij}=\sqrt{2}\overline{q}_{j,L}\,\Gamma q_{i,L}\text{ ,
}R_{\Gamma,ij}=\sqrt{2}\overline{q}_{j,R}\,\Gamma q_{i,R}$ (2.177)

that can be expressed as:

$L_{\Gamma}=M_{1}+M_{2}\text{ , }R_{\Gamma}=M_{1}-M_{2}\text{ .}$ (2.178)

Under chiral transformations, the left-handed and right-handed currents
transform as:

$L_{\Gamma}\rightarrow U_{L}L_{\Gamma}U_{L}^{\dagger}\text{ ,
}R_{\Gamma}\rightarrow U_{R}R_{\Gamma}U_{R}^{\dagger}\text{ ,}$ (2.179)

hence they are called homochiral. For these multiplets, the chiral partners
$M_{1}$ and $M_{2}$ have opposite $\mathcal{C}$, see Table 2.5.

#### 2.4.2 List of chiral multiplets

Below we list various quark-antiquark chiral multiplets by increasing values
of $J$ and by following the grouping of the previous subsection where chiral
partners were presented one after the other. In the end, we also present one
chiral multiplet for vectorial hybrid mesons. Tables 2.5 and 2.6 summarizes
the multiplets and their most important properties.

1) $J^{\mathcal{P}\mathcal{C}}=0^{\pm+},$ heterochiral multiplet of
(pseudo)scalar states, with pseudoscalar nonet $P$
($J^{\mathcal{P}\mathcal{C}}=0^{-+}$; $L=S=0$; $n^{\;2S+1}L_{J}\equiv$
$1^{\;1}S_{0})$ and the scalar meson nonet $S$
($J^{\mathcal{P}\mathcal{C}}=0^{++}$; $L=S=1$; $n$ ${}^{2S+1}L_{J}\equiv$ $1$
${}^{3}P_{0}).$ In short: heteroscalars.

The most important multiplet for chiral models is the one that involves
(pseudo)scalar states. It is obtained from the general equations above by
setting $\Gamma=1,$ hence:

$\displaystyle\Phi_{ij}$
$\displaystyle\equiv\Phi_{ij}=\sqrt{2}\overline{q}_{j,R}\,\Gamma
q_{i,L}=S_{ij}+iP_{ij}\text{ ,}$ $\displaystyle\Phi$ $\displaystyle=S+iP$

where $S$ and $P$ refer to the scalar and pseudoscalar ground state nonets.

This is clearly a heterochiral multiplet with $\Phi\rightarrow U_{L}\Phi
U_{R}^{\dagger}$ under chiral transformations. Thanks to this simple
transformation, it is very easy to construct $SU(3)_{L}\times SU(3)_{R}$
chirally invariant objects by using the property of the trace, such as
$Tr\left[\Phi\Phi^{\dagger}\right]$ or
$Tr\left[\Phi\Phi^{\dagger}\Phi\Phi^{\dagger}\right]$. Another possibility is
to use the determinant, $\det\Phi,$ which however is invariant only under
$SU(3)_{L}\times SU(3)_{R}$ but not $U(1)_{A}.$ This is why this multiplet, as
well as other heterochiral ones, contain interaction terms that explicitly
break the axial symmetry $U(1)_{A},$ in agreement with the axial anomaly. In
particular, a mass contribution and mixing among the isoscalar members is
possible (as for the renowned $\eta$-$\eta^{\prime}$ system of the
pseudoscalar nonet). In the next section, we will make use of both the trace
and the determinant, as well as an extension of the latter.

Moreover, using the previous results for parity and charge conjugation, we
obtain

$\Phi(t,\mathbf{x})\xrightarrow{\mathcal{P}}\Phi^{\dagger}(t,-\mathbf{x})\text{
, }\Phi\xrightarrow{\mathcal{C}}\Phi^{T}\text{ .}$ (2.180)

It is also useful to express the matrix $\Phi$ in explicit form:

$\Phi=\sum_{k=0}^{8}(S_{k}+iP_{k})t^{k}=\frac{1}{\sqrt{2}}\begin{pmatrix}\frac{(\sigma_{N}+a_{0}^{0})+i(\eta_{N}+\pi^{0})}{\sqrt{2}}&a_{0}^{+}+i\pi^{+}&K_{0}^{\star+}+iK^{+}\\\
a_{0}^{-}+i\pi^{-}&\frac{(\sigma_{N}-a_{0}^{0})+i(\eta_{N}-\pi^{0})}{\sqrt{2}}&K_{0}^{\star
0}+iK^{0}\\\
K_{0}^{\star-}+iK^{-}&\bar{K}_{0}^{\star}+i\bar{K}^{0}&\sigma_{S}+i\eta_{S}\end{pmatrix}\,\,.$
(2.181)

The matrix $\Phi$ develops nonzero v.e.v. because of the isoscalar-scalar
fields $\sigma_{N}$ and $\sigma_{S}$. Any LSM needs to contain (at least some
of) the fields above.

2) $J^{\mathcal{P}\mathcal{C}}=1^{\pm\pm},$ homochiral multiplet of
(axial-)vector states, with the vector meson nonet $V^{\mu}$
($J^{\mathcal{P}\mathcal{C}}=1^{--}$; $L=0,$ $S=1$; $n$ ${}^{2S+1}L_{J}\equiv$
$1$ ${}^{3}S_{1})$ and the axial-vector meson nonet $A_{1}^{\mu}$
($J^{\mathcal{P}\mathcal{C}}=1^{++}$; $L=S=1$; $n$ ${}^{2S+1}L_{J}\equiv$ $1$
${}^{3}P_{1}).$ In short: homovectors.

By setting $\Sigma=\gamma^{\mu}$ we get the homochiral left-handed and right-
handed currents (referred to as $L^{\mu}$ and $R^{\mu}$), which are a
combination of vector and axial-vector nonets:

$\displaystyle\left(L_{\gamma^{\mu}}\right)_{ij}^{\mu}$ $\displaystyle\equiv
L_{ij}^{\mu}=\sqrt{2}\overline{q}_{j,L}\gamma^{\mu}q_{i,L}\,\text{ ,
}\left(R_{\gamma^{\mu}}\right)_{ij}^{\mu}\equiv
R_{ij}^{\mu}=\sqrt{2}\overline{q}_{j,R}\gamma^{\mu}q_{i,R}\,,$ $\displaystyle
L^{\mu}$ $\displaystyle=V^{\mu}+A_{1\,}^{\mu}\text{ ,
}R^{\mu}=V^{\mu}-A_{1\,}^{\mu}\text{ .}$

Under $U(3)_{L}\times U(3)_{R}$ they change as

$L^{\mu}\rightarrow U_{L}L^{\mu}U_{L}^{\dagger}\text{ , }R^{\mu}\rightarrow
U_{R}R^{\mu}U_{R}^{\dagger}\text{ .}$ (2.182)

Interestingly, these objects are invariant under the axial group $U(1)_{A}$,
so interaction terms involving $L^{\mu}$ and $R^{\mu}$ do not result in axial
anomalous terms. In general homochiral nonets share this property, hence the
isoscalar members are expected to be closer to nonstrange and strange members,
as it is indeed for the well-known vector meson nonet, see Eq. (2.129). On the
other hand, it is quite easy to write chirally invariant terms, e.g.
$Tr[L_{\mu}]Tr[L^{\mu}]$, but these terms are suppressed in the large-$N_{c}$
limit (the coupling constant in front of them scales as $N_{c}^{-1}$ instead
of $N_{c}^{0}$ as a usual mass term).

Under parity and charge conjugation one gets:

$\displaystyle
L^{\mu}(t,\mathbf{x})\xrightarrow{\mathcal{P}}R_{\mu}(t,-\mathbf{x})\text{ ,
}R^{\mu}(t,\mathbf{x})\xrightarrow{\mathcal{P}}-L_{\mu}(t,-\mathbf{x})\text{
;}$ $\displaystyle L^{\mu}\xrightarrow{\mathcal{C}}-R^{\mu,T}\text{ ,
}R^{\mu}\xrightarrow{\mathcal{C}}-L^{\mu,T}\text{ .}$

The explicit expressions read:

$L^{\mu}:=\sum_{k=1}^{8}(V_{k}^{\mu}+A_{1\,k}^{\mu})t^{k}=\frac{1}{\sqrt{2}}\begin{pmatrix}\frac{(\omega_{N}+\rho^{0})}{\sqrt{2}}+\frac{(f_{1N}+a_{1}^{0})}{\sqrt{2}}&\rho^{+}+a_{1}^{+}&K^{\star+}+K_{1A}^{+}\\\
\rho^{-}+a_{1}^{-}&\frac{(\omega_{N}-\rho^{0})}{\sqrt{2}}+\frac{(f_{1N}-a_{1}^{0})}{\sqrt{2}}&K^{\star
0}+K_{1A}^{0}\\\ K^{\star-}+K_{1A}^{-}&\bar{K}^{\star
0}+\bar{K}_{1A}^{0}&\omega_{S}+f_{1S}\end{pmatrix}^{\mu}\,\,,$ (2.183)
$R^{\mu}:=\sum_{k=0}^{8}(V_{k}^{\mu}-A_{1\,k}^{\mu})t^{k}=\frac{1}{\sqrt{2}}\begin{pmatrix}\frac{(\omega_{N}+\rho^{0})}{\sqrt{2}}-\frac{(f_{1N}+a_{1}^{0})}{\sqrt{2}}&\rho^{+}-a_{1}^{+}&K^{\star+}-K_{1A}^{+}\\\
\rho^{-}-a_{1}^{-}&\frac{(\omega_{N}-\rho^{0})}{\sqrt{2}}-\frac{(f_{1N}-a_{1}^{0})}{\sqrt{2}}&K^{\star
0}-K_{1A}^{0}\\\ K^{\star-}-K_{1A}^{-}&\bar{K}^{\star
0}-\bar{K}_{1A}^{0}&\omega_{S}-f_{1S}\end{pmatrix}^{\mu}\,\,.$ (2.184)

3) $J^{\mathcal{P}\mathcal{C}}=1^{\pm-},$ heterochiral multiplet of
pseudovector and orbitally excited vector states, with the pseudovector meson
nonet $B^{\mu}$ ($J^{\mathcal{P}\mathcal{C}}=1^{+-}$; $L=1,$ $S=0$; $n$
${}^{2S+1}L_{J}\equiv$ $1$ ${}^{1}P_{1})$ and the orbitally excited vector
meson nonet $V_{D}^{\mu}$ ($J^{\mathcal{P}\mathcal{C}}=1^{--}$; $L=2$ , $S=1$;
$n$ ${}^{2S+1}L_{J}\equiv$ $1$ ${}^{3}D_{1})$. In short: heterovectors.

The next $\bar{q}q$ multiplet deals also with vector states. It arises from
the choice $\Sigma=i\overleftrightarrow{D}^{\mu},$ thus

$\displaystyle\left(\Phi_{\overleftrightarrow{D}^{\mu}}\right)_{ij}^{\mu}$
$\displaystyle=\Phi_{ij}^{\mu}=\sqrt{2}\,\overline{q}_{j,R}\,\overleftrightarrow{D}^{\mu}\,q_{i,L}\,,$
$\displaystyle\Phi^{\mu}$ $\displaystyle=B^{\mu}+iV_{D}^{\mu}\text{ .}$

The structure is heterochiral: $\Phi^{\mu}\rightarrow
U_{L}\Phi^{\mu}U_{R}^{\dagger}$ under $SU(3)_{L}\times SU(3)_{R},$ just as for
the ground-state (pseudo)scalar multiplet. That implies that the axial anomaly
may be realized, see later on. Under parity and charge conjugation:

$\Phi^{\mu}(t,\mathbf{x})\xrightarrow{\mathcal{P}}\Phi_{\mu}^{\dagger}(t,-\mathbf{x})\text{
, }\Phi^{\mu}\xrightarrow{\mathcal{C}}-{\Phi^{\mu}}^{T}\text{ .}$ (2.185)

The explicit form is given by:

$\Phi^{\mu}:=\sum_{k=0}^{8}(B_{k}^{\mu}+iV_{D\,,k}^{\mu})t^{k}=\frac{1}{\sqrt{2}}\begin{pmatrix}\frac{(\omega_{D}+\rho_{D}^{0})+i(h_{1N}+b_{1}^{0})}{\sqrt{2}}&\rho_{D}^{+}+ib_{1}^{+}&K_{1D}^{\star+}+iK_{1b}^{+}\\\
\rho_{D}^{-}+ib_{1}^{-}&\frac{(\omega_{D}-\rho_{D}^{0})+i(h_{1N}-b_{1}^{0})}{\sqrt{2}}&K_{1D}^{\star
0}+iK_{1b}^{0}\\\
K_{1D}^{\star-}+iK_{1b}^{-}&\bar{K}_{1D}^{\star}+i\bar{K}_{1b}^{0}&\phi_{D}+ih_{1S}\end{pmatrix}^{\mu}\,\,.$
(2.186)

4) $J^{\mathcal{P}\mathcal{C}}=2^{\pm\pm},$ homochiral multiplet of
(axial-)tensor mesons, with the tensor meson nonet $T^{\mu\nu}$
($J^{\mathcal{P}\mathcal{C}}=2^{++}$; $L=S=1$; $n$ ${}^{2S+1}L_{J}\equiv$ $1$
${}^{3}P_{2})$ and the axial-tensor meson nonet $A_{2}^{\mu\nu}$
($J^{\mathcal{P}\mathcal{C}}=2^{--}$; $L=2$ , $S=1$; $n$
${}^{2S+1}L_{J}\equiv$ $1$ ${}^{3}D_{2})$. In short: homotensors.

The choice
$\Sigma=\gamma^{\mu}\,\overleftrightarrow{D}^{\nu}+\gamma^{\nu}\,\overleftrightarrow{D}^{\mu}$
generates tensorial left-handed and right-handed ‘homochiral’ currents that
involve tensor meson nonet $T^{\mu\nu}$:

$\displaystyle\left(L_{\gamma^{\mu}\,\overleftrightarrow{D}^{\nu}+...}\right)_{ij}^{\mu\nu}$
$\displaystyle\equiv\mathbf{L}_{ij}^{\mu\nu}=\overline{q}_{j,L}(\gamma^{\mu}\,\overleftrightarrow{D}^{\nu}+\gamma^{\nu}\,\overleftrightarrow{D}^{\mu})q_{i,L}\,,\text{
}\left(R_{\gamma^{\mu}\,\overleftrightarrow{D}^{\nu}+...}\right)_{ij}^{\mu\nu}\equiv\mathbf{R}_{ij}^{\mu\nu}=\overline{q}_{j,R}(\gamma^{\mu}\,\overleftrightarrow{D}^{\nu}+\gamma^{\nu}\,\overleftrightarrow{D}^{\mu})q_{i,R}\,;$
$\displaystyle\mathbf{L}^{\mu\nu}$
$\displaystyle=T^{\mu\nu}+A_{2}^{\mu\nu}\text{ ,
}\mathbf{R}^{\mu\nu}=T^{\mu\nu}-A_{2}^{\mu\nu}\text{ .}$

Note, the currents are marked as bold to distinguish them from
$L^{\mu\nu}=\partial^{\mu}L^{\nu}-\partial^{\nu}L^{\mu}$ that appear as
kinetic terms of the right(left)-handed homochiral vector mesons.

Under chiral transformations, $\mathbf{L}^{\mu\nu}\rightarrow
U_{L}\mathbf{L}^{\mu\nu}U_{L}^{\dagger},$ $\mathbf{R}^{\mu\nu}\rightarrow
U_{R}\mathbf{R}^{\mu\nu}U_{R}^{\dagger},$ and under parity and charge
conjugation:

$\displaystyle\mathbf{L}^{\mu\nu}(t,\mathbf{x})\xrightarrow{\mathcal{P}}\mathbf{R}^{\mu\nu}(t,-\mathbf{x})\text{
,
}\mathbf{R}^{\mu\nu}(t,\mathbf{x})\xrightarrow{\mathcal{P}}\mathbf{R}^{\mu\nu}(t,-\mathbf{x})\text{
;}$
$\displaystyle\mathbf{L}^{\mu\nu}\xrightarrow{\mathcal{C}}{\rightarrow}{\mathbf{R}^{\mu\nu}}^{T}\text{
, }\mathbf{R}^{\mu\nu}\xrightarrow{\mathcal{C}}{\mathbf{L}^{\mu\nu}}^{T}\text{
.}$

The matrix expressions read:

$\mathbf{L}^{\mu\nu}:=\sum_{k=0}^{8}(T_{k}^{\mu\nu}+A_{2,\,k}^{\mu\nu})t^{k}=\frac{1}{\sqrt{2}}\begin{pmatrix}\frac{(\omega_{2N}+\rho_{2}^{0})}{\sqrt{2}}+\frac{(f_{2N}+a_{2}^{0})}{\sqrt{2}}&\rho_{2}^{+}+a_{2}^{+}&K_{2}^{\star+}+K_{2A}^{+}\\\
\rho_{2}^{-}+a_{2}^{-}&\frac{(\omega_{2N}-\rho_{2}^{0})}{\sqrt{2}}+\frac{(f_{2N}-a_{2}^{0})}{\sqrt{2}}&K_{2}^{\star
0}+K_{2A}^{0}\\\ K_{2}^{\star-}+K_{2A}^{-}&\bar{K}_{2}^{\star
0}+\bar{K}_{2A}^{0}&\omega_{2S}+f_{2S}\end{pmatrix}^{\mu\nu},\,\,$ (2.187)
$\mathbf{R}^{\mu\nu}:=\sum_{k=0}^{8}(T_{k}^{\mu\nu}-A_{2,\,k}^{\mu\nu})t^{k}=\frac{1}{\sqrt{2}}\begin{pmatrix}\frac{(\omega_{2N}+\rho_{2}^{0})}{\sqrt{2}}-\frac{(f_{2N}+a_{2}^{0})}{\sqrt{2}}&\rho_{2}^{+}-a_{2}^{+}&K_{2}^{\star+}-K_{2A}^{+}\\\
\rho_{2}^{-}-a_{2}^{-}&\frac{(\omega_{2N}-\rho_{2}^{0})}{\sqrt{2}}-\frac{(f_{2N}-a_{2}^{0})}{\sqrt{2}}&K_{2}^{\star
0}-K_{2A}^{0}\\\ K_{2}^{\star-}-K_{2A}^{-}&\bar{K}_{2}^{\star
0}-\bar{K}_{2A}^{0}&\omega_{2S}-f_{2S}\end{pmatrix}^{\mu\nu}\,\,.$ (2.188)

5) $J^{\mathcal{P}\mathcal{C}}=2^{\pm+},$ heterochiral multiplet of
(pseudo)tensor states and their chiral partners, with the pseudotensor meson
nonet $P_{2}^{\mu\nu}$ ($J^{\mathcal{P}\mathcal{C}}=2^{-+}$; $L=2$,$S=0$;
$n^{\;2S+1}L_{J}\equiv$ $1^{\;1}D_{2})$ and the excited tensor meson nonet
$T_{2F}^{\mu\nu}$ ($J^{\mathcal{P}\mathcal{C}}=2^{++}$; $L=3$ , $S=1$;
$n^{\;2S+1}L_{J}\equiv$ $1^{\;3}F_{2})$. In short: heterotensors.

This heterochiral nonet is obtained for
$\Sigma=\frac{g^{\mu\nu}}{4}\overleftrightarrow{D}^{\alpha}\overleftrightarrow{D}_{\alpha}-\overleftrightarrow{D}^{\mu}\overleftrightarrow{D}^{\nu}$
, out of which:

$\displaystyle\left(\Phi_{\overleftrightarrow{D}^{\mu}\,\overleftrightarrow{D}^{\nu}+...}\right)_{ij}^{\mu\nu}\equiv\Phi_{ij}^{\mu\nu}$
$\displaystyle=\overline{q}_{j,R}\,(\frac{g^{\mu\nu}}{4}\overleftrightarrow{D}^{\alpha}\overleftrightarrow{D}_{\alpha}-\overleftrightarrow{D}^{\mu}\overleftrightarrow{D}^{\nu})\,q_{i,L}\,,$
$\displaystyle\Phi^{\mu\nu}$
$\displaystyle=T^{\mu\nu}_{2F}+iP^{\mu\nu}_{2}\text{ .}$

Under the chiral group $\Phi^{\mu\nu}\rightarrow
U_{L}\Phi^{\mu\nu}U_{R}^{\dagger},$ while under parity and charge conjugation:

$\Phi^{\mu\nu}(t,\mathbf{x})\xrightarrow{\mathcal{P}}\Phi^{\mu\nu,\dagger}(t,-\mathbf{x})\text{
, }\Phi^{\mu\nu}\xrightarrow{\mathcal{C}}\Phi^{\mu\nu,T}\text{ .}$ (2.189)

In matrix form:

$\Phi^{\mu\nu}=\sum_{k=0}^{8}(T_{2F,\,k}^{\mu\nu}{}+iP_{2,\,k}^{\mu\nu})t^{k}=\frac{1}{\sqrt{2}}\begin{pmatrix}\frac{(f_{2F,N}+a_{2F}^{0})+i(\eta_{2N}+\pi_{2}^{0})}{\sqrt{2}}&a_{2F}^{+}+i\pi_{2}^{+}&K_{2F}^{\star+}+iK_{2P}^{+}\\\
a_{2F}^{-}+i\pi_{2}^{-}&\frac{(f_{2F,N}-a_{2F}^{0})+i(\eta_{2N}-\pi_{2}^{0})}{\sqrt{2}}&K_{2F}^{\star
0}+iK_{2P}^{0}\\\
K_{2F}^{\star-}+iK_{2P}^{-}&\bar{K}_{2F}^{\star}+i\bar{K}_{2P}^{0}&f_{2F,S}+i\eta_{2S}\end{pmatrix}^{\mu\nu}\,\,.$
(2.190)

6) $J^{\mathcal{P}\mathcal{C}}=1^{\pm\mp},$ homochiral multiplet of vectorial
hybrid mesons, with the exotic hybrid nonet $\Pi^{\text{hyb},\mu}$
($J^{\mathcal{P}\mathcal{C}}=1^{-+})$ and the cryptoexotic hybrid nonet
$B^{\text{hyb},\mu}$ ($J^{\mathcal{P}\mathcal{C}}=1^{+-}$). In short: hybrid
homovectors.

Upon choosing the hybrid current $\Sigma=G_{\mu\nu}\gamma^{\nu},$ we obtain
the right-handed and left-handed hybrid currents:

$\displaystyle L_{\mu}^{\text{hyb}}$
$\displaystyle=\overline{q}_{L}G_{\mu\nu}\gamma^{\nu}q_{L}\,,\text{
}R_{\mu}^{\text{hyb}}=\overline{q}_{R}G_{\mu\nu}\gamma^{\nu}q_{R}\,,\qquad$
$\displaystyle L_{\mu}^{\text{hyb}}$
$\displaystyle=\Pi_{\mu}^{\text{hyb}}+B_{\mu}^{\text{hyb}}\text{ ,
}R_{\mu}^{\text{hyb}}=\Pi_{\mu}^{\text{hyb}}-B_{\mu}^{\text{hyb}}\text{ . }$

Under chiral transformations $L_{\mu}^{\text{hyb}}\rightarrow
U_{L}L_{\mu}^{\text{hyb}}U_{L}^{\dagger}$ , $R_{\mu}^{\text{hyb}}\rightarrow
U_{R}R_{\mu}^{\text{hyb}}U_{R}^{\dagger},$ while under parity and charge
conjugation:

$\displaystyle
L_{\mu}^{\text{hyb}}(t,\mathbf{x})\xrightarrow{\mathcal{P}}R^{\text{hyb,}\mu}(t,-\mathbf{x})\text{
,
}R_{\mu}^{\text{hyb}}(t,\mathbf{x})\xrightarrow{\mathcal{P}}L^{\text{hyb,}\mu}(t,-\mathbf{x})\text{
;}$ $\displaystyle
L_{\mu}^{\text{hyb}}\xrightarrow{\mathcal{C}}R_{\mu}^{\text{hyb,}T}\text{ ,
}R_{\mu}^{\text{hyb}}\xrightarrow{\mathcal{C}}L_{\mu}^{\text{hyb,}T}\text{ .}$

The hybrid multiplets can be written via matrices just as regular quark-
antiquark nonets:

$L_{\mu}^{\text{hyb}}=P_{\mu}^{\text{hyb}}+B_{\mu}^{\text{hyb}}=\frac{1}{\sqrt{2}}\left(\begin{array}[c]{ccc}\frac{\eta_{1,N}^{\text{hyb}}+\pi_{1}^{\text{hyb}\,,0}}{\sqrt{2}}+\frac{h_{1N,B}^{\text{hyb}}+b_{1}^{\text{hyb}\,,0}}{\sqrt{2}}&\pi_{1}^{\text{hyb}\,,+}+b_{1}^{\text{hyb}\,,+}&K_{1}^{\text{hyb}\,,+}+K_{1,B}^{\text{hyb}\,,+}\\\
\pi_{1}^{\text{hyb}\,,-}+b_{1}^{\text{hyb}\,,-}&\frac{\eta_{1,N}^{\text{hyb}}-\pi_{1}^{\text{hyb}\,,0}}{\sqrt{2}}+\frac{h_{1N,B}^{\text{hyb}}-b_{1}^{\text{hyb}\,,0}}{\sqrt{2}}&K_{1}^{\text{hyb}\,,0}+K_{1,B}^{\text{hyb}\,,0}\\\
K_{1}^{\text{hyb}\,,-}+K_{1,B}^{\text{hyb}\,,-}&\bar{K}_{1}^{\text{hyb}\,,0}+\bar{K}_{1,B}^{\text{hyb}\,,0}&\eta_{1,S}^{\text{hyb}}+h_{1S,B}^{\text{hyb}}\end{array}\right)_{\mu}\text{
,}$ (2.191)
$R_{\mu}^{\text{hyb}}=P_{\mu}^{\text{hyb}}-B_{\mu}^{\text{hyb}}=\frac{1}{\sqrt{2}}\left(\begin{array}[c]{ccc}\frac{\eta_{1,N}^{\text{hyb}}+\pi_{1}^{\text{hyb}\,,0}}{\sqrt{2}}-\frac{h_{1N,B}^{\text{hyb}}+b_{1}^{\text{hyb}\,,0}}{\sqrt{2}}&\pi_{1}^{\text{hyb}\,,+}-b_{1}^{\text{hyb}\,,+}&K_{1}^{\text{hyb}\,,+}-K_{1,B}^{\text{hyb}\,,+}\\\
\pi_{1}^{\text{hyb}\,,-}-b_{1}^{\text{hyb}\,,-}&\frac{\eta_{1,N}^{\text{hyb}}-\pi_{1}^{\text{hyb}\,,0}}{\sqrt{2}}-\frac{h_{1N,B}^{\text{hyb}}-b_{1}^{\text{hyb}\,,0}}{\sqrt{2}}&K_{1}^{\text{hyb}\,,0}-K_{1,B}^{\text{hyb}\,,0}\\\
K_{1}^{\text{hyb}\,,-}-K_{1,B}^{\text{hyb}\,,-}&\bar{K}_{1}^{\text{hyb}\,,0}-\bar{K}_{1,B}^{\text{hyb}\,,0}&\eta_{1,S}^{\text{hyb}}-h_{1S,B}^{\text{hyb}}\end{array}\right)_{\mu}\text{
.}$ (2.192)

Chiral Multiplet | Parity $(\mathcal{P})$ | Charge conjugation $(\mathcal{C})$ | $U(3)_{L}\times U(3)_{R}$ | Type
---|---|---|---|---
$\Phi=S+iP$ | $\Phi^{\dagger}(t,-\vec{x})$ | $\Phi^{T}$ | $U_{L}\Phi U^{\dagger}_{R}$ | Heterochiral
$R^{\mu}=V^{\mu}-A_{1}^{\mu}$ | $L_{\mu}(t,-\vec{x})$ | $-(L^{\mu})^{T}$ | $U_{R}R^{\mu}U_{R}^{\dagger}$ | Homochiral
$L^{\mu}=V^{\mu}+A_{1}^{\mu}$ | $R_{\mu}(t,-\vec{x})$ | $-(R^{\mu})^{T}$ | $U_{L}L^{\mu}U_{L}^{\dagger}$
$\Phi^{\mu}=S^{\mu}+iP^{\mu}$ | $\Phi_{\mu}^{\dagger}(t,-\vec{x})$ | $-(\Phi^{\mu})^{T}$ | $U_{L}\Phi^{\mu}U^{\dagger}_{R}$ | Heterochiral
$\mathbf{R}^{\mu\nu}=T^{\mu\nu}-A_{2}^{\mu\nu}$ | $\mathbf{L}_{\mu\nu}(t,-\vec{x})$ | $(\mathbf{L}^{\mu\nu})^{T}$ | $U_{R}\mathbf{R}^{\mu\nu}U^{\dagger}_{R}$ | Homochiral
$\mathbf{L}^{\mu\nu}=T^{\mu\nu}+A_{2}^{\mu\nu}$ | $\mathbf{R}_{\mu\nu}(t,-\vec{x})$ | $(\mathbf{R}^{\mu\nu})^{T}$ | $U_{L}\mathbf{L}^{\mu\nu}U^{\dagger}_{L}$
$\Phi^{\mu\nu}=T_{2F}^{\mu\nu}+iP_{2}^{\mu\nu}$ | $\Phi_{\mu\nu}^{\dagger}(t,-\vec{x})$ | $(\Phi^{\mu\nu})^{T}$ | $U_{L}\Phi^{\mu\nu}U^{\dagger}_{R}$ | Heterochiral
$R^{\text{hyb}\,,\mu}=P^{\text{hyb}\,,\mu}-B^{\text{hyb}\,,\mu}$ | $L_{\mu}^{\text{hyb}}(t,-\vec{x})$ | $-(L^{\text{hyb}\,,\mu})^{T}$ | $U_{R}R^{\text{hyb}\,,\mu}U_{R}^{\dagger}$ | Homochiral
$L^{\text{hyb}\,,\mu}=P^{\text{hyb}\,,\mu}+B^{\text{hyb}\,,\mu}$ | $R_{\mu}^{\text{hyb}}(t,-\vec{x})$ | $-(R^{\text{hyb}\,\mu})^{T}$ | $U_{L}L^{\text{hyb}\,,\mu}U_{L}^{\dagger}$
Table 2.6: Heterochiral and homochiral multiplets and their chiral + axial,
$\mathcal{C}$ and $\mathcal{P}$ transformation properties.

## 3 The Lagrangian of the eLSM and its consequences

### 3.1 General considerations

Effective approaches of QCD, such as the eLSM, are constructed to possess the
global symmetries of QCD, i.e. the $SU(3)_{L}\times SU(3)_{R}$ chiral
symmetry, and the $\mathcal{C}$ and $\mathcal{P}$ discrete symmetries. Since
the chiral symmetry is not exact symmetry, violating terms are also needed.

It is worth noting that in earlier versions of linear sigma models, when
vector and axial-vector fields were added to the Lagrangian, local chiral
invariance was usually considered. This local invariance was broken to a
global one with the introduction of (axial-)vector mass terms, see [118, 124]
and refs. therein. This approach was used to reflect the vector meson
dominance hypothesis [228] and to be consistent with the current-field
identity [229]. It also had the advantage of requiring fewer coupling
constants, since local symmetry forced certain coefficients to coincide.
However, it turned out that in order to correctly describe the meson decay
widths and the pion-pion scattering length(s), the introduction of six-
dimensional terms was also necessary [124]. Since this effective model can be
regarded as a low-energy theory, valid only up to a certain scale, this is not
a problem in itself, but in this way it is somehow hard to explain why one
does not include power eight, ten, etc. terms in the Lagrangian. On the other
hand, there is no deep reason to make the chiral symmetry local, since it is
global in QCD. It has been shown in [120] in a two-flavor LSM that
experimental data can be well reproduced by using global chiral symmetry.
Consequently, we follow this approach for the eLSM, i.e. we consider the
chiral symmetry as a global symmetry as in QCD. Moreover, we request the
validity of dilatation invariance, meaning that -in the chiral limit and in
absence of the chiral anomaly- the allowed terms have exactly order 4
(dimensionless coupling, besides one parameter in the dilaton sector). This is
the condition (and not renormalizability) that imposes that only a limited
number of terms is retained in the eLSM [230]. The eLSM constructed under
these requirements is able to reproduce experimental masses and decay widths
up to $\sim 2$ GeV with good precision [6].

In the next subsection, we shall present the three-flavor form of the eLSM
with (pseudo)scalar and (axial-)vector mesons. Here we merely present some
general arguments that are at the basis of its building blocks

  * 1.

An example of a chirally symmetric and dilatation (as well as $\mathcal{C}$
and $\mathcal{P}$ ) invariant term is given by the quartic term:

$-\lambda_{2}Tr\left[\Phi\Phi^{{\dagger}}\Phi\Phi^{{\dagger}}\right]$ (3.1)

Namely, $\lambda_{2}$ is dimensionless (dilatation invariant). Moreover, it
should be $\lambda_{2}>0$ so that the effective potential is bounded from
below. Chiral symmetry can be easily proved by applying $\Phi\rightarrow
U_{L}\Phi U_{R}^{\dagger},$ leading to

$-\lambda_{2}Tr\left[\Phi\Phi^{{\dagger}}\Phi\Phi^{{\dagger}}\right]\rightarrow-\lambda_{2}Tr\left[U_{L}\Phi
U_{R}^{\dagger}U_{R}\Phi^{{\dagger}}U_{L}^{\dagger}U_{L}\Phi
U_{R}^{\dagger}\Phi^{{\dagger}}U_{R}\Phi^{{\dagger}}U_{L}^{\dagger}\right]=-\lambda_{2}Tr\left[\Phi\Phi^{{\dagger}}\Phi\Phi^{{\dagger}}\right]\text{
.}$ (3.2)

Invariance under parity and charge conjugation can be shown in a similar way.
Moreover, it turns out that in the large-$N_{c}$ limit $\lambda_{2}\propto
N_{c}^{-1}$ is the dominant term of this type of quartic interaction [19].

  * 2.

Another dilatation, $\mathcal{P}$, $\mathcal{C}$, and chirally invariant term
is given by $-\lambda_{1}Tr\left[\Phi\Phi^{{\dagger}}\right]^{2},$ where
$\lambda_{1}\propto N_{c}^{-2}$ in the large-$N_{c}$ limit, thus $\lambda_{1}$
is suppressed w.r.t. the $\lambda_{2}$. Even if subleading, this type of terms
may be important for some specific processes, such as suppressed decays and
mixing patterns.

  * 3.

The “mass” term for (pseudo)scalar mesons is given by the dilatation and
chirally (as well as $\mathcal{C}$ and $\mathcal{P}$ ) invariant term

$-\lambda_{G\Phi}G^{2}Tr\left[\Phi\Phi^{{\dagger}}\right]$ (3.3)

where $\lambda_{G\Phi}\propto N_{c}^{-2}$ is the dominant term for this type
of glueball-glueball-meson-meson interaction. The important point is that
$\lambda_{G\Phi}<0,$ implying the ‘wrong’ mass sign for (pseudo)scalar fields
when the dilaton/glueball field $G$ condenses. This feature is at the basis of
the Mexican-hat form of the (pseudo)scalar potential. We shall discuss the
form of the potential in Sec. 3.3. Note, since $G_{0}\propto N_{c}$, this
contribution to the squared masses arising from this term go as $N_{c}^{0}$,
as expected (see B).

  * 4.

We stress that terms of the type
$G^{-2}Tr\left[\Phi\Phi^{{\dagger}}\Phi\Phi^{{\dagger}}\Phi\Phi^{{\dagger}}\right]$
or $G^{4}Tr\left[\Phi\Phi^{{\dagger}}\right]^{-2}$ fulfill all the symmetries
above, but are non-analytic in $G=0.$ These terms are not considered, since a
smoothness of the potential for any finite value of the fields is a natural
requirement for effective models. Moreover, a vanishing $G$ corresponds to
restoration of dilatation invariance, while a vanishing $\Phi$ a restoration
of chiral invariance, both of them expected to take place at high $T$ and
$\mu$.

  * 5.

When right-handed and left-handed homovector fields are introduced,
corresponding invariant terms can be introduced, e.g.:

$2h_{3}Tr\left[\Phi R_{\mu}\Phi^{{\dagger}}L^{\mu}\right]$ (3.4)

with $h_{3}$ being dimensionless and $\propto N_{c}^{-1}$ (dominant).

  * 6.

The mass term for (axial-)vector homochiral fields fulfilling the required
symmetries is given by

$\lambda_{GLR}G^{2}Tr\left[R_{\mu}R^{\mu}+L_{\mu}L^{\mu}\right]$ (3.5)

Here, $\lambda_{GLR}>0$ guarantees the standard (non tachyonic) sign for the
masses of (axial-)vector fields.

  * 7.

Besides traces, one may generate chirally, $\mathcal{P}$, and $\mathcal{C}$
invariant but $U(1)_{A}$ anomalous terms, such as

$c_{2}\left(\det\Phi-\det\Phi^{\dagger}\right)^{2}\text{ .}$ (3.6)

In fact, $\det\Phi$ transforms as

$\det\Phi\rightarrow\det U_{L}\Phi U_{R}^{\dagger}=\det U_{L}\det\Phi\det
U_{R}^{\dagger}\text{ ,}$ (3.7)

which is invariant under $SU(3)_{R}\times SU(3)_{L}$ transformations since
$\det U_{L}=\det U_{R}=1,$ but not under $U(1)_{A}$. Namely, the latter
amounts to $U_{L}=U_{R}^{\dagger}=e^{i\alpha}\mathbb{1}_{3\times 3},$ implying
$\det\Phi\rightarrow e^{6i\alpha}\det\Phi\neq\det\Phi.$ There are important
phenomenological consequence of such terms for the isoscalar-pseudoscalar
mesons $\eta$ and $\eta^{\prime}.$ New anomalous terms that break the axial
anomaly and involve also fields with $J>0$ were recently considered [48, 49].
Note that in general anomalous terms also break dilatation invariance and
involve dimensional coupling, which can be understood as an average over
instanton ensembles [49].

  * 8.

There are terms that break explicitly $SU_{L}(3)\times SU_{R}(3)$ due to
nonzero and unequal quark masses. An example is provided by

$Tr\left[H\left(\Phi+\Phi^{\dagger}\right)\right]\;$ (3.8)

with $H=\mathrm{diag}\\{h_{0N}/2,h_{0N}/2,h_{0S}/\sqrt{2}\\}$, $h_{0N}\propto
m_{n}$. and $h_{0S}\propto m_{s}$. In fact,

$Tr\left[H\left(\Phi+\Phi^{\dagger}\right)\right]\rightarrow
Tr\left[H\left(U_{L}\Phi
U_{R}^{\dagger}+U_{R}\Phi^{{\dagger}}U_{L}^{\dagger}\right)\right]$ (3.9)

which is in general not chirally invariant. In the limit in which $H$ is
proportional to the identity, $SU_{V}(3)$ flavor symmetry (realized for
$U_{R}=U_{L}$) is still fulfilled. Yet, the breaking due to $m_{s}>m_{n}$ is
non negligible. For the choice of $H$ above, isospin symmetry $SU_{V}(2)$ is
still retained (the isospin violating case is discussed in Section 7.1).

### 3.2 The explicit form of the eLSM Lagrangian

The eLSM Lagrangian with the dilaton/glueball field and with scalar,
pseudoscalar, vector, and axial-vector $\bar{q}q$ mesonic nonets (or, more
shortly, with heteroscalars and homovectors), is constructed under the
requirement of being –in the chiral limit– invariant under global chiral
transformations $SU(3)_{L}\times SU(3)_{R}$ and being -besides the dilatation
potential and axial anomalous terms- invariant under dilatation
transformations. Additional terms that break chiral symmetry and dilatation
invariance due to the nonzero quark masses, and the special case of the axial
anomaly $U(1)_{A}$ terms that fulfill $SU(3)_{L}\times SU(3)_{R}$ but break
$U(1)_{A}$, are also introduced. The explicit form of the eLSM Lagrangian
reads:

$\displaystyle\mathcal{L}$
$\displaystyle=\mathcal{L}_{\text{dil}}+\mathcal{L}_{\Phi}+\mathcal{L}_{U(1)_{A}}+\mathcal{L}_{LR}+\mathcal{L}_{\Phi
LR}\text{ ,}$ (3.10) with $\displaystyle\mathcal{L}_{\text{dil}}$
$\displaystyle=\frac{1}{2}(\partial_{\mu}G)^{2}-\frac{1}{4}\frac{m_{G}^{2}}{\Lambda_{G}^{2}}\left(G^{4}\ln\frac{G^{2}}{\Lambda_{G}^{2}}-\frac{G^{4}}{4}\right)\text{
,}$ (3.11) $\displaystyle\mathcal{L}_{\Phi}$
$\displaystyle=\mathop{\mathrm{Tr}}[(D_{\mu}\Phi)^{\dagger}(D_{\mu}\Phi)]-m_{0}^{2}\left(\frac{G}{G_{0}}\right)^{2}\mathop{\mathrm{Tr}}(\Phi^{\dagger}\Phi)-\lambda_{1}[\mathop{\mathrm{Tr}}(\Phi^{\dagger}\Phi)]^{2}-\lambda_{2}\mathop{\mathrm{Tr}}(\Phi^{\dagger}\Phi)^{2}+\mathop{\mathrm{Tr}}[H(\Phi+\Phi^{\dagger})]\text{
,}$ (3.12) $\displaystyle\mathcal{L}_{U(1)_{A}}$
$\displaystyle=c_{2}(\det\Phi-\det\Phi^{\dagger})^{2}\text{ ,}$ (3.13)
$\displaystyle\mathcal{L}_{LR}$
$\displaystyle=-\frac{1}{4}\mathop{\mathrm{Tr}}(L_{\mu\nu}^{2}+R_{\mu\nu}^{2})+\mathop{\mathrm{Tr}}\left[\left(\left(\frac{G}{G_{0}}\right)^{2}+\Delta\right)\frac{m_{1}^{2}}{2}(L_{\mu}^{2}+R_{\mu}^{2})\right]+i\frac{g_{2}}{2}(\mathop{\mathrm{Tr}}\\{L_{\mu\nu}[L^{\mu},L^{\nu}]\\}+\mathop{\mathrm{Tr}}\\{R_{\mu\nu}[R^{\mu},R^{\nu}]\\})$
$\displaystyle+g_{3}[\mathop{\mathrm{Tr}}(L_{\mu}L_{\nu}L^{\mu}L^{\nu})+\mathop{\mathrm{Tr}}(R_{\mu}R_{\nu}R^{\mu}R^{\nu})]+g_{4}[\mathop{\mathrm{Tr}}\left(L_{\mu}L^{\mu}L_{\nu}L^{\nu}\right)+\mathop{\mathrm{Tr}}\left(R_{\mu}R^{\mu}R_{\nu}R^{\nu}\right)]{}$
$\displaystyle+g_{5}\mathop{\mathrm{Tr}}\left(L_{\mu}L^{\mu}\right)\mathop{\mathrm{Tr}}\left(R_{\nu}R^{\nu}\right)+g_{6}[\mathop{\mathrm{Tr}}(L_{\mu}L^{\mu})\,\mathop{\mathrm{Tr}}(L_{\nu}L^{\nu})+\mathop{\mathrm{Tr}}(R_{\mu}R^{\mu})\,\mathop{\mathrm{Tr}}(R_{\nu}R^{\nu})]\text{
,}$ (3.14) $\displaystyle\mathcal{L}_{\Phi LR}$
$\displaystyle=\frac{h_{1}}{2}\mathop{\mathrm{Tr}}(\Phi^{\dagger}\Phi)\mathop{\mathrm{Tr}}(L_{\mu}^{2}+R_{\mu}^{2})+h_{2}\mathop{\mathrm{Tr}}[|L_{\mu}\Phi|^{2}+|\Phi
R_{\mu}|^{2}]+2h_{3}\mathop{\mathrm{Tr}}(L_{\mu}\Phi
R^{\mu}\Phi^{\dagger})\text{ ,}$ (3.15)

where

$\displaystyle D^{\mu}\Phi$ $\displaystyle\equiv\partial^{\mu}\Phi-
ig_{1}(L^{\mu}\Phi-\Phi R^{\mu})-ieA^{\mu}[t_{3},\Phi]\;,$ $\displaystyle
L^{\mu\nu}$
$\displaystyle\equiv\partial^{\mu}L^{\nu}-ieA^{\mu}[t_{3},L^{\nu}]-\left\\{\partial^{\nu}L^{\mu}-ieA^{\nu}[t_{3},L^{\mu}]\right\\}\;\text{,}$
$\displaystyle R^{\mu\nu}$
$\displaystyle\equiv\partial^{\mu}R^{\nu}-ieA^{\mu}[t_{3},R^{\nu}]-\left\\{\partial^{\nu}R^{\mu}-ieA^{\nu}[t_{3},R^{\mu}]\right\\}\;\text{,}$

and

$\displaystyle H$
$\displaystyle=H_{0}t_{0}+H_{8}t_{8}=\left(\begin{array}[c]{ccc}\frac{h_{0N}}{2}&0&0\\\
0&\frac{h_{0N}}{2}&0\\\ 0&0&\frac{h_{0S}}{\sqrt{2}}\end{array}\right)\;\text{
, }$ (3.19) $\displaystyle\Delta$
$\displaystyle=\Delta_{0}t_{0}+\Delta_{8}t_{8}=\left(\begin{array}[c]{ccc}\frac{\tilde{\delta}_{N}}{2}&0&0\\\
0&\frac{\tilde{\delta}_{N}}{2}&0\\\
0&0&\frac{\tilde{\delta}_{S}}{\sqrt{2}}\end{array}\right)\equiv\left(\begin{array}[c]{ccc}\delta_{N}&0&0\\\
0&\delta_{N}&0\\\ 0&0&\delta_{S}\end{array}\right)\text{ .}$ (3.26)

The latter are the constant external fields for the (pseudo)scalar and
(axial)vector fields that describe the effect of the nonzero quark masses and
thus break chiral symmetry and also flavour symmetry (yet they still satisfy
isospin symmetry). The $\Phi$, $L^{\mu}$ and $R^{\mu}$ nonets are defined in
Eqs. (2.181),(2.183),(2.184), respectively, $A^{\mu}$ is the electromagnetic
field and $t^{3}$ is the third generator of $U(3)$ (see A).

Let us go thorough the different parts of the Lagrangian and their most
important properties.

(i) Eq. (3.11) is the dilaton Lagrangian, which was already discussed in
Section 2.2 and can be related to the scalar glueball.

(ii) Eq. (3.12) is the chirally symmetric (pseudo)scalar part, that is all
terms, except the last one, are invariant under the heterochiral
transformation given in Table 2.6. The first term is the kinetic term, with a
derivative that also includes the $L^{\mu}/R^{\mu}$ vector and the $A^{\mu}$
electromagnetic fields, and is similar to the covariant derivatives in local
gauge theories, except that here there are only global transformations. The
second term is a mass term via the condensation of the dilaton field
$G=G_{0}$, the third and fourth terms are (pseudo)scalar self-interactions,
while the last term is the explicit symmetry breaking term with
$H\propto\text{diag}(m_{n},m_{n},m_{s})$.

(iii) Eq. (3.13) is an $U(1)_{A}$ anomalous term, which represents one
possible implementation, e.g. Ref. [6]. However, other anomalous terms are
also possible, see Section 3.5.

(iv) Eq. (3.14) includes the kinetic, the mass, the self-interaction and
explicit breaking terms for the left- and right-handed vector fields $L^{\mu}$
and $R^{\mu}$.

(v) Finally, the term of Eq. (3.14) consists of the interaction terms between
the (pseudo)scalars and (axial)vectors. As discussed previously, the minimum
of the dilaton potential occurs at $G=G_{0}$. Therefore, if we set $G=G_{0}$,
the dilaton field decouples from the rest of the Lagrangian.

After fixing the Lagrangian, the standard procedure is to assume non-zero
condensates – that is spontaneous symmetry breaking (SSB) – for certain fields
(besides, of course, the field $G$, that condenses due to the dilaton
potential). In the next Section we discuss which conditions the parameters
must fulfill to have SSB. Once it takes place, the fields that condense are
those with the quantum numbers of the vacuum, i.e. the scalar-isoscalar states
$\sigma_{N}$, $\sigma_{S}$ (or equivalently $\sigma_{0}$, $\sigma_{8}$) and
$\sigma_{3}$. The last one, $\sigma_{3}$, is responsible for isospin breaking
and is usually neglected since its effect is much smaller compared to the
others (for more details on isospin breaking444It worth noting that other
condensates are also considered in the literature, like the pion [231] and
kaon condensates [232], or vector condensates [233]. see Sec. 7.1). Next, the
$\sigma_{N/S}$ fields are shifted by their vacuum expectation values:

$\sigma_{N/S}\to\sigma_{N/S}+\phi_{N/S}\text{ ,
}\quad\text{with}\quad\phi_{N/S}\equiv\langle\sigma_{N/S}\rangle\text{ .}$
(3.27)

After the shifts, the quadratic terms generally give the tree-level masses of
all particles, i.e. the spectrum, and the decay widths can be calculated.
However, due to the inclusion of the (axial-)vector meson fields in the
covariant derivative of the (pseudo)scalar fields, a complication arises,
namely there will be quadratic mixing terms between the members of the
(pseudo)scalar and (axial)vector nonets. For example, one such possible term
is of the form

$\sim g_{1}a_{1}^{\mu\pm,0}\partial_{\mu}\pi^{\mp,0}$ (3.28)

This mixing can be eliminated by redefining the axial vector field,

$a_{1}^{\mu\pm,0}\to
a_{1}^{\mu\pm,0}+Z_{\pi}w_{a_{1}}\partial^{\mu}\pi^{\pm,0}.$ (3.29)

Requiring the disappearance of the mixing term and also the canonical form of
the kinetic term for the pion field – i.e.
$1/2\partial_{\mu}\pi^{\pm,0}\partial^{\mu}\pi^{\mp,0}$ – determines the
coefficients $w_{a_{1}}$ and $Z_{\pi}$. This procedure with explicit formulas
for the masses and decay widths can be found in Ref. [121] for the two flavor
case, and in Ref. [6] for the three flavor case.

### 3.3 The Mexican hat potential

In order to discuss the shape of the potential for the (pseudo)scalar sector
and the emergence of chiral condensates, it is convenient to rewrite one part
of the eLSM Lagrangian as it follows

$\mathcal{L}_{dil}+\mathcal{L}_{\Phi}=\mathcal{L}_{G\Phi}=\mathcal{L}_{G\Phi}^{\text{(kin)}}-V_{G\Phi}\text{
,}$ (3.30)

where $V_{G\Phi}\equiv
V_{G\Phi}(G,\sigma_{N},\sigma_{S},...,\mathbf{\pi},...)$ is the potential that
depends on all the 19 (pseudo)scalar fields. Among all of them, only those
having the same quantum numbers of the vacuum
$J^{\mathcal{P}\mathcal{C}}=0^{++}$ can condensate: $G,\sigma_{N},\sigma_{S}.$
The explicit form of the potential by retaining only these fields is given by:

$V_{G\Phi}(G,\sigma_{N},\sigma_{S},0,...,0)=\frac{\lambda_{G}}{4}G^{4}\left(\ln\frac{G}{\Lambda_{G}}-\frac{1}{4}\right)+\frac{\lambda_{G\Phi}G^{2}}{2}\left(\sigma_{N}^{2}+\sigma_{S}^{2}\right)+\frac{\lambda_{2}}{8}\left(\sigma_{N}^{4}+2\sigma_{S}^{4}\right)+\frac{\lambda_{1}}{4}\left(\sigma_{N}^{2}+\sigma_{S}^{2}\right)^{2}-h_{N}\sigma_{N}-h_{S}\sigma_{S}$
(3.31)

The minimum is found by setting the derivatives to zero:

$\displaystyle\partial_{G}V_{G\Phi}$
$\displaystyle=\lambda_{G}G^{3}\ln\frac{G}{\Lambda_{G}}+\lambda_{G\Phi}G\left(\sigma_{N}^{2}+\sigma_{S}^{2}\right)=0\text{
, }$ (3.32) $\displaystyle\partial_{\sigma_{N}}V_{G\Phi}$
$\displaystyle=\lambda_{1}\sigma_{N}(\sigma_{N}^{2}+\sigma_{S}^{2})+\frac{\lambda_{2}\sigma_{N}^{3}}{2}+\lambda_{G\phi}G^{2}\sigma_{N}-h_{N}=0\text{
, }$ (3.33) $\displaystyle\partial_{\sigma_{S}}V_{G\Phi}$
$\displaystyle=\lambda_{1}\sigma_{S}(\sigma_{N}^{2}+\sigma_{S}^{2})+\lambda_{2}\sigma_{S}^{3}+\lambda_{G\phi}G^{2}\sigma_{S}-h_{S}=0\text{
.}$ (3.34)

We also refer to Refs. [30, 19] for a pedagogical introduction on the subject,
including the search for minima (with and without the dilaton) and the
simplified one-flavor case.

If $\lambda_{G\Phi}>0$ and in the limit $h_{N,S}\rightarrow 0$, only the
dilaton field $G$ condenses, while the v.e.v. of $\sigma_{N}$ and $\sigma_{S}$
vanish:

$G_{0}=\Lambda_{G}\text{ , }\sigma_{N}=\text{ }\phi_{N}=0\text{ ,
}\sigma_{S}=\text{ }\phi_{S}=0\text{ .}$ (3.35)

In this case, the potential along the $\sigma_{N}$ and $\pi^{0}$ directions
looks like

$V_{G\Phi}(G=G_{0}=\Lambda_{G},\sigma_{N},0,0,.,\pi^{0},0,...)=\frac{m_{0}^{2}}{2}\left(\sigma_{N}^{2}+(\pi^{0})^{2}\right)+\frac{\lambda_{2}}{2}\left(\sigma_{N}^{2}+(\pi^{0})^{2}\right)-h_{N}\sigma_{N}+const\text{
.}$ (3.36)

The masses read:

$m_{\sigma_{N}}^{2}=m_{\pi}^{2}=m_{0}^{2}=\lambda_{G\Phi}G_{0}^{2}=\lambda_{G\Phi}\Lambda_{G}^{2}>0\text{
,}$ (3.37)

thus the scalar-isoscalar field $\sigma_{N}$ and its chiral partner, the
pseudoscalar $\pi^{0},$ carry the same mass. The schematic form of the
potential is depicted in Fig. 3.1. As well known, this case is not realized in
Nature, since chiral partners are not degenerate in mass. Note that allowing
for small but nonzero $h_{N,S}$ modifies slightly the potential, and
$\phi_{N},\phi_{S}$ are small but nonzero with $\phi_{N,S}\propto m_{n,s}$,
yet the investigation of this case is not relevant for our purposes.

$V(G_{0},\sigma_{N},\pi^{0})$$\sigma_{N}$$\pi^{0}$ Figure 3.1: The Potential
term in Eq. (3.36) for $\lambda_{G\Phi}>0$ in the chiral limit. A unique
minimum in the origin is present.

Next, we turn to the physically relevant case $\lambda_{G\Phi}<0,$ for which
all condensates $G_{0},\phi_{N},\phi_{S}$ do not vanish, even not in in the
limit $h_{N,S}\rightarrow 0^{+}$. The potential along $\sigma_{N}$ and
$\pi^{0}$ has the same form as in Eq. (3.36), but now with

$m_{0}^{2}=\lambda_{G\Phi}G_{0}^{2}<0\text{ .}$ (3.38)

Upon expanding around the minimum, the masses of the chiral partner of the
pion reads (neglecting the large-$N_{c}$ suppressed $\lambda_{1}$-term):
$m_{\sigma_{N}}^{2}=m_{0}^{2}+3\lambda_{2}\sigma_{N}^{2}/3=-2m_{0}^{2}+3m_{\pi}^{2}$
with $m_{\pi}^{2}=h_{N}/\phi_{N}\rightarrow 0^{+}$ for $h_{N}\rightarrow
0^{+}$ (Goldstone boson). Note,
$m_{\sigma_{N}}^{2}-m_{\pi}^{2}=-2m_{0}^{2}+2m_{\pi}^{2}=\lambda_{2}\phi_{N}^{2}>0$
show that the mass difference between chiral partners is proportional to the
chiral condensate $\phi_{N}$, which in turn is proportional to the gluon
condensate $G_{0}\sim\Lambda_{G}$. This is a recurring pattern in the eLSM
that appears also in much more lengthy mass expressions.

The form of the potential in the $\\{\sigma_{N},\pi^{0}\\}$-subspace is
depicted in Fig 3.2 for the case $h_{N}\rightarrow 0^{+}$ (left panel), in
which the form of the potential is the well-known Mexican-hat shape with a
circle of minima, and for $h_{N}>0$ (right panel), where an additional titling
due to nonzero quark light masses takes place: a unique minimum for
$\sigma_{N}=\phi_{N}>0$ and $\pi^{0}=0$ is present. Note that as long as
$h_{N}>0$, no matter how small, the potential has a unique minimum, since it
is always tilted, even if very little. In this case the pion field can never
condense. Only for $h_{N}=0$ (strictly zero) multiple minima are actually
present.

$V(G_{0},\sigma_{N},\pi^{0})$$\pi^{0}$$\sigma_{N}$

$\sigma_{N}$$V(G_{0},\sigma_{N},\pi^{0})$$\pi^{0}$

Figure 3.2: Left: Mexican-hat potential with minima along the chiral circle in
the chiral limit. Right: the potential in presence of nonzero light quark mass
with a unique minimum.

### 3.4 Parameterizations and tree-level masses

To compute predictions from the model, the parameters of the eLSM Lagrangian
of Sec. 3.2 must be determined. This can be done with the following procedure,
called parameterization, which consists of calculating physical quantities,
such as tree-level masses and decay widths, from the model and comparing them
with the corresponding experimental values taken from the PDG [13] using a
$\chi^{2}$ minimization. The $\chi^{2}$ is defined as

$\chi^{2}(x_{1},\dots,x_{N})=\sum_{i=1}^{M}\left[\frac{Q_{i}(x_{1},\dots,x_{N})-Q_{i}^{\text{exp}}}{\delta
Q_{i}}\right]^{2},$ (3.39)

where $x_{1},\dots,x_{N}$ are the parameters of the Lagrangian,
$Q_{1}(x_{1},\dots,x_{N}),\ldots,Q_{M}(x_{1},\dots,x_{N})$ are the physical
quantities, such as masses, decay widths, etc., calculated from the model,
$Q_{1}^{\text{exp}},\ldots,Q_{M}^{\text{exp}}$ are the corresponding
experimental values, while $\delta Q_{1},\ldots,\delta Q_{M}$ are the errors.
In principle, the errors should also be chosen simply from the PDG, but this
would usually lead to fits that are too biased towards quantities that are
known very precisely (e.g. the $\eta(547)$ mass). Thus, one usually uses
instead prescriptions like $\delta Q_{i}=\max\\{5\%,Q_{i}^{\text{exp}}\\}$,
and in the case of very uncertain scalars sometimes $10\%$ or even $20\%$ is
used instead of $5\%$. This procedure is also motivated by the fact that
isospin breaking processes (typically few percent effect) are here neglected.

Next, one searches for the absolute minimum of the $\chi^{2}$ in this
multidimensional parameter space, which is in general not an easy task.
Usually, due to the highly nonlinear nature of the equations, there are many
local minima (hundreds). For this task a MINUIT code can be used [234]. One of
the advantages of this parameterization is that it works for both $M<N$ and
$M\geq N$, but if the number of conditions $M$ is smaller than the number of
unknown parameters $N$, then the resulting fit will not be restrictive. In
practice, this would mean finding many very different parameter sets with very
similar $\chi^{2}$ values and with large variances of the fitted values. Note
that this procedure is not necessary in simpler versions of the model (e.g.
without the vector mesons), or in cases where vacuum fluctuations are
neglected. In these cases it may be even possible to solve the
parameterization equations analytically.

In the following we make an important simplification, namely we decouple the
dilaton field by setting $G=G_{0}$ in Eq. (3.10). Accordingly, we show the
tree-level masses and parameterization results in the eLSM presented in Refs.
[6, 235]. The Table 3.1 shows the assignments of the fields entering the model
(Eq. (2.181)) and their PDG counterparts, along with their quark contents and
experimental values for their masses.

Field | PDG | Quark content | $I^{\mathcal{G}}$ | $J^{\mathcal{P}\mathcal{C}}$ | Mass (MeV)
---|---|---|---|---|---
$\pi^{+},\pi^{-},\pi^{0}$ | $\pi$ | $u\bar{d},d\bar{u},\frac{u\bar{u}-d\bar{d}}{\sqrt{2}}$ | $1^{-}$ | $0^{-+}$ | $138.04$
$K^{+},K^{-},K^{0},\bar{K}^{0}$ | $K$ | $u\bar{s},s\bar{u},d\bar{s},s\bar{d}$ | $1/2$ | $0^{-}$ | $495.64$
$\text{Eigenv}^{-}(\eta_{N},\eta_{S})$ | $\eta$ |  $\frac{u\bar{u}+d\bar{d}}{\sqrt{2}}$, $s\bar{s}$ | $0^{+}$ | $0^{-+}$ | $547.86$
$\text{Eigenv}^{+}(\eta_{N},\eta_{S})$ | $\eta^{\prime}(958)$ |  $\frac{u\bar{u}+d\bar{d}}{\sqrt{2}}$, $s\bar{s}$ | $0^{+}$ | $0^{-+}$ | $957.78$
$a_{0}^{+},a_{0}^{-},a_{0}^{0}$ | $a_{0}(1450)$ | $u\bar{d},d\bar{u},\frac{u\bar{u}-d\bar{d}}{\sqrt{2}}$ | $1^{-}$ | $0^{++}$ | $1474$
$K_{S}^{+},K_{S}^{-},K_{S}^{0},\bar{K}_{S}^{0}$ | $K_{0}^{\ast}(1430)$ | $u\bar{s},s\bar{u},d\bar{s},s\bar{d}$ | $1/2$ | $0^{+}$ | $1425$
$\text{Eigenv}^{-}(\sigma_{N},\sigma_{S})$ | $f_{0}(1370)$ |  $\frac{u\bar{u}+d\bar{d}}{\sqrt{2}}$, $s\bar{s}$ | $0^{+}$ | $0^{++}$ | $1350$
$\text{Eigenv}^{+}(\sigma_{N},\sigma_{S})$ | $f_{0}(1500)$ |  $\frac{u\bar{u}+d\bar{d}}{\sqrt{2}}$, $s\bar{s}$ | $0^{+}$ | $0^{++}$ | $1505$
$\rho^{+},\rho^{-},\rho^{0}$ | $\rho(770)$ | $u\bar{d},d\bar{u},\frac{u\bar{u}-d\bar{d}}{\sqrt{2}}$ | $1^{+}$ | $1^{--}$ | $775.16$
$K^{\ast+},K^{\ast-},K^{\ast 0},\bar{K}^{\ast 0}$ | $K^{\ast}(892)$ | $u\bar{s},s\bar{u},d\bar{s},s\bar{d}$ | $1/2$ | $1^{-}$ | $895.51$
$\omega_{N}$ | $\omega(782)$ | $\frac{u\bar{u}+d\bar{d}}{\sqrt{2}}$ | $0^{-}$ | $1^{--}$ | $782.66$
$\omega_{S}$ | $\phi(1020)$ | $s\bar{s}$ | $0^{-}$ | $1^{--}$ | $1019.46$
$a_{1}^{+},a_{1}^{-},a_{1}^{0}$ | $a_{1}(1230)$ | $u\bar{d},d\bar{u},\frac{u\bar{u}-d\bar{d}}{\sqrt{2}}$ | $1^{-}$ | $1^{++}$ | $1230$
$K_{1}^{+},K_{1}^{-},K_{1}^{0},\bar{K}_{1}^{0}$ | $K_{1}(1270)$ | $u\bar{s},s\bar{u},d\bar{s},s\bar{d}$ | $1/2$ | $1^{+}$ | $1253$
$f_{1,N}$ | $f_{1}(1285)$ | $\frac{u\bar{u}+d\bar{d}}{\sqrt{2}}$ | $0^{+}$ | $1^{++}$ | $1281.9$
$f_{1,S}$ | $f_{1}(1420)$ | $s\bar{s}$ | $0^{+}$ | $1^{++}$ | $1426.3$
Table 3.1: eLSM fields, PDG correspondence, quark content, quantum numbers and
PDG mass values from [13]. In the case of charged particles isospin-averaged
values are used. Here $\text{Eigenv}^{\pm}$ stands for the larger/smaller
eigenvalue of the non-strange and strange states.

In both the scalar and pseudoscalar sectors there is a mixing between the non-
strange and strange (or similarly the $0$ and $8$) states, which can be
resolved by $2\times 2$ orthogonal transformations. The tree-level mass
expressions calculated in the model are listed in Table 3.2 for the scalar and
pseudoscalar (spin-0) mesons and in Table 3.3 for the (axial-)vector mesons.

Mass squares | Analytical expressions
---|---
$m_{\pi}^{2}$ | $Z_{\pi}^{2}\left[m_{0}^{2}+\left(\lambda_{1}+\frac{\lambda_{2}}{2}\right)\phi_{N}^{2}+\lambda_{1}\phi_{S}^{2}\right]\equiv\frac{Z_{\pi}^{2}h_{0N}}{\phi_{N}}$
$m_{K}^{2}$ | $Z_{K}^{2}\left[m_{0}^{2}+\left(\lambda_{1}+\frac{\lambda_{2}}{2}\right)\phi_{N}^{2}-\frac{\lambda_{2}}{\sqrt{2}}\phi_{N}\phi_{S}+\left(\lambda_{1}+\lambda_{2}\right)\phi_{S}^{2}\right]$
$m_{\eta_{N}}^{2}$ | $Z_{\pi}^{2}\left[m_{0}^{2}+\left(\lambda_{1}+\frac{\lambda_{2}}{2}\right)\phi_{N}^{2}+\lambda_{1}\phi_{S}^{2}+c_{2}\,\phi_{N}^{2}\phi_{S}^{2}\right]\equiv Z_{\pi}^{2}\left(\frac{h_{0N}}{\phi_{N}}+c_{2}\,\phi_{N}^{2}\phi_{S}^{2}\right)$
$m_{\eta_{S}}^{2}$ | $Z_{\eta_{S}}^{2}\left[m_{0}^{2}+\lambda_{1}\phi_{N}^{2}+\left(\lambda_{1}+\lambda_{2}\right)\phi_{S}^{2}+\frac{c_{2}}{4}\phi_{N}^{4}\right]\equiv Z_{\eta_{S}}^{2}\left(\frac{h_{0S}}{\phi_{S}}+\frac{c_{2}}{4}\phi_{N}^{4}\right)$
$m_{\eta_{NS}}^{2}$ | $Z_{\eta_{N}}Z_{\eta_{S}}\frac{c_{2}}{2}\phi_{N}^{3}\phi_{S}$
$m_{a_{0}}^{2}$ | $m_{0}^{2}+\left(\lambda_{1}+\frac{3}{2}\lambda_{2}\right)\phi_{N}^{2}+\lambda_{1}\phi_{S}^{2}$
$m_{K_{0}^{\star}}^{2}$ | $Z_{K_{0}^{\star}}^{2}\left[m_{0}^{2}+\left(\lambda_{1}+\frac{\lambda_{2}}{2}\right)\phi_{N}^{2}+\frac{\lambda_{2}}{\sqrt{2}}\phi_{N}\phi_{S}+\left(\lambda_{1}+\lambda_{2}\right)\phi_{S}^{2}\right]$
$m_{\sigma_{N}}^{2}$ | $m_{0}^{2}+3\left(\lambda_{1}+\frac{\lambda_{2}}{2}\right)\phi_{N}^{2}+\lambda_{1}\phi_{S}^{2}$
$m_{\sigma_{S}}^{2}$ | $m_{0}^{2}+\lambda_{1}\phi_{N}^{2}+3\left(\lambda_{1}+\lambda_{2}\right)\phi_{S}^{2}$
$m^{2}_{\sigma_{NS}}$ | $2\lambda_{1}\phi_{N}\phi_{S}$
Table 3.2: Mass expressions of spin-0 mesons (scalars and pseudoscalars) within the eLSM. Mass squares | Analytical expressions
---|---
$m_{\rho}^{2}$ | $m_{1}^{2}+\frac{1}{2}(h_{1}+h_{2}+h_{3})\phi_{N}^{2}+\frac{h_{1}}{2}\phi_{S}^{2}+2\delta_{N}$
$m_{K^{\star}}^{2}$ | $m_{1}^{2}+\frac{1}{4}\left(g_{1}^{2}+2h_{1}+h_{2}\right)\phi_{N}^{2}+\frac{1}{\sqrt{2}}\phi_{N}\phi_{S}(h_{3}-g_{1}^{2})+\frac{1}{2}(g_{1}^{2}+h_{1}+h_{2})\phi_{S}^{2}+\delta_{N}+\delta_{S}$
$m_{\omega_{N}}^{2}$ | $m_{1}^{2}+\frac{1}{2}(h_{1}+h_{2}+h_{3})\phi_{N}^{2}+\frac{h_{1}}{2}\phi_{S}^{2}+2\delta_{N}=m_{\rho}^{2}$
$m_{\omega_{S}}^{2}$ | $m_{1}^{2}+\frac{h_{1}}{2}\phi_{N}^{2}+\left(\frac{h_{1}}{2}+h_{2}+h_{3}\right)\phi_{S}^{2}+2\delta_{S}$
$m_{a_{1}}^{2}$ | $m_{1}^{2}+\frac{1}{2}(2g_{1}^{2}+h_{1}+h_{2}-h_{3})\phi_{N}^{2}+\frac{h_{1}}{2}\phi_{S}^{2}+2\delta_{N}$
$m_{K_{1}}^{2}$ | $m_{1}^{2}+\frac{1}{4}\left(g_{1}^{2}+2h_{1}+h_{2}\right)\phi_{N}^{2}-\frac{1}{\sqrt{2}}\phi_{N}\phi_{S}(h_{3}-g_{1}^{2})+\frac{1}{2}\left(g_{1}^{2}+h_{1}+h_{2}\right)\phi_{S}^{2}+\delta_{N}+\delta_{S}$
$m_{f_{1N}}^{2}$ | $m_{1}^{2}+\frac{1}{2}(2g_{1}^{2}+h_{1}+h_{2}-h_{3})\phi_{N}^{2}+\frac{h_{1}}{2}\phi_{S}^{2}+2\delta_{N}=m_{a_{1}}^{2}$
$m_{f_{1S}}^{2}$ | $m_{1}^{2}+\frac{h_{1}}{2}\phi_{N}^{2}+\left(2g_{1}^{2}+\frac{h_{1}}{2}+h_{2}-h_{3}\right)\phi_{S}^{2}+2\delta_{S}$
Table 3.3: Mass expressions of spin-1 mesons (vectors and axial vectors)
within the eLSM.

In the case of (pseudo)scalars, as mentioned before, due to the mixing with
(axial-)vectors, so-called $Z$-factors appear in the mass expressions (for
more details see Ref. [6]), which are given by

$\displaystyle Z_{\pi}$
$\displaystyle=Z_{\eta_{N}}=\frac{m_{a_{1}}}{\sqrt{m_{a_{1}}^{2}-g_{1}^{2}\phi_{N}^{2}}}\text{
,}$ $\displaystyle Z_{K}$
$\displaystyle=\frac{2m_{K_{1}}}{\sqrt{4m_{K_{1}}^{2}-g_{1}^{2}(\phi_{N}+\sqrt{2}\phi_{S})^{2}}}\text{
,}$ (3.40) $\displaystyle Z_{\eta_{S}}$
$\displaystyle=\frac{m_{f_{1S}}}{\sqrt{m_{f_{1S}}^{2}-2g_{1}^{2}\phi_{S}^{2}}}\text{
,}$ $\displaystyle Z_{K_{0}^{\star}}$
$\displaystyle=\frac{2m_{K^{\star}}}{\sqrt{4m_{K^{\star}}^{2}-g_{1}^{2}(\phi_{N}-\sqrt{2}\phi_{S})^{2}}}\;.$
(3.41)

Moreover, in the N-S sectors of the scalars and pseudoscalars there are
$2\times 2$ mixings with off-diagonal mass squares $m^{2}_{\sigma_{NS}}$ and
$m^{2}_{\eta_{NS}}$ respectively. In these sectors the mass eigenvalues are
given by

$\displaystyle m_{f_{0}^{H}/f_{0}^{L}}^{2}$
$\displaystyle=\frac{1}{2}\left[m_{\sigma_{N}}^{2}+m_{\sigma_{S}}^{2}\pm\sqrt{(m_{\sigma_{N}}^{2}-m_{\sigma_{S}}^{2})^{2}+4m_{\sigma_{NS}}^{4}}\right]\;.$
(3.42) $\displaystyle m_{\eta^{\prime}/\eta}^{2}$
$\displaystyle=\frac{1}{2}\left[m_{\eta_{N}}^{2}+m_{\eta_{S}}^{2}\pm\sqrt{(m_{\eta_{N}}^{2}-m_{\eta_{S}}^{2})^{2}+4m_{\eta_{NS}}^{4}}\right]\;.$
(3.43)

It is worth noting that, in contrast to the (pseudo)scalar sector, the eLSM
Lagrangian of Sec. 3.2 does not contain a mixing of vectors and axial-vectors
isoscalar states at tree-level. As discussed in Sec. 2, this mixing is rather
small, so it can be in first approximation safely neglected (or included as a
phenomenological corrections to some specific decay channels). Yet, one may
add large-$N_{c}$ suppressed terms of the type $Tr[L_{\mu}]Tr[L^{\mu}]+...$,
that would generate an isoscalar mixing in the (axial-)vector sectors as well.

As an example, a fit result is shown in Table 3.4, while the corresponding fit
parameters are given in Table 3.5 (both taken from [6]). The comparison of
model results for certain observables with experimental data is presented in
Fig. 3.3.

Observable | Fit $\left[\text{MeV}\right]$ | Experiment $\left[\text{MeV}\right]$ | Observable | Fit $\left[\text{MeV}\right]$ | Experiment $\left[\text{MeV}\right]$
---|---|---|---|---|---
$f_{\pi}$ | $96.3\pm 0.7$ | $92.2\pm 4.6$ | $f_{K}$ | $106.9\pm 0.6$ | $110.4\pm 5.5$
$m_{\pi}$ | $141.0\pm 5.8$ | $138\pm 6.9$ | $m_{K}$ | $485.6\pm 3.0$ | $495.6\pm 24.8$
$m_{\eta}$ | $509.4\pm 3.0$ | $547.9\pm 27.4$ | $m_{\eta^{\prime}}$ | $962.5\pm 5.6$ | $957.8\pm 47.9$
$m_{\rho}$ | $783.1\pm 7.0$ | $775.5\pm 38.8$ | $m_{K^{\star}}$ | $885.1\pm 6.3$ | $893.8\pm 44.7$
$m_{\phi}$ | $975.1\pm 6.4$ | $1019.5\pm 51.0$ | $m_{a_{1}}$ | $1186\pm 6.0$ | $1230\pm 62$
$m_{f_{1}\left(1420\right)}$ | $1372.4\pm 5.3$ | $1426\pm 71$ | $m_{a_{0}}$ | $1363\pm 1$ | $1474\pm 74$
$m_{K_{0}^{\star}}$ | $1450\pm 1$ | $1425\pm 71$ | $\Gamma_{\rho\rightarrow\pi\pi}$ | $160.9\pm 4.4$ | $149.1\pm 7.4$
$\Gamma_{K^{\star}\rightarrow K\pi}$ | $44.6\pm 1.9$ | $46.2\pm 2.3$ | $\Gamma_{\phi\rightarrow\bar{K}K}$ | $3.34\pm 0.14$ | $3.54\pm 0.18$
$\Gamma_{a_{1}\rightarrow\rho\pi}$ | $549\pm 43$ | $425\pm 175$ | $\Gamma_{a_{1}\rightarrow\pi\gamma}$ | $0.66\pm 0.01$ | $0.64\pm 0.25$
$\Gamma_{f_{1}\left(1420\right)\rightarrow K^{\star}K}$ | $44.6\pm 39.9$ | $43.9\pm 2.2$ | $\Gamma_{a_{0}}$ | $266\pm 12$ | $265\pm 13$
$\Gamma_{K_{0}^{\star}\rightarrow K\pi}$ | $285\pm 12$ | $270\pm 80$ |  |  |
Table 3.4: An example of fit results from [6], together with the experimental
values taken from [13].

Figure 3.3: eLSM results compared to the experiment from Ref. [6]. The figure on the left represents the masses, while the one on the right represents the decay rates. The $x$-axis lists the masses ( decay width) of the resonance, while the $y$-axis represents the deviations from the experimental value as compared to the experimental error. Vertical lines on the data points represent the fit errors. Those decay channels located between the green dashed lines are described well within the model. Parameter | Value |  $N_{c}$-scaling
---|---|---
$C_{1}$ $\left[\text{GeV}^{2}\right]$ | $-0.9183\pm 0.0006$ | $N_{c}^{0}$
$C_{2}$ $\left[\text{GeV}^{2}\right]$ | $0.4135\pm 0.0147$ | $N_{c}^{0}$
$c_{2}$ $\left[\text{GeV}^{-2}\right]$ | $450.5420\pm 7.033$ | $N_{c}^{-3}$
$\delta_{S}$ $\left[\text{GeV}^{2}\right]$ | $0.1511\pm 0.0038$ | $N_{c}^{0}$
$g_{1}$ | $5.843\pm 0.018$ | $N_{c}^{-1/2}$
$g_{2}$ | $3.0250\pm 0.2329$ | $N_{c}^{-1/2}$
$\phi_{N}$ $\left[\text{GeV}\right]$ | $0.1646\pm 0.0001$ | $N_{c}^{\frac{1}{2}}$
$\phi_{S}$ $\left[\text{GeV}\right]$ | $0.1262\pm 0.0001$ | $N_{c}^{\frac{1}{2}}$
$h_{2}$ | $9.8796\pm 0.6627$ | $N_{c}^{-1}$
$h_{3}$ | $4.8667\pm 0.0864$ | $N_{c}^{-1}$
$\lambda_{2}$ | $68.2972\pm 0.0435$ | $N_{c}^{-1}$
Table 3.5: Parameter values for the fit in Table 3.4 and the $N_{c}$-scaling
of the parameters. Note that in the case of $C_{1}$ and $C_{2}$ the first
terms, i.e. $m_{0}^{2}$ and $m_{1}^{2}$, scale with $N_{c}^{0}$, while the
second terms are suppressed by a factor of $N_{c}$.

In this particular fit, the masses and decay widths of the two $f_{0}$ scalar-
isoscalar states are not used. Consequently, not all of the parameters can be
determined separately. Actually there are two combinations of parameters that
appear in every expression, namely
$C_{1}=m_{0}^{2}+\lambda_{1}(\phi_{N}^{2}+\phi_{S}^{2})$ and
$C_{2}=m_{1}^{2}+h_{1}/2(\phi_{N}^{2}+\phi_{S}^{2})$. The number of $x_{i}$
parameters is $N=11$, namely $C_{1}$, $C_{2}$, $c_{2}$, $\delta_{S}$, $g_{1}$,
$g_{2}$, $\phi_{N}$, $\phi_{S}$, $h_{2}$, $h_{3}$ and $\lambda_{2}$ and the
number of physical $Q_{i}$ quantities is $M=21$. Note that the determination
of the two external fields $h_{0N}$ and $h_{0S}$ is exchanged for the
determination of the condensates $\phi_{N}$ and $\phi_{S}$ by the field
equations (3.33) and (3.34) in vacuum. Moreover, $\delta_{N}$ can be merged
into $m_{1}^{2}$, and thus can be set to zero without loss of generality. The
fitted physical quantities include masses, decay widths and the pion and kaon
decay constants from the Partially Conserved Axial Current (PCAC) relations.
The two-particle decay widths ($A\to BC$) can be calculated as follows

$\Gamma_{A\rightarrow BC}=\mathcal{I}\frac{|\mathbf{k}|}{8\pi
m_{A}^{2}}\left|\mathcal{M}_{A\rightarrow BC}\right|^{2}\;,$ (3.44)

where $\mathcal{M}_{A\rightarrow BC}$ is the transition matrix element for the
decay, while $\mathbf{k}$ is the three-momentum of one of the product
particles in the rest frame of particle $A$. All expressions for the two-
particle decay widths can be found in [6]. For illustrative purposes, we
provide here only the decay width for the process $\rho\to\pi\pi$, which is

$\Gamma_{\rho\rightarrow\pi\pi}=\frac{m_{\rho}^{5}}{48\pi
m_{a_{1}}^{4}}\left[1-\left(\frac{2m_{\pi}}{m_{\rho}}\right)^{2}\right]^{3/2}\left[g_{1}Z_{\pi}^{2}-\frac{g_{2}}{2}\left(Z_{\pi}^{2}-1\right)\right]^{2}.$
(3.45)

The value of the reduced chi-square
$\chi^{2}_{\text{red}}\equiv\chi^{2}/N_{\text{dof}}$, where
$N_{\text{dof}}=M-N=10$ is the number of degrees of freedom of the fit, at the
minimum is $1.2$. There is an important consequence of this fit, namely that
the scalar quark-antiquark mesons ($a_{0}$ and $K_{0}^{\star}$) are above $1$
GeV and therefore the light scalar mesons below $1$ GeV ($a_{0}(980)$ and
$K_{0}^{\star}(800)$) are something else (four-quark states). In particular,
they may appear as dynamically generated companion poles of the mostly
$\bar{q}q$ resonances $a_{0}(1450)$ and $K_{0}^{\star}(1430)$ [135, 136].
Another parameterization, with the inclusion of $f_{0}$’s will be presented in
Section 8.

### 3.5 General $U(1)_{A}$ anomaly terms

The axial anomaly $U(1)_{A}$ has been described in Sec. 3.2 by a specific
term, but other terms are possible. A more general Lagrangian can be written
as

$\mathcal{L}_{U(1)_{A}}^{\text{gen}}=c_{1}(\det\Phi+\det\Phi^{\dagger})+c_{2}(\det\Phi-\det\Phi^{\dagger})^{2}+c_{m}(\det\Phi+\det\Phi^{\dagger})\mathop{\mathrm{Tr}}(\Phi^{\dagger}\Phi)\text{
.}$ (3.46)

The first term is described in Ref. [46] and is also used within the eLSM in
Refs. [8, 236], the second term, that first appeared in Refs. [237, 238] and
is called Veneziano-Witten term, is also used in Refs. [121, 6, 36, 132, 130]
(and in Sec. 2.3), while the last one is a mixed term [239]. The first term
contains three, the second six and the last five mesonic fields, so $c_{1}$,
$c_{2}$ and $c_{m}$ have energy dimensions $1$, $-2$ and $-1$ respectively.
Since the parameters are not dimensionless, these terms also break dilatation
invariance. Note that instead of the $c_{2}(\det\Phi-\det\Phi^{\dagger})^{2}$
term, a $\bar{c}_{2}((\det\Phi)^{2}+(\det\Phi^{\dagger})^{2})$ term could also
be used, such as in [239]. The difference of the two terms is
$\sim\det\Phi\det\Phi^{\dagger}$, which is chirally invariant (not anomalous)
and can be transformed out by redefining the coefficients of some of the
chirally invariant terms, so both $\sim\det\Phi^{2}$ terms can be used
interchangeably.

In [126] it is investigated how these terms are able to reproduce the particle
spectrum. It was found that each term ($c_{1}$, $c_{2}$ and $c_{m}$) can give
an equally good description of the masses. Moreover, in principle these terms
can be used simultaneously and still reproduce the particle spectrum with a
good $\chi^{2}$.

For the sake of completeness, we list in Table 3.6 how these terms modify the
tree-level mass expressions of Table 3.2.

Mass squares | Analytical expressions
---|---
$m_{\pi}^{2}$ | $Z_{\pi}^{2}\left[m_{0}^{2}+\left(\lambda_{1}+\frac{\lambda_{2}}{2}\right)\phi_{N}^{2}+\lambda_{1}\phi_{S}^{2}-{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(c_{1}+c_{m}(\phi_{N}^{2}+\frac{1}{2}\phi_{S}^{2}))\frac{\phi_{S}}{\sqrt{2}}}\right]$
$m_{K}^{2}$ | $Z_{K}^{2}\left[m_{0}^{2}+\left(\lambda_{1}+\frac{\lambda_{2}}{2}\right)\phi_{N}^{2}-\frac{\lambda_{2}}{\sqrt{2}}\phi_{N}\phi_{S}+\left(\lambda_{1}+\lambda_{2}\right)\phi_{S}^{2}-{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(2c_{1}+c_{m}(\phi_{N}^{2}+\phi_{S}^{2}+\sqrt{2}\phi_{N}\phi_{S}))\frac{\phi_{N}}{4}}\right]$
$m_{\eta_{N}}^{2}$ | $Z_{\pi}^{2}\left[m_{0}^{2}+\left(\lambda_{1}+\frac{\lambda_{2}}{2}\right)\phi_{N}^{2}+\lambda_{1}\phi_{S}^{2}+c_{2}\,\phi_{N}^{2}\phi_{S}^{2}-{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(c_{1}+\sqrt{2}c_{2}\phi_{N}^{2}\phi_{S}+c_{m}\frac{1}{2}\phi_{S}^{2})\frac{\phi_{S}}{\sqrt{2}}}\right]$
$m_{\eta_{S}}^{2}$ | $Z_{\eta_{S}}^{2}\left[m_{0}^{2}+\lambda_{1}\phi_{N}^{2}+\left(\lambda_{1}+\lambda_{2}\right)\phi_{S}^{2}+\frac{c_{2}}{4}\phi_{N}^{4}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(c_{2}\phi_{N}^{2}-\sqrt{2}c_{m}\phi_{S})\frac{\phi_{N}^{2}}{4}}\right]$
$m_{\eta_{NS}}^{2}$ | $Z_{\eta_{N}}Z_{\eta_{S}}\left[{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(2c_{1}+\sqrt{2}c_{2}\phi_{N}^{2}\phi_{S}+c_{m}(\phi_{N}^{2}+\phi_{S}^{2}))\frac{\phi_{N}}{2\sqrt{2}}}\right]$
$m_{a_{0}}^{2}$ | $m_{0}^{2}+\left(\lambda_{1}+\frac{3}{2}\lambda_{2}\right)\phi_{N}^{2}+\lambda_{1}\phi_{S}^{2}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(c_{1}+\frac{1}{2}c_{m}\phi_{S}^{2})\frac{\phi_{S}}{\sqrt{2}}}$
$m_{K_{0}^{\star}}^{2}$ | $Z_{K_{0}^{\star}}^{2}\left[m_{0}^{2}+\left(\lambda_{1}+\frac{\lambda_{2}}{2}\right)\phi_{N}^{2}+\frac{\lambda_{2}}{\sqrt{2}}\phi_{N}\phi_{S}+\left(\lambda_{1}+\lambda_{2}\right)\phi_{S}^{2}+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(2c_{1}+c_{m}(\phi_{N}^{2}+\phi_{S}^{2}-\sqrt{2}\phi_{N}\phi_{S}))\frac{\phi_{N}}{4}}\right]$
$m_{\sigma_{N}}^{2}$ | $m_{0}^{2}+3\left(\lambda_{1}+\frac{\lambda_{2}}{2}\right)\phi_{N}^{2}+\lambda_{1}\phi_{S}^{2}-{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(c_{1}+\frac{1}{2}c_{m}(6\phi_{N}^{2}+\phi_{S}^{2})\frac{\phi_{S}}{\sqrt{2}}}$
$m_{\sigma_{S}}^{2}$ | $m_{0}^{2}+\lambda_{1}\phi_{N}^{2}+3\left(\lambda_{1}+\lambda_{2}\right)\phi_{S}^{2}-{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\frac{3}{2\sqrt{2}}c_{m}\phi_{N}^{2}\phi_{S}}$
$m^{2}_{\sigma_{NS}}$ | $2\lambda_{1}\phi_{N}\phi_{S}-{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}(c_{1}+c_{m}(\phi_{N}^{2}+\frac{3}{2}\phi_{S}^{2}))\frac{\phi_{N}}{\sqrt{2}}}$
Table 3.6: Mass expressions of spin-0 mesons (scalars and pseudoscalars)
within the eLSM with the general $U(1)_{A}$ anomaly term. The anomaly
contributions are written in blue (online).

Note that in the case of the scalars there is a non-zero mixing in the
nonstrange-strange (N-S) sector even if all $U(1)_{A}$ anomaly terms are zero
(described by the large-$N_{c}$ suppressed $\lambda_{1}$-term), while in the
pseudoscalar N-S sector there is no mixing if there is no $U(1)_{A}$ anomaly.
Another interesting property is that if we were to use only the $c_{2}$ type
of term, it would only modify the masses in the pseudoscalar sector, but not
in the scalar sector. In the recent paper [239] , Pisarski and Rennecke
conjecture that the $\sim\det\Phi^{2}$ term dominates the $\sim\det\Phi$ term,
i.e. it is possible that $c_{2}>>c_{1}$. This term may affect other
properties, such as the order of the phase transition in the chiral limit,
which changes by using different $U(1)_{A}$ anomaly terms.

### 3.6 The eLSM at large-$N_{c}$

As discussed in detail in Refs. [16, 17, 18, 19], in the limit of a large
number of colors, the so-called large-$N_{c}$, interesting and useful
simplifications occur in the ’t Hooft limit [15], in which
$\Lambda_{\text{QCD}}$ is $N_{c}$-independent and $g_{\text{QCD}}$ scales as
$1/\sqrt{N_{c}}$: conventional quark-antiquark mesons, hybrids, and glueballs
retain their mass and become long-lived. In particular, the decay width of
conventional mesons and hybrids scales as $1/N_{c}$, while that of glueballs
as $1/N_{c}^{2}$ [16]. Moreover, the dominant four-leg interaction between
conventional mesons and hybrids involves a single trace and scale as
$1/N_{c}$, while the subdominant ones (product of two traces) goes as
$1/N_{c}^{3}$. This is an important mean to distinguish between different
terms, especially when few data are available (see Sec. 4). The four-leg
glueball scaling behaves as $1/N_{c}^{2}$, just as the four-leg interaction
with two glueballs and two conventional mesons or hybrids (see Sec. 5 and 6
for applications).

The condensates scale as well: the dilaton/glueball as
$G_{0}\sim\Lambda_{G}\propto N_{c}$ , while the quark ones $\phi_{N,S}\propto
N_{c}^{1/2}$. A special case regards the chiral anomaly, where an additional
large-$N_{c}$ suppression takes place [238]. For a detailed description of all
the scaling behaviors mentioned above, with special focus on chiral models, we
refer to the recent lectures of Ref. [19].

In Table 3.5 we report the scaling of the parameters of the eLSM, while in
Table 3.7 we discuss some of its main consequences, that well agree with the
general expected rules. For a brief recall of large-$N_{c}$, see B.

Quantity | Scaling
---|---
Gluon condensate $G_{0}$ | $N_{c}$
Quark condensates $\phi_{N,S}$ | $N_{c}^{1/2}$
Mass parameter $m_{0}^{2}=\lambda_{G\Phi}G_{0}<0$ | $N_{c}^{0}$
Masses of mesons ($\bar{q}q,$ glueballs, hybrids) | $N_{c}^{0}$
Exception ($U_{A}(1)$ anomaly): mass of $\eta_{0}$ (split into $\eta,\eta^{\prime}$) | $N_{c}^{-1}$
Decay of $\bar{q}q$ states into $\bar{q}q,$ e.g. $\Gamma_{\rho\rightarrow\pi\pi}$ | $N_{c}^{-1}$
Decay of glueballs, e.g. $\Gamma_{G\rightarrow\pi\pi}$ | $N_{c}^{-2}$
Exception ($U_{A}(1)$ anomaly): $\tilde{G},$ e.g. $\Gamma_{\tilde{G}\rightarrow\eta f_{0}}$ | $N_{c}^{-4}$
Four-leg $\bar{q}q$ scattering amplitude, e.g. $A_{\pi\pi\rightarrow\pi\pi}$ | $N_{c}^{-1}$
Four-leg $GG$ scattering amplitude, e.g. $A_{GG\rightarrow GG}$ | $N_{c}^{-2}$
Table 3.7: $N_{c}$-scaling of some phenomenologically relevant eLSM
quantities.

## 4 Inclusion of other conventional mesons with $J=1,2$ into the eLSM

In this section, we discuss the inclusion of additional chiral multiplets into
the eLSM: (i) the heterochiral multiplet (heterovectors) including the
pseudovector nonet ($J^{\mathcal{P}\mathcal{C}}=1^{+-}$, ${}^{2S+1}L_{J}$ =
${}^{1}P_{1}$) and the orbitally excited vector one
($J^{\mathcal{P}\mathcal{C}}=1^{--}$, ${}^{2S+1}L_{J}$ =${}^{3}D_{1}$); (ii)
the homochiral multiplet (homotensors) with tensor mesons
($J^{\mathcal{P}\mathcal{C}}=2^{++}$,${}^{2S+1}L_{J}=^{3}P_{2}$) and with
axial-tensor mesons ($J^{\mathcal{P}\mathcal{C}}=2^{--}$, ${}^{2S+1}L_{J}$
=${}^{3}D_{2}$); (iii) the heterochiral multiplet (heterotensors) of
pseudotensor mesons ($J^{\mathcal{P}\mathcal{C}}=2^{-+}$, ${}^{2S+1}L_{J}$
=${}^{1}D_{2}$) and their chiral partners. The corresponding resonances are
listed in Table 4.1.

Field | PDG | Quark content | $I^{\mathcal{G}}$ | $J^{\mathcal{P}\mathcal{C}}$ | PDG Mass (MeV)
---|---|---|---|---|---
$\rho_{D}^{+},\rho_{D}^{-},\rho_{D}^{0}$ | $\rho(1700)$ | $u\bar{d},d\bar{u},\frac{u\bar{u}-d\bar{d}}{\sqrt{2}}$ | $1^{+}$ | $1^{--}$ | $1720\pm 20$
$K_{D}^{\ast+},K_{D}^{\ast-},K_{D}^{\ast 0},\bar{K}_{D}^{\ast 0}$ | $K^{\ast}(1680)$ | $u\bar{s},s\bar{u},d\bar{s},s\bar{d}$ | $1/2$ | $1^{-}$ | $1718\pm 18$
$\omega_{DN}$ | $\omega(1650)$ | $\frac{u\bar{u}+d\bar{d}}{\sqrt{2}}$ | $0^{-}$ | $1^{--}$ | $1670\pm 30$
$\omega_{DS}$ | $\phi(???)$ | $s\bar{s}$ | $0^{-}$ | $1^{--}$ | $-$
$b_{1}^{+},b_{1}^{-},b_{1}^{0}$ | $b_{1}(1235)$ | $u\bar{d},d\bar{u},\frac{u\bar{u}-d\bar{d}}{\sqrt{2}}$ | $1^{+}$ | $1^{+-}$ | $1229.5\pm 3.2$
$K_{1B}^{+},K_{1B}^{-},K_{1B}^{0},\bar{K}_{1B}^{0}$ |  | $K_{1}(1400)$
---
$K_{1}(1270)$
$u\bar{s},s\bar{u},d\bar{s},s\bar{d}$ | $1/2$ | $1^{-}$ |  | $1368\pm 38$
---
$1253\pm 7$
$h_{1,N}$ | $h_{1}(1170)$ | $\frac{u\bar{u}+d\bar{d}}{\sqrt{2}}$ | $0^{-}$ | $1^{+-}$ | $1166\pm 6$
$h_{1,S}$ | $h_{1}(1415)$ | $s\bar{s}$ | $0^{-}$ | $1^{+-}$ | $1419^{+9}_{-8}$
$\pi_{2}^{+},\pi_{2}^{-},\pi_{2}^{0}$ | $\pi_{2}(1670)$ | $u\bar{d},d\bar{u},\frac{u\bar{u}-d\bar{d}}{\sqrt{2}}$ | $1^{-}$ | $2^{-+}$ | $1670.6^{+2.9}_{-1.2}$
$K_{2P}^{+},K_{2P}^{-},K_{2P}^{0},\bar{K}_{2P}^{0}$ |  | $K_{2}(1770)$
---
$K_{2}(1820)$
$u\bar{s},s\bar{u},d\bar{s},s\bar{d}$ | $1/2$ | $2^{-}$ |  | $1773\pm 8$
---
$1819\pm 12$
$\eta_{2,N}$ | $\eta_{2}(1645)$ | $\frac{u\bar{u}+d\bar{d}}{\sqrt{2}}$ | $0^{+}$ | $2^{-+}$ | $1617\pm 5$
$\eta_{2,S}$ | $\eta_{2}(1870)$ | $s\bar{s}$ | $0^{+}$ | $2^{-+}$ | $1842\pm 8$
$a_{2}^{+},a_{2}^{-},a_{2}^{0}$ | $a_{2}(1320)$ | $u\bar{d},d\bar{u},\frac{u\bar{u}-d\bar{d}}{\sqrt{2}}$ | $1^{-}$ | $2^{++}$ | $1317.7\pm 1.4$
$K_{2}^{+},K_{2}^{-},K_{2}^{0},\bar{K}_{2}^{0}$ | $K_{2}^{\ast}(1430)$ | $u\bar{s},s\bar{u},d\bar{s},s\bar{d}$ | $1/2$ | $2^{+}$ | $1424\pm 4$
$f_{2,N}$ | $f_{2}(1270)$ | $\frac{u\bar{u}+d\bar{d}}{\sqrt{2}}$ | $0^{+}$ | $2^{++}$ | $1275.4\pm 0.8$
$f_{2,S}$ | $f_{2}^{\prime}(1525)$ | $s\bar{s}$ | $0^{+}$ | $2^{++}$ | $1517.3\pm 2.4$
Table 4.1: Resonances studied within the eLSM in this Section: PDG name, quark
content, quantum numbers and PDG mass values from [13]. One potential
candidate for the missing orbitally excited vector meson $\phi(?)$ is the
resonance $\phi(2170)$. However, it is disfavored due to its higher mass.

### 4.1 Heterochiral mesons with $J=1$

The heterochiral multiplet $\Phi^{\mu}=B^{\mu}+iV_{D}^{\mu}$, that includes
the pseudovector nonet $B^{\mu}$ with $J^{\mathcal{P}\mathcal{C}}=1^{+-}$ and
the orbitally excited vector one with $J^{\mathcal{P}\mathcal{C}}=1^{--}$, can
be easily embedded into the eLSM. Namely, the Lagrangian term that is
responsible for the mass generation of pseudovector and orbitally vector
mesons can be written as:

$\displaystyle\mathcal{L}^{\Phi_{\mu}}_{\text{mass}}=$
$\displaystyle\text{Tr}\Big{[}\Big{(}\frac{m_{1}^{2}\,G^{2}}{2\,G_{0}^{2}}+\Delta^{\text{pv}}\Big{)}\Big{(}\Phi_{\mu}^{\dagger}\Phi^{\mu}\Big{)}\Big{]}+\frac{\lambda_{\Phi_{\mu},1}}{2}\text{Tr}\Big{[}\Phi^{\dagger}\Phi\Big{]}\text{Tr}\Big{[}\Phi_{\mu}^{\dagger}\Phi^{\mu}\Big{]}+\lambda_{\Phi_{\mu},2}\text{Tr}\Big{[}\Phi_{\mu}^{\dagger}\Phi\Phi^{\mu\dagger}\Phi+\Phi_{\mu}\Phi^{\dagger}\Phi^{\mu}\Phi^{\dagger}\Big{]}$
(4.1)
$\displaystyle+\lambda_{\Phi_{\mu},3}\text{Tr}\Big{[}\Phi_{\mu}\Phi^{\dagger}\Phi\Phi^{\mu\,\dagger}+\Phi_{\mu}^{\dagger}\Phi\Phi^{\dagger}\Phi^{\mu}\Big{]}\text{
,}$

where the chiral fields are $\Phi$ (2.181) and $\Phi_{\mu}$ (2.186) have been
introduced. The explicit chiral symmetry and dilatation breaking terms is the
one proportional to
$\Delta^{\text{pv}}=\text{diag}(\delta_{N}^{\text{pv}},\delta_{N}^{\text{pv}},\delta_{S}^{\text{pv}})$.
As usual, one may set $\delta_{N}^{\text{pv}}=0$ and one expects that
$\delta_{S}^{\text{pv}}\sim\delta_{S}$, see Table 3.5. Note, the field
$\Phi^{\mu}$ has been included in the eLSM in Ref. [129], but the Lagrangian
above is here presented for the first time. Its effect is to introduce a mass
splitting between pseudovector and orbitally excited vector mesons, but its
detailed study was not yet carried out in detail and is left as an outlook.

Next, a chiral invariant interaction Lagrangian leading to the decay of
heterovectors reads: reads:

$\displaystyle\mathcal{L}^{\text{int}}_{\Phi^{\mu}}$
$\displaystyle=g_{\Phi^{\mu}\Phi\Phi}\text{Tr}\Big{[}\Phi^{\mu}\,\Phi\partial_{\mu}\Phi+\text{c.c}\Big{]}+g_{\Phi^{\mu}LR}\text{Tr}\Big{[}\Phi_{\alpha}^{\dagger}L_{\beta}L^{\alpha}\partial^{\beta}\Phi+R_{\alpha}\Phi_{\beta}^{\dagger}\partial^{\alpha}\Phi
R^{\beta}+L_{\alpha}\partial^{\beta}\Phi\Phi_{\alpha}^{\dagger}L^{\beta}+\partial_{\alpha}\Phi^{\dagger}R_{\beta}R^{\alpha}\Phi^{\beta}\Big{]}\text{
.}$ (4.2)

It reduces to the following two dominant terms which describe the decays of
the orbitally excited vector mesons into two pseudoscalar and vector
pseudoscalar mesons:

$\displaystyle\mathcal{L}_{V_{D}}=ig_{v_{d}pp}\text{Tr}\Bigg{[}\Big{[}\partial_{\mu}P,V_{D}^{\mu}\Big{]}P\Bigg{]}+g_{v_{d}vp}\text{Tr}\Bigg{[}V_{D}^{\mu\nu}\Big{\\{}V_{\mu\nu},P\Big{\\}}\Bigg{]}\text{
,}$ (4.3)

see Ref. [197] and Table 4.2 for numerical results. For pseudovector mesons,
it reduces to a term proportional to $\text{Tr}[B_{\mu}\\{V^{\mu},P\\}]$ that
has been studied in Ref. [195]. Yet, it is important to remark that a joint
study of decays of both the orbitally excited vector mesons and pseudovector
mesons has also not been yet undertaken.

Decay process | Width (MeV) | Decay process | Width (MeV)
---|---|---|---
$\rho(1700)\rightarrow\overline{K}K$ | $40\pm 11$ | $\rho(1700)\rightarrow\pi\pi$ | $140\pm 37$
$K^{\star}(1680)\rightarrow K\pi$ | $82\pm 22$ | $K^{\star}(1680)\rightarrow K\eta$ | $52\pm 14$
$\omega(1650)\rightarrow\overline{K}K$ | $37\pm 10$ | $\rho(1700)\rightarrow\omega\pi$ | $140\pm 59$
$\rho(1700)\rightarrow K^{\ast}(892)K$ | $56\pm 23$ | $\rho(1700)\rightarrow\rho\eta$ | $41\pm 17$
$K^{\ast}(1680)\rightarrow K\rho$ | $64\pm 27$ | $K^{\ast}(1680)\rightarrow K\phi$ | $13\pm 6$
$K^{\ast}(1680)\rightarrow K\omega$ | $21\pm 9$ | $K^{\ast}(1680)\rightarrow K^{\ast}(892)\pi$ | $81\pm 34$
$K^{\ast}(1680)\rightarrow K^{\ast}(892)\eta$ | $0.5\pm 0.2$ | $\omega(1650)\rightarrow\rho\pi$ | $370\pm 156$
$\omega(1650)\rightarrow K^{\ast}(892)K$ | $42\pm 18$ | $\omega(1650)\rightarrow\omega(782)\eta$ | $32\pm 13$
$\phi(1930)\rightarrow K\bar{K}^{\ast}(892)$ | $260\pm 109$ | $\phi(1930)\rightarrow\phi(1020)\eta$ | $67\pm 28$
Table 4.2:  Decays of the radially excited vector mesons [197].

Very interestingly, the chiral anomaly can be introduced for this chiral
multiplet. To this end, a mathematical object that we denote as
‘$\epsilon$-product’ was originally introduced implicitly by Giacosa, Pisarski
and Koenigstein (GPK) in Ref. [48] and explictly by Giacosa, Pisarski, and
Jafarzade (GPJ) in Ref. [49] (thus GPKJ in short). For $3\times 3$ matrices it
reads::

$\epsilon[A,B,C]=\epsilon^{ijk}\epsilon^{i^{\prime}j^{\prime}k^{\prime}}A_{ii^{\prime}}B_{jj^{\prime}}C_{kk^{\prime}}/3!\;.$
(4.4)

It has quite interesting properties such as

$\displaystyle\epsilon[A,A,A]=\det
A\,,\qquad\epsilon[A,1,1]=\frac{1}{3}\text{Tr}A\ \text{ ,}$ (4.5)

and is invariant under exchange of the matrices,
$\epsilon[A,B,C]=\epsilon[B,A,C],$ etc. It can be also easily generalized to
the $N$-case with:

$\epsilon[A_{1},A_{2},...,A_{N}]=\epsilon^{i_{1}i_{2}...i_{N}}\epsilon^{i_{1}^{\prime}i_{2}^{\prime}...i_{N}^{\prime}}{A_{1}}_{i_{1}i_{1}^{\prime}}{A_{2}}_{j_{2}i_{2}^{\prime}}...{A_{N}}_{i_{N}i_{N}^{\prime}}/N!\;.$
(4.6)

Using this object, one may write down a Lagrangian that breaks $U(1)_{A}$
axial symmetry but retains chiral symmetry (just as the determinant term in
the (pseudo)scalar sector, see Sec. 3.2 and especially Sec. 3.5 and refs.
therein). Following Refs. [48, 49]:

$\displaystyle\mathcal{L}^{U(1)_{A}-\text{anom}}_{\Phi^{\mu}}=a_{\Phi^{\mu}}\Big{(}\epsilon[\Phi\,,\Phi_{\mu}\,,\Phi^{\mu}]+\text{c.c.}\Big{)}\
\text{ ,}$ (4.7)

where the coupling is estimated $a_{\Phi^{\mu}}\approx-0.14$ GeV in Ref. [49]
by using the eLSM implemented with Dilute Gas of Instantons (DGI) approach.
Anomalous decay channels following the Lagrangian (4.7) are listed in Table
4.3. Even though they are quite small, their experimental measurement can
still play a significant role in understanding the axial anomaly in spin-1
heterochiral mesons.

Decay process | Width (MeV) | Decay process | Width (MeV)
---|---|---|---
$\rho(1700)\rightarrow h_{1}(1415)\,\pi$ | $0.027$ | $\phi(2170)\rightarrow b_{1}(1235)\,\pi$ | $0.071$
$\phi(2170)\rightarrow h_{1}(1170)\,\eta$ | $0.012$ | $\phi(2170)\rightarrow h_{1}(1170)\,\eta^{\prime}(958)$ | $0.010$
Table 4.3:  Decay rates for the anomalous decay channels of spin-1 mesons
estimated in a DGI [49].

The mixing angle $\beta_{pv}$ between $h_{1}(1170)$ and $h_{1}(1415)$
introduced in (2.138) to be within the range $0$-$10^{\circ}$ in the eLSM+DGI
approach of Ref. [49], see Fig. 4.1 for a comparison to the experiments in
Refs. [240, 241, 242, 194] and to lattice (LQCD) [98].

Figure 4.1: The estimate of the mixing angle $\beta_{pv}$ in a eLSM + DGI
approach [49] is compared to the experiments [240, 241, 242, 194] and to LQCD
[98]. Here $M_{1}$ is a DGI parameter, which links the mesonic and quark
fields. The theoretical prediction lies between the two horizontal lines
(solid orange and dashed yellow).

The term in Eq. (4.7) represents an interesting new type of anomalous
interactions, which in general are possible for heterochiral multiplets (it
affects in particular the masses and the mixing of the isoscalar members). A
new plethora of such terms is possible, and a promising outlook is the study
of their detailed role in the mesonic phenomenology.

### 4.2 Homochiral mesons with $J=2$

The tensor homochiral multiplets (homotensors), involving a tensor nonet
$T^{\mu\nu}$ (2.146) (with $J^{\mathcal{P}\mathcal{C}}=2^{++}$) and and
(axial-)tensor one $A_{2}^{\mu\nu}$ (2.150) (with
$J^{\mathcal{P}\mathcal{C}}=2^{--}$), were studied within the eLSM in Ref.
[10]. The chiral Lagrangian describing the masses of (axial-)tensor states
reads:

$\displaystyle\mathcal{L}_{\textbf{L},\textbf{R}}^{\text{mass}}=\text{Tr}\Big{[}$
$\displaystyle\Big{(}\frac{m_{2}^{2}\,G^{2}}{2\,G_{0}^{2}}+\Delta^{\text{ten}}\Big{)}\Big{(}\mathbf{L}_{\mu\nu}^{2}+\mathbf{R}_{\mu\nu}^{2}\Big{)}\Big{]}+\frac{h_{1}^{\text{ten}}}{2}\text{Tr}\Big{[}\Phi^{\dagger}\Phi\Big{]}\text{Tr}\Big{[}\mathbf{L}^{\mu\nu}\mathbf{L}_{\mu\nu}+\mathbf{R}^{\mu\nu}\mathbf{R}_{\mu\nu}\Big{]}$
$\displaystyle+h_{2}^{\text{ten}}\text{Tr}\Big{[}\Phi^{\dagger}\mathbf{L}^{\mu\nu}\mathbf{L}_{\mu\nu}\Phi+\Phi\mathbf{R}^{\mu\nu}\mathbf{R}_{\mu\nu}\Phi^{\dagger}\Big{]}+2h_{3}^{\text{ten}}\text{Tr}\Big{[}\Phi\mathbf{R}^{\mu\nu}\Phi^{\dagger}\mathbf{L}_{\mu\nu}\Big{]}\text{.}$
(4.8)

Above, the explicit symmetry breaking term parameterized by the diagonal
matrix
$\Delta^{\text{ten}}=\text{diag}(\delta_{N}^{\text{ten}},\delta_{N}^{\text{ten}},\delta_{S}^{\text{ten}})$
is the only dimensionful one. By assuming (without loss of generality)
$\delta_{N}^{\text{ten}}\simeq 0$, we obtain $\delta_{S}^{\text{ten}}$ via the
following expression:

$\delta_{S}^{\text{ten}}=m_{K_{2}}^{2}-m_{\mathbb{a}_{2}}^{2}\simeq
0.3\,\text{GeV}^{2}\,.$ (4.9)

The masses arising from Eq. (4.2) are listed in Table 4.4. In particular,
within the nonets we have:

$m_{\rho_{2}}^{2}=m_{\omega_{2,N}}^{2}\,\,\,\text{and}\,\,\,m_{\mathbf{a}_{2}}^{2}=m_{f_{2,N}}^{2}\text{.}$
(4.10)

The following relations between chiral partners follow:

$m_{\mathbf{a}_{2}}^{2}-m_{\rho_{2}}^{2}=h_{3}^{\text{ten}}\phi_{N}^{2}\text{
,}m_{K_{2}}^{2}-m_{K_{2A}}^{2}=\sqrt{2}h_{3}^{\text{ten}}\phi_{N}\phi_{S}\text{
, }m_{f_{2,S}}^{2}-m_{\omega_{2,S}}^{2}=2h_{3}^{\text{ten}}\phi_{S}^{2}\text{
,}$ (4.11)

showing that the mass differences between chiral partners are proportional to
the chiral condensate(s), as expected. Using the parameters from Table 3.5 and
the masses of $K_{2}(1820)$ and $K_{2}^{*}(1430)$ from Table 4.1, we obtain
the parameter $h_{3}^{\text{ten}}$ as:

$h_{3}^{\text{ten}}=\frac{m_{K_{2A}}^{2}-m_{K_{2}}^{2}}{\sqrt{2}\phi_{N}\phi_{S}}\simeq-41\,\text{
.}$ (4.12)

Mass squares | Analytical expressions | Estimates of masses (MeV)
---|---|---
$m_{\mathbf{a}_{2}}^{2}$ | $\big{(}m^{2}_{2}+2\delta_{N}^{\text{ten}}\big{)}+\frac{h_{3}^{\text{ten}}\phi_{N}^{2}}{2}+\frac{h_{2}^{\text{ten}}\phi_{N}^{2}}{2}+\frac{h_{1}^{\text{ten}}}{2}(\phi_{N}^{2}+\phi_{S}^{2})$ | 1317
$m_{K_{2}}^{2}$ | $\big{(}m^{2}_{2}+\delta_{N}^{\text{ten}}+\delta_{S}^{\text{ten}}\big{)}+\frac{h_{1}^{\text{ten}}}{2}(\phi_{N}^{2}+\phi_{S}^{2})+\frac{h_{3}^{\text{ten}}}{\sqrt{2}}\phi_{N}\phi_{S}+\frac{h_{2}^{\text{ten}}}{4}(\phi_{N}^{2}+2\phi_{S}^{2})$ | 1427
$m_{f_{2,N}}^{2}$ | $\big{(}m^{2}_{2}+2\delta_{N}^{\text{ten}}\big{)}+\frac{h_{3}^{\text{ten}}\phi_{N}^{2}}{2}+\frac{h_{1}^{\text{ten}}}{2}(\phi_{N}^{2}+\phi_{S}^{2})+\frac{h_{2}^{\text{ten}}\phi_{N}^{2}}{2}=m_{\mathbf{a}_{2}}^{2}$ | $1317$
$m_{f_{2,S}}^{2}$ | $\big{(}m^{2}_{2}+2\delta_{S}^{\text{ten}}\big{)}+\frac{h_{1}^{\text{ten}}}{2}(\phi_{N}^{2}+\phi_{S}^{2})+h_{3}^{\text{ten}}\phi_{S}^{2}+h_{2}^{\text{ten}}\phi_{S}^{2}$ | $1524$
$m_{\rho_{2}}^{2}$ | $\big{(}m^{2}_{2}+2\delta_{N}^{\text{ten}}\big{)}-\frac{h_{3}^{\text{ten}}\phi_{N}^{2}}{2}+\frac{h_{2}^{\text{ten}}\phi_{N}^{2}}{2}+\frac{h_{1}^{\text{ten}}}{2}(\phi_{N}^{2}+\phi_{S}^{2})$ | $1663$
$m_{K_{2A}}^{2}$ | $\big{(}m^{2}_{2}+\delta_{N}^{\text{ten}}+\delta_{S}^{\text{ten}}\big{)}+\frac{h_{1}^{\text{ten}}}{2}(\phi_{N}^{2}+\phi_{S}^{2})-\frac{h_{3}^{\text{ten}}}{\sqrt{2}}\phi_{N}\phi_{S}+\frac{h_{2}^{\text{ten}}}{4}(\phi_{N}^{2}+2\phi_{S}^{2})$ | 1819
$m_{\omega_{2,N}}^{2}$ | $\big{(}m^{2}_{2}+2\delta_{N}^{\text{ten}}\big{)}-\frac{h_{3}^{\text{ten}}\phi_{N}^{2}}{2}+\frac{h_{2}^{\text{ten}}\phi_{N}^{2}}{2}+\frac{h_{1}^{\text{ten}}}{2}(\phi_{N}^{2}+\phi_{S}^{2})=m_{\rho_{2}}^{2}$ | $1663$
$m_{\omega_{2,S}}^{2}$ | $\big{(}m^{2}_{2}+2\delta_{S}^{\text{ten}}\big{)}-h_{3}^{\text{ten}}\phi_{S}^{2}+h_{2}^{\text{ten}}\phi_{S}^{2}+\frac{h_{1}^{\text{ten}}}{2}(\phi_{N}^{2}+\phi_{S}^{2})$ | $1971$
Table 4.4:  Masses of $2^{++}$ and $2^{--}$ mesons within eLSM. Bold masses
are taken from PDG.

Our estimate for the (still experimentally missing) isovector member
$\rho_{2}$ of the axial-tensor mesons is $1.67$ GeV, consistent with the
prediction of Ref. [50]. The other members of the axial-tensor nonet have
masses below $2$ GeV.

The Lagrangian responsible for the decays of (axial-)tensor mesons reads:

$\displaystyle\mathcal{L}^{\text{int}}_{\textbf{L},\textbf{R}}=\frac{g_{2}^{\text{ten}}}{2}\Big{(}\text{Tr}\Big{[}\mathbf{L}_{\mu\nu}\\{L^{\mu},L^{\nu}\\}\Big{]}+\text{Tr}\Big{[}\mathbf{R}_{\mu\nu}\\{R^{\mu},R^{\nu}\\}\Big{]}\Big{)}+\frac{g_{2}^{\prime\,\text{ten}}}{6}\text{Tr}\Big{[}\mathbf{L}_{\mu\nu}+\mathbf{R}_{\mu\nu}\Big{]}\text{Tr}\Big{[}\\{L^{\mu},L^{\nu}\\}+\\{R^{\mu},R^{\nu}\\}\Big{]}$
(4.13)
$\displaystyle+\frac{a^{\text{ten}}}{2}\text{Tr}\Big{[}\mathbf{L}_{\mu\nu}\\{L^{\mu}_{\beta},L^{\nu\beta}\\}+\mathbf{R}_{\mu\nu}\\{R^{\mu}_{\beta},R^{\nu\beta}\\}\Big{]}+\frac{a^{\prime\,\text{ten}}}{6}\text{Tr}\Big{[}\mathbf{L}_{\mu\nu}+\mathbf{R}_{\mu\nu}\big{]}\text{Tr}\Big{[}\\{L^{\mu}_{\beta},L^{\nu\beta}\\}+\\{R^{\mu}_{\beta},R^{\nu\beta}\\}\Big{]}$
$\displaystyle+c_{1}^{\text{ten}}\,\text{Tr}\Big{[}\partial^{\mu}\textbf{L}^{\nu\alpha}\tilde{L}_{\mu\nu}\,\partial_{\alpha}\Phi\,\Phi^{\dagger}-\partial^{\mu}\textbf{R}^{\nu\alpha}\Phi^{\dagger}\,\partial_{\alpha}\Phi\tilde{R}_{\mu\nu}-\partial^{\mu}\textbf{R}^{\nu\alpha}\tilde{R}_{\mu\nu}\partial_{\alpha}\Phi^{\dagger}\Phi+\partial^{\mu}\textbf{L}^{\nu\alpha}\Phi\partial_{\alpha}\Phi^{\dagger}\tilde{L}_{\mu\nu}\Big{]}$
$\displaystyle+c_{2}^{\text{ten}}\,\text{Tr}\Big{[}\partial^{\mu}\textbf{L}^{\nu\alpha}\partial_{\alpha}\Phi\tilde{R}_{\mu\nu}\,\,\Phi^{\dagger}-\partial^{\mu}\textbf{R}^{\nu\alpha}\Phi^{\dagger}\,\tilde{L}_{\mu\nu}\partial_{\alpha}\Phi-\partial^{\mu}\textbf{R}^{\nu\alpha}\partial_{\alpha}\Phi^{\dagger}\tilde{L}_{\mu\nu}\Phi+\partial^{\mu}\textbf{L}^{\nu\alpha}\Phi\tilde{R}_{\mu\nu}\partial_{\alpha}\Phi^{\dagger}\Big{]}\text{.}$

This Lagrangian enables us to test eLSM for the tensor mesons and estimate the
decay rates for the axial-tensor mesons. Its parameters with their fit values
and large-$N_{c}$ scaling properties are listed in Table 4.5.

Parameters | Numerical values |  $N_{c}$ scaling
---|---|---
$h_{3}^{\text{ten}}$ | $-41$ | $N_{c}^{-1}$
$\delta_{S}^{\text{ten}}$ | $0.3\,\text{GeV}^{2}$ | $N_{c}^{0}$
$g_{2}^{\text{ten}}$ | $(1.392\pm 0.024)\cdot 10^{4}(\text{MeV})$ | $N_{c}^{-\frac{1}{2}}$
$g_{2}^{\prime\,\text{ten}}$ | $(0.024\pm 0.041)\cdot 10^{4}(\text{MeV})$ | $N_{c}^{-\frac{3}{2}}$
$c^{\text{ten}}\equiv c_{1}^{\text{ten}}+c_{2}^{\text{ten}}$ | $(4.8\pm 0.9)\cdot 10^{-7}(\text{MeV})^{-3}$ | $N_{c}^{-\frac{1}{2}}$
$a^{\text{ten}}$ | $(-2.09\pm 0.06)\cdot 10^{-2}(\text{MeV})^{-1}$ | $N_{c}^{-\frac{1}{2}}$
$a^{\prime\,\text{ten}}$ | $(3.5\pm 0.4)\cdot 10^{-3}(\text{MeV})^{-1}$ | $N_{c}^{-\frac{3}{2}}$
Table 4.5: Parameters of the spin-2 homochiral eLSM Lagrangian(s) describing
(axial-)tensor mesons.

Results for the dominant decay channels of the tensor mesons are compared to
the PDG values in Table 4.6. This comparison is also illustrated in Figure
4.2. Apart from some specific channels like
$f_{2}^{\prime}(1525)\rightarrow\overline{K}K$ that deserve further
clarification, most of the results are in qualitative agreement with the PDG.
The deviations can be caused by large-$N_{c}$ suppressed terms, isospin
breaking, and uncertainties in the resonance masses, which are not included in
our calculations. Note, the results are also quantitatively in agreement with
one obtained in the flavor invariant model of Ref. [199].

Decay process | eLSM (MeV) | PDG (MeV) | Decay process | eLSM (MeV) | PDG (MeV)
---|---|---|---|---|---
$\,\;a_{2}(1320)\rightarrow\bar{K}\,K$ | $4.06\pm 0.14$ | $7.0^{+2.0}_{-1.5}$ | $\,\;a_{2}(1320)\rightarrow\pi\,\eta$ | $25.37\pm 0.87$ | $18.5\pm 3.0$
$\,\;a_{2}(1320)\rightarrow\pi\,\eta^{\prime}(958)$ | $1.01\pm 0.03$ | $0.58\pm 0.10$ | $\,\;K_{2}^{\ast}(1430)\rightarrow\pi\,\bar{K}$ | $44.82\pm 1.54$ | $49.9\pm 1.9$
$\,\;f_{2}(1270)\rightarrow\bar{K}\,K$ | $3.54\pm 0.29$ | $8.5\pm 0.8$ | $\,\;f_{2}(1270)\rightarrow\pi\,\pi$ | $168.82\pm 3.89$ | $157.2^{+4.0}_{-1.1}$
$\,\;f_{2}(1270)\rightarrow\eta\,\eta$ | $0.67\pm 0.03$ | $0.75\pm 0.14$ | $\,\;f_{2}^{\prime}(1525)\rightarrow\bar{K}\,K$ | $23.72\pm 0.60$ | $75\pm 4$
$\,\;f_{2}^{\prime}(1525)\rightarrow\pi\,\pi$ | $0.67\pm 0.14$ | $0.71\pm 0.14$ | $\,\;f_{2}^{\prime}(1525)\rightarrow\eta\,\eta$ | $1.81\pm 0.05$ | $9.9\pm 1.9$
$\,\;a_{2}(1320)\rightarrow\rho(770)\,\pi$ | $71.0\pm 2.6$ | $73.61\pm 3.35$ | $\,\;K_{2}^{\ast}(1430)\rightarrow\bar{K}^{\ast}(892)\,\pi$ | $27.9\pm 1.0$ | $26.92\pm 2.14$
$\,\;K_{2}^{\ast}(1430)\rightarrow\rho(770)\,K$ | $10.3\pm 0.4$ | $9.48\pm 0.97$ | $\,\;K_{2}^{\ast}(1430)\rightarrow\omega(782)\,\bar{K}$ | $3.5\pm 0.1$ | $3.16\pm 0.88$
$\,\;f_{2}^{\prime}(1525)\rightarrow\bar{K}^{\ast}(892)\,K+\mathrm{c}.\mathrm{c}.$ | $19.89\pm 0.73$ |  |  |  |
Table 4.6: Decay rates of the ground-state $2^{++}$ resonances.  Figure 4.2:
Results for tensor mesons within the eLSM [10]. The $x$-axis lists the decay
width type, while the $y$-axis represents the deviations from the experimental
value as compared to the experimental error. Vertical lines on the data points
represent the fit errors. Those decay channels located between the dashed
lines are described well within the model.

Chiral symmetry relates chiral partners, in this specific case the well-known
tensor mesons to the poorly known axial-tensor ones. Here, within the eLSM we
use the parameters in Table 4.5 to estimate the decays of the axial-tensor
mesons. The results are presented in Table 4.7 and are compared to the lattice
QCD ones of Ref. [100] in Table 4.8, showing that they are qualitatively in
agreement with each other. We observe that, even when taking the lower limit
for the decay rates in Table 4.7, the ground-state axial-tensor mesons are
very broad. This can be a viable and rather simple explanation for the missing
status of $\rho_{2}$, $\omega_{2,N}$ and $\omega_{2,S}\sim\phi_{2}$ in the
meson list of Table 4.1: they might be too broad to be measured in
experiments.

Decay process | eLSM (MeV) | Decay process | eLSM (MeV)
---|---|---|---
$\,\;\rho_{2}(?)\rightarrow\rho(770)\,\eta$ | $\approx 99\pm 50$ | $\,\;\rho_{2}(?)\rightarrow\bar{K}^{\ast}(892)\,K+\mathrm{c}.\mathrm{c}.$ | $\approx 85\pm 43$
$\,\;\rho_{2}(?)\rightarrow\omega(782)\,\pi$ | $\approx 419\pm 210$ | $\,\;\rho_{2}(?)\rightarrow\phi(1020)\,\pi$ | $\approx 0.8$
$\,\;K_{2,A}\rightarrow\rho(770)\,K$ | $\approx 195\pm 98$ | $\,\;K_{2,A}\rightarrow\bar{K}^{\ast}(892)\,\pi$ | $\approx 316\pm 158$
$\,\;K_{2,A}\rightarrow\bar{K}^{\ast}(892)\,\eta$ | $\approx 0.01$ | $\,\;K_{2,A}\rightarrow\omega(782)\,\bar{K}$ | $\approx 51\pm 26$
$\,\;K_{2,A}\rightarrow\phi(1020)\,\bar{K}$ | $\approx 50\pm 25$ | $\,\;\omega_{2,N}\rightarrow\rho(770)\,\pi$ | $\approx 1314\pm 657$
$\,\;\omega_{2,N}\rightarrow\bar{K}^{\ast}(892)\,K+\mathrm{c}.\mathrm{c}.$ | $\approx 85\pm 43$ | $\,\;\omega_{2,N}\rightarrow\omega(782)\,\eta$ | $\approx 93\pm 47$
$\,\;\omega_{2,N}\rightarrow\phi(1020)\,\eta$ | $\approx 0.06$ | $\,\;\omega_{2,S}\rightarrow\omega(782)\,\eta^{\prime}(958)$ | $\approx 0.3$
$\,\;\omega_{2,S}\rightarrow\bar{K}^{\ast}(892)\,K+\mathrm{c}.\mathrm{c}.$ | $\approx 510\pm 255$ | $\,\;\omega_{2,S}\rightarrow\omega(782)\,\eta$ | $\approx 1.0\pm 0.5$
$\,\;\omega_{2,S}\rightarrow\phi(1020)\,\eta$ | $\approx 101\pm 51$ |  |
Table 4.7: Esimates of the axial-tensor meson decays. Decay process | eLSM (MeV) | LQCD (MeV) | Decay process | eLSM (MeV) | LQCD (MeV)
---|---|---|---|---|---
$\,\;\rho_{2}(?)\rightarrow\rho(770)\,\eta$ | $\approx 30$ |  | $\,\;\rho_{2}(?)\rightarrow\bar{K}^{\ast}(892)\,K+\mathrm{c}.\mathrm{c}.$ | $\approx 27$ | $36$
$\,\;\rho_{2}(?)\rightarrow\omega(782)\,\pi$ | $\approx 122$ | $125$ | $\,\;\rho_{2}(?)\rightarrow\phi(1020)\,\pi$ | $\approx 0.3$ |
$\,\;K_{2,A}\rightarrow\rho(770)\,K$ | $\approx 53$ |  | $\,\;K_{2,A}\rightarrow\bar{K}^{\ast}(892)\,\pi$ | $\approx 87$ |
$\,\;K_{2,A}\rightarrow\bar{K}^{\ast}(892)\,\eta$ | $\approx 0.004$ |  | $\,\;K_{2,A}\rightarrow\omega(782)\,\bar{K}$ | $\approx 13.8$ |
$\,\;K_{2,A}\rightarrow\phi(1020)\,\bar{K}$ | $\approx 13.7$ |  | $\,\;\omega_{2,N}\rightarrow\rho(770)\,\pi$ | $\approx 363$ | $365$
$\,\;\omega_{2,N}\rightarrow\bar{K}^{\ast}(892)\,K+\mathrm{c}.\mathrm{c}.$ | $\approx 25$ | $36$ | $\,\;\omega_{2,N}\rightarrow\omega(782)\,\eta$ | $\approx 27$ | $17$
$\,\;\omega_{2,N}\rightarrow\phi(1020)\,\eta$ | $\approx 0.02$ |  | $\,\;\omega_{2,S}\rightarrow\bar{K}^{\ast}(892)\,K+\mathrm{c}.\mathrm{c}.$ | $\approx 100$ | $148$
$\,\;\omega_{2,S}\rightarrow\omega(782)\,\eta$ | $\approx 0.2$ |  | $\,\;\omega_{2,S}\rightarrow\omega(782)\,\eta^{\prime}(958)$ | $\approx 0.02$ |
$\,\;\omega_{2,S}\rightarrow\phi(1020)\,\eta$ | $\approx 17$ | $44$ |  |  |
Table 4.8:  Decay rates of axial-tensor mesons fitted to LQCD Ref. [100].

### 4.3 Heterohiral mesons with $J=2$

The spin-2 heterochiral mesons (heterotensors) contain the rather well-
established pseudotensor mesons described by the nonet $P_{2}^{\mu\nu}$
(2.154) and their, up to now missing, chiral partners, the orbitally excited
tensor mesons. For this reason, a joint detailed study of masses and decays of
both nonets could not yet be undertaken.

The mass term for the heterochiral spin-2 mesons can be constructed by
following the standard procedure:

$\displaystyle\mathcal{L}_{\Phi_{\mu\nu}}^{\text{mass}}=$
$\displaystyle\text{Tr}\Big{[}\Big{(}\frac{m_{2^{\ast}}^{2}\,G^{2}}{2\,G_{0}^{2}}+\Delta^{\text{pt}}\Big{)}\Big{(}\Phi_{\mu\nu}^{\dagger}\Phi^{\mu\nu}\Big{)}\Big{]}+\frac{\lambda_{\Phi_{\mu\nu}\,,1}}{2}\text{Tr}\Big{[}\Phi^{\dagger}\Phi\Big{]}\text{Tr}\Big{[}\Phi_{\mu\nu}^{\dagger}\Phi^{\mu\nu}\Big{]}+\lambda_{\Phi_{\mu\nu}\,,2}\text{Tr}\Big{[}\Phi_{\mu\nu}^{\dagger}\Phi\Phi^{\mu\nu\dagger}\Phi+\Phi_{\mu\nu}\Phi^{\dagger}\Phi^{\mu\nu}\Phi^{\dagger}\Big{]}$
(4.14)
$\displaystyle+\lambda_{\Phi_{\mu\nu}\,,3}\text{Tr}\Big{[}\Phi_{\mu\nu}\Phi^{\dagger}\Phi\Phi^{\mu\nu\,\dagger}+\Phi_{\mu\nu}^{\dagger}\Phi\Phi^{\dagger}\Phi^{\mu\nu}\Big{]}\text{
,}$

where the parameter $\Delta^{\text{pt}}$ describes, as usual, a mass-driven
contribution. Due to the poor knowledge of the orbitally excited tensor
mesons, this Lagrangian has been not yet used to calculate masses.

Next, we turn to additional interactions that lead to decays of heterotensors
via chirally invariant under $U(3)_{L}\times U(3)_{R}$, giving:

$\displaystyle\mathcal{L}^{\text{int}}_{\Phi^{\mu\nu}}=g^{(1)}_{\Phi^{\mu\nu}}\text{Tr}\Big{[}\Phi^{\mu\nu}\,L_{\mu}\Phi^{\dagger}R_{\nu}+\text{c.c}\Big{]}+g^{(2)}_{\Phi^{\mu\nu}}\text{Tr}\Big{[}\Phi^{\mu\nu}\,\mathbf{L}_{\mu\nu}\Phi^{\dagger}+\text{c.c}\Big{]}\,\text{
.}$ (4.15)

For pseudotensor mesons only, it reduces to the following Lagrangian

$\mathcal{L}_{P_{2}}=g_{p_{2}vp}\text{Tr}\Bigg{[}P_{2}^{\mu\nu}\Big{[}V_{\mu},\partial_{\nu}P\Big{]}\Bigg{]}+g_{p_{2}t_{2}p}\text{Tr}\Bigg{[}P_{2}^{\mu\nu}\Big{\\{}T_{\mu\nu},P\Big{\\}}\Bigg{]}\,\text{
,}$ (4.16)

whose decay properties were studied in Ref. [209], see also Table 4.9. The
coupling constants $g_{p_{2}vp}$ and $g_{p_{2}t_{2}p}$ were fixed using the
PDG value for $\pi_{2}(1670)\rightarrow\rho(770)\,\pi$ and
$K_{2}(1770)\rightarrow\bar{K}_{2}^{\ast}(1430)\pi$, respectively. The other
entries of table LABEL:ab:pseudotensor were not yet reported as branching
rations in the PDG, see however Ref. [209] for details.

Decay process | theory (MeV) | Decay process | theory (MeV)
---|---|---|---
$\pi_{2}(1670)\rightarrow\rho(770)\,\pi$ | $80.6\pm 10.8$ | $\pi_{2}(1670)\rightarrow\bar{K}^{\ast}(892)\,K+\textbf{c}.\mathrm{c}$ | $11.7\pm 1.6$
$K_{2}(1770)\rightarrow\rho(770)K$ | $22.2\pm 3.0$ | $K_{2}(1770)\rightarrow\bar{K}^{\ast}(892)\pi$ | $25.5\pm 3.4$
$K_{2}(1770)\rightarrow\bar{K}^{\ast}(892)\eta$ | $20.5\pm 1.4$ | $K_{2}(1770)\rightarrow\omega(782)K$ | $8.3\pm 1.1$
$K_{2}(1770)\rightarrow\phi(1020)K$ | $4.2\pm 0.6$ | $\pi_{2}(1670)\rightarrow f_{2}(1270)\pi$ | $146.4\pm 9.7$
$K_{2}(1770)\rightarrow\bar{K}_{2}^{\ast}(1430)\pi$ | $84.5\pm 5.6$ |  |
Table 4.9: Decay rates of pseudotensor mesons estimated in Ref. [209].

Next, since this chiral multiplet is heterochiral, we can use the ‘GPKJ’
mathematical object defined in Eq. (4.4) to write down $U(1)_{A}$ anomalous
but chirally invariant terms. Following Refs. [48, 49], one has:

$\mathcal{L}^{U(1)_{A}-\text{anom}}_{\Phi^{\mu\nu}}=-b_{\Phi^{\mu\nu}}\Big{(}\epsilon\Big{[}\big{(}\partial_{\mu}\Phi\big{)},\Phi_{\nu}\,,\Phi^{\mu\nu}\Big{]}+\text{c.c.}\Big{)}-c_{\Phi^{\mu\nu}}\Big{(}\epsilon\Big{[}\big{(}\partial_{\mu}\Phi\big{)},\big{(}\partial_{\nu}\Phi\big{)},\Phi^{\mu\nu}\Big{]}+\text{c.c.}\Big{)}\,.$
(4.17)

The anomalous couplings $b_{\Phi^{\mu\nu}}$ and $c_{\Phi^{\mu\nu}}$ and
corresponding decay rates are estimated within DGI in Ref. [49]. The anomalous
interaction also allows to estimate the isoscalar mixing angle $\beta_{pt}$,
that turns out to be negative and quite small (smaller than $\beta_{pv}$), in
agreement with lattice QCD results [99]. Note, based on a study of decay
ratios, $\beta_{pt}$ was estimated to be large in Ref. [209]
$\beta_{pt}=-42^{\circ}$ . Such a large mixing has been later on confirmed in
the fit of Ref. [243]: for what concerns this important point, a tension
between different results is present and novel studies are needed.

## 5 Hybrid mesons within the eLSM

In this section we focus on the ground-state hybrid mesons with the exotic
quantum numbers $J^{\mathcal{P}\mathcal{C}}=1^{-+}$ and their chiral partners,
the pseudovector hybrid mesons $J^{\mathcal{P}\mathcal{C}}=1^{+-}$, embedded
into a homochiral multiplet (hybrid homovectors). Following Ref. [9], we
couple them to the eLSM and study their masses and decay processes. Thus, this
chapter shows that the eLSM can be applied to non-conventional hybrid mesons
in pretty much the same way as for regular $\bar{q}q$ multiplets.

### 5.1 Masses of hybrid mesons

The chiral Lagrangian describing the mass of the homochiral hybrid mesons
reads:

$\displaystyle\mathcal{L}^{\text{mass}}_{\text{hyb}}=\text{Tr}\Big{[}$
$\displaystyle\Big{(}\frac{m_{\text{hyb}}^{2}\,G^{2}}{2\,G_{0}^{2}}+\Delta^{\text{hyb}}\Big{)}\Big{(}L_{\mu}^{\text{hyb}\,2}+R_{\mu}^{\text{hyb}\,2}\Big{)}\Big{]}+\frac{h_{1}^{\text{hyb}}}{2}\text{Tr}\Big{[}\Phi^{\dagger}\Phi\Big{]}\text{Tr}\Big{[}\Big{(}L_{\mu}^{\text{hyb}\,2}+R_{\mu}^{\text{hyb}\,2}\Big{)}\Big{]}$
(5.1)
$\displaystyle+h_{2}^{\text{hyb}}\text{Tr}\Big{[}\Phi^{\dagger}L_{\mu}^{\text{hyb}}L^{\mu\,,\text{hyb}}\Phi+\Phi
R_{\mu}^{\text{hyb}}R^{\mu\,,\text{hyb}}\Phi^{\dagger}\Big{]}+2h_{3}^{\text{hyb}}\text{Tr}\Big{[}\Phi
R^{\mu\,,\text{hyb}}\Phi^{\dagger}L_{\mu}^{\text{hyb}}\Big{]}\text{ ,}$

where
$\Delta^{\text{hyb}}=\text{diag}(\delta_{N}^{\text{hyb}},\delta_{N}^{\text{hyb}},\delta_{S}^{\text{hyb}})$
is the standard contribution due to bare quark masses with
$\delta_{N,S}^{\text{hyb}}\propto m_{n,s}^{2}$. As usual, we sent
$\delta_{N}^{\text{hyb}}=0$. We assume that $\pi_{1}(1600)$ belongs to the
$J^{\mathcal{P}\mathcal{C}}=1^{-+}$ and we set the mass of the chiral partner
as $b_{1}^{\text{hyb}}$ at about $2\,\text{GeV}$. Then, using the previously
determined eLSM parameters from Table 3.5 (and fixing
$\delta^{\text{hyb}}_{S}\simeq\delta_{S}=0.151\text{GeV}^{2}$) we obtain the
hybrid masses in Table 5.1. In particular, the following relations between
chiral partners hold:

$m_{b_{1}^{\text{hyb}}}^{2}=m_{\pi_{1}^{\text{hyb}}}^{2}-2h_{3}^{\text{hyb}}\phi_{N}^{2}\text{
,
}m_{K_{1,B}^{\text{hyb}}}^{2}=m_{K_{1}^{\text{hyb}}}^{2}-\sqrt{2}\phi_{N}\phi_{S}h_{3}^{\text{hyb}}\text{
,
}m_{h_{1S}^{\text{hyb}}}^{2}=m_{\eta_{1,S}^{\text{hyb}}}^{2}-h_{3}^{\text{hyb}}\phi_{S}^{2}\text{
.}$ (5.2)

They are, as usual, proportional to the chiral condensates and the parameter
$h_{3}^{\text{hyb}}$. Note, the heavy $\bar{s}sg$ hybrid
$\eta_{1,S}^{\text{hyb}}$ with a mass of about 1.75 GeV is part of the
lightest hybrid nonet. In 2022, the resonance $\eta_{1}(1855)$ with exotic
quantum numbers $J^{\mathcal{P}\mathcal{C}}=1^{-+}$ has been discovered by the
BESIII collaboration in Ref. [223]. Just shortly after this discovery, in Ref.
[224] it was shown that, when a small but nonnegligible mixing in the
isoscalar sector is considered, the resonance $\eta_{1}(1855)$ can be
interpreted as predominantly $\eta_{1,S}^{\text{hyb}}=\bar{s}sg$.

Mass squares | Analytical expressions | Estimates of masses (MeV)
---|---|---
$m_{\pi_{1}^{\text{hyb}}}^{2}$ | $\big{(}m^{2}_{\text{hyb}}+2\delta_{N}^{\text{hyb}}\big{)}+\frac{h_{3}^{\text{hyb}}\phi_{N}^{2}}{2}+\frac{h_{2}^{\text{hyb}}\phi_{N}^{2}}{2}+\frac{h_{1}^{\text{hyb}}}{2}(\phi_{N}^{2}+\phi_{S}^{2})$ | 1660
$m_{K_{1}^{\text{hyb}}}^{2}$ | $\big{(}m^{2}_{\text{hyb}}+\delta_{N}^{\text{hyb}}+\delta_{S}^{\text{hyb}}\big{)}+\frac{h_{1}^{\text{hyb}}}{2}(\phi_{N}^{2}+\phi_{S}^{2})+\frac{h_{3}^{\text{hyb}}}{\sqrt{2}}\phi_{N}\phi_{S}+\frac{h_{2}^{\text{hyb}}}{4}(\phi_{N}^{2}+2\phi_{S}^{2})$ | $1707$
$m_{\eta_{1,N}^{\text{hyb}}}^{2}$ | $\big{(}m^{2}_{\text{hyb}}+2\delta_{N}^{\text{hyb}}\big{)}+\frac{h_{3}^{\text{hyb}}\phi_{N}^{2}}{2}+\frac{h_{1}^{\text{hyb}}}{2}(\phi_{N}^{2}+\phi_{S}^{2})+\frac{h_{2}^{\text{hyb}}\phi_{N}^{2}}{2}=m_{\pi_{1}^{\text{hyb}}}^{2}$ | $1660$
$m_{\eta_{1,S}^{\text{hyb}}}^{2}$ | $\big{(}m^{2}_{\text{hyb}}+2\delta_{S}^{\text{hyb}}\big{)}+\frac{h_{1}^{\text{hyb}}}{2}(\phi_{N}^{2}+\phi_{S}^{2})+h_{3}^{\text{hyb}}\phi_{S}^{2}+h_{2}^{\text{hyb}}\phi_{S}^{2}$ | $1751$
$m_{b_{1}^{\text{hyb}}}^{2}$ | $\big{(}m^{2}_{\text{hyb}}+2\delta_{N}^{\text{hyb}}\big{)}-\frac{h_{3}^{\text{hyb}}\phi_{N}^{2}}{2}+\frac{h_{2}^{\text{hyb}}\phi_{N}^{2}}{2}+\frac{h_{1}^{\text{hyb}}}{2}(\phi_{N}^{2}+\phi_{S}^{2})$ | 2000
$m_{K_{1,B}^{\text{hyb}}}^{2}$ | $\big{(}m^{2}_{\text{hyb}}+\delta_{N}^{\text{hyb}}+\delta_{S}^{\text{hyb}}\big{)}+\frac{h_{1}^{\text{hyb}}}{2}(\phi_{N}^{2}+\phi_{S}^{2})-\frac{h_{3}^{\text{hyb}}}{\sqrt{2}}\phi_{N}\phi_{S}+\frac{h_{2}^{\text{hyb}}}{4}(\phi_{N}^{2}+2\phi_{S}^{2})$ | $2063$
$m_{h_{1,N}^{\text{hyb}}}^{2}$ | $\big{(}m^{2}_{\text{hyb}}+2\delta_{N}^{\text{hyb}}\big{)}-\frac{h_{3}^{\text{hyb}}\phi_{N}^{2}}{2}+\frac{h_{2}^{\text{hyb}}\phi_{N}^{2}}{2}+\frac{h_{1}^{\text{hyb}}}{2}(\phi_{N}^{2}+\phi_{S}^{2})=m_{b_{1}^{\text{hyb}}}^{2}$ | $2000$
$m_{h_{1,S}^{\text{hyb}}}^{2}$ | $\big{(}m^{2}_{\text{hyb}}+2\delta_{S}^{\text{hyb}}\big{)}-h_{3}^{\text{hyb}}\phi_{S}^{2}+h_{2}^{\text{hyb}}\phi_{S}^{2}+\frac{h_{1}^{\text{hyb}}}{2}(\phi_{N}^{2}+\phi_{S}^{2})$ | $2126$
Table 5.1:  Masses of hybrid mesons within the eLSM. Parameters |  $N_{c}$-scale | Parameters |  $N_{c}$-scale
---|---|---|---
$h_{1,2,3}^{\text{hyb}}$ | $N_{c}^{-2}$ | $m_{\text{hyb}}$ | $N_{c}^{0}$
$\lambda_{1,2}^{\text{hyb}}$ | $N_{c}^{-\frac{1}{2}}$ | $\alpha^{\text{hyb}}$ | $N_{c}^{-1}$
$\beta^{\text{hyb}}$ | $N_{c}^{-1}$ |  |
Table 5.2: Parameters of the hybrid mesons within eLSM.

### 5.2 Decays of hybrid mesons

The resonance $\pi_{1}(1600)$ was observed in the decay channels
$\pi_{1}\rightarrow\rho\pi,b_{1}\pi,f_{1}\pi,\eta^{\prime}\pi,\eta\pi$ [14].
Following the usual steps, a chiral Lagrangian that leads to these decay
channels can be easily constructed:

$\displaystyle\mathcal{L}^{\text{int}}_{\text{hyb}}$
$\displaystyle=i\lambda_{1}^{\text{hyb}}G\text{Tr}\Big{[}L_{\mu}^{\text{hyb}}\Big{(}\Phi^{\mu}\Phi^{\dagger}-\Phi\Phi^{\mu\dagger}\Big{)}+R_{\mu}^{\text{hyb}}\Big{(}\Phi^{\mu\dagger}\Phi-\Phi^{\dagger}\Phi^{\mu}\Big{)}\Big{]}$
(5.3)
$\displaystyle+i\lambda_{2}^{\text{hyb}}\text{Tr}\Big{[}[L_{\mu}^{\text{hyb}},L^{\mu}]\Big{(}\Phi\Phi^{\dagger}\Big{)}+[R_{\mu}^{\text{hyb}}R^{\mu}]\Big{(}\Phi^{\dagger}\Phi\Big{)}\Big{]}$
$\displaystyle+\alpha^{\text{hyb}}\text{Tr}\Big{[}\tilde{L}_{\mu\nu}^{\text{hyb}}\Phi
R^{\mu\nu}\Phi^{\dagger}-\tilde{R}_{\mu\nu}^{\text{hyb}}\Phi^{\dagger}L^{\mu\nu}\Phi\Big{]}$
$\displaystyle+\beta^{\text{hyb}}_{A}\Big{(}\det\Phi-\det\Phi^{\dagger}\Big{)}\text{Tr}\Big{[}L_{\mu}^{\text{hyb}}\Big{(}\partial^{\mu}\Phi\cdot\Phi^{\dagger}-\Phi\cdot\partial^{\mu}\Phi^{\dagger}\Big{)}-R_{\mu}^{\text{hyb}}\Big{(}\partial^{\mu}\Phi^{\dagger}\cdot\Phi-\Phi^{\dagger}\cdot\partial^{\mu}\Phi\Big{)}\Big{]}\text{
,}$

where the terms proportional to $\lambda_{1}^{\text{hyb}}$ and
$\lambda_{2}^{\text{hyb}}$ are dilatation invariant, but the one proportional
to $\alpha^{\text{hyb}}$ (involving the Levi-Civita tensor) and the axial-
symmetry anomalous term proportional to $\beta^{\text{hyb}}_{A}$ break it.

The first three terms of the Lagrangian in (5.3) lead to the decay ratios in
Table 5.3, in which both the ground-state hybrids and their chiral partners
are listed. Among these decay channels, $\pi_{!}(1600)\rightarrow b_{1}\pi$ is
dominant, as well confirmed by data. Another interesting decay channel is the
process $\pi_{1}\rightarrow\rho\pi$, which decays further into three pions,
was observed at COMPASS [217]. The last term in (5.3) contains the interaction
term breaking $U(1)_{A}$ axial symmetry and gives rise to new channels that
show the effect manifestations of chiral anomaly in the exotic mesons sector.
In particular, the eLSM prediction for the decay ratio
$(\pi_{1}\rightarrow\pi\eta^{\prime})/(\pi_{1}\rightarrow\pi\eta)\sim 12.7$ is
roughly comparable with the result $6.8$ of Ref. [244]. We list other
anomalous decay ratios in Table 5.4.

In Ref. [224] a throughout discussion of the masses and strong decays of the
lightest ground-state exotic nonet with $J^{\mathcal{PC}}=1^{-+}$ (including
the not-yet found light $\eta_{1}$ and the $K_{1}$ states). The model, based
on flavor symmetry only, can be seen as a special case of the Lagrangian
(5.3). A rather clear picture emerges, in agreement with both experimental and
lattice QCD results of Ref. [106]. Finally, a detailed discussion of their
radiative production and decays within the same flavor-symmetry based approach
was put forward in Ref. [225].

Decay Ratios | Estimates | Decay Ratios | Estimates
---|---|---|---
$\Gamma_{K_{1}^{\text{hyb}}\rightarrow Kh_{1}(1170)}/\Gamma_{\pi_{1}^{\text{hyb}}\rightarrow\pi b_{1}}$ | 0.050 | $\Gamma_{b_{1}^{\text{hyb}}\rightarrow\pi\omega(1650)}/\Gamma_{\pi_{1}^{\text{hyb}}\rightarrow\pi b_{1}}$ | 0.065
$\Gamma_{K_{1B}^{\text{hyb}}\rightarrow\pi K^{\ast}(1680)}/\Gamma_{\pi_{1}^{\text{hyb}}\rightarrow\pi b_{1}}$ | 0.19 | $\Gamma_{h_{1,N}^{\text{hyb}}\rightarrow\pi\rho(1700)}/\Gamma_{\pi_{1}^{\text{hyb}}\rightarrow\pi b_{1}}$ | 0.16
$\Gamma_{\pi_{1}^{0hyb}\rightarrow\overline{K}^{0}K^{\ast 0}}/\Gamma_{\pi_{1}^{-hyb}\rightarrow\rho^{0}\pi^{-}}$ | $0.61$ | $\Gamma_{\eta_{1N}^{\text{hyb}}\rightarrow\overline{K}^{0}K^{\ast 0}}/\Gamma_{\pi_{1}^{-hyb}\rightarrow\rho^{0}\pi^{-}}$ | $0.61$
$\Gamma_{\eta_{1S}^{\text{hyb}}\rightarrow\overline{K}^{0}K^{\ast 0}}/\Gamma_{\pi_{1}^{-hyb}\rightarrow\rho^{0}\pi^{-}}$ | $1.6$ | $\Gamma_{K_{1}^{0hyb}\rightarrow\overline{K}^{\ast 0}\eta}/\Gamma_{\pi_{1}^{-hyb}\rightarrow\rho^{0}\pi^{-}}$ | $0.0011$
$\Gamma_{b_{1}^{0hyb}\rightarrow\pi^{-}a_{1}^{+}}/\Gamma_{\pi_{1}^{-hyb}\rightarrow\rho^{0}\pi^{-}}$ | $3.8$ | $\Gamma_{b_{1}^{0hyb}\rightarrow\overline{K}_{1}^{0}K^{0}}/\Gamma_{\pi_{1}^{-hyb}\rightarrow\rho^{0}\pi^{-}}$ | $0.60$
$\Gamma_{h_{1N,B}^{\text{hyb}}\rightarrow\overline{K}_{1}^{0}K^{0}}/\Gamma_{\pi_{1}^{-hyb}\rightarrow\rho^{0}\pi^{-}}$ | $0.59$ | $\Gamma_{h_{1S,B}^{\text{hyb}}\rightarrow\overline{K}_{1}^{0}K^{0}}/\Gamma_{\pi_{1}^{-hyb}\rightarrow\rho^{0}\pi^{-}}$ | $1.7801$
$\Gamma_{K_{1,B}^{0hyb}\rightarrow K_{1}^{0}\eta}/\Gamma_{\pi_{1}^{-hyb}\rightarrow\rho^{0}\pi^{-}}$ | $0.010$ | $\Gamma_{K_{1,B}^{0hyb}\rightarrow\overline{K}_{1}^{0}\pi^{0}}/\Gamma_{\pi_{1}^{-hyb}\rightarrow\rho^{0}\pi^{-}}$ | $0.029$
$\Gamma_{K_{1,B}^{0hyb}\rightarrow\overline{K}^{0}a_{1}^{0}}/\Gamma_{\pi_{1}^{-hyb}\rightarrow\rho^{0}\pi^{-}}$ | $0.046$ | $\Gamma_{K_{1}^{0hyb}\rightarrow K^{\ast 0}\pi^{0}}/\Gamma_{\pi_{1}^{-hyb}\rightarrow\rho^{0}\pi^{-}}$ | $0.00022$
Table 5.3: Decay ratios of the hybrid mesons within eLSM. Decay Ratios | Estimates | Decay Ratios | Estimates
---|---|---|---
$\Gamma_{\pi_{1}^{\text{hyb}}\rightarrow\eta^{\prime}\pi}/\Gamma_{\pi_{1}^{\text{hyb}}\rightarrow\eta\pi}$ | $12.7$ | $\Gamma_{K_{1}^{\text{hyb}}\rightarrow K\eta}/\Gamma_{\pi_{1}^{\text{hyb}}\rightarrow\eta\pi}$ | $0.69$
$\Gamma_{K_{1}^{\text{hyb}}\rightarrow K\eta^{\prime}}/\Gamma_{\pi_{1}^{\text{hyb}}\rightarrow\eta\pi}$ | $5.3$ | $\Gamma_{\eta_{1,N}^{\text{hyb}}\rightarrow\eta^{\prime}\eta}/\Gamma_{\pi_{1}^{\text{hyb}}\rightarrow\eta\pi}$ | $2.2$
$\Gamma_{\eta_{1,S}^{\text{hyb}}\rightarrow\eta^{\prime}\eta}/\Gamma_{\pi_{1}^{\text{hyb}}\rightarrow\eta\pi}$ | $1.57$ |  |
Table 5.4: Anomalous decay ratios of hybrid mesons.

## 6 Glueballs within the eLSM

### 6.1 General considerations

According to lattice QCD, glueballs with various quantum numbers are expected
to exist, see Refs. [102, 105, 101] and Table 6.1 for the compilation of the
results of Refs. [102, 105]. These lattice glueball masses have been recently
used to calculate the pressure within the Glueball Resonance Gas model in Ref.
[159]. The pressure has been then compared to the results obtained in
independent and purely thermodynamic lattice QCD simulations [245]. This
comparison shows that the glueball mass spectrum obtained in Ref. [105] is
slightly favored.

The existence of glueballs is confirmed by various models, see e.g. Refs. [55,
181, 246, 68]. However, up to now, no glueball state has been unambiguously
identified, even if some, eventually strong, candidates exist. The eLSM can
help to elucidate some features of glueballs.

The first glueball that we present is the already introduced scalar glueball
$G$ that appears as a dilaton field in the eLSM, see Sec. 6.2 and Refs. [122,
7]. This is the lightest glueball with a mass of about 1.7 GeV. Next, we move
to further glueballs with $\mathcal{C}=1$ (that is, two constituent gluons),
that were studied within the eLSM: the tensor glueball in Sec. 6.3 (Ref.
[128]), and the pseudoscalar glueball in Sec. 6.4 (Ref. [127]), which are the
second and the third lightest according to lattice with masses of about 2.2
GeV and 2.6 GeV respectively. Finally, we present the results for the vector
glueball in Sec. 6.5 (Ref. [129]), which carries $\mathcal{C}=-1$,
representing an example of a three-gluon object.

Figure 6.1: Glueball spectrum obtained in Refs.[158, 105]. $n\,J^{\mathcal{P}\mathcal{C}}$ | M[MeV] |  $n\,J^{\mathcal{P}\mathcal{C}}$ | M[MeV]
---|---|---|---
| Chen et.al. [158] | A & T [105] |  | Chen et.al. [158] | A & T [105]
$\textbf{1\, 0}^{++}$ | 1710(50)(80) | 1653(26) | $\textbf{1\,1}^{--}$ | 3830(40)(190) | 4030(70)
$\textbf{2\,0}^{++}$ |  | 2842(40) | $\textbf{1\,2}^{--}$ | 4010(45)(200) | 3920(90)
$\textbf{3\,0}^{++}$ |  |  | $\textbf{1\,3}^{--}$ | 4200(45)(200) |
$\textbf{1\,2}^{++}$ | 2390(30)(120) | 2376(32) | $\textbf{1\,0}^{+-}$ | 4780(60)(230) |
$\textbf{2\,2}^{++}$ |  | 3300(50) | $\textbf{1\,1}^{+-}$ | 2980(30)(140) | 2944(42)
$\textbf{1\,3}^{++}$ | 3670(50)(180) | 3740(70) | $\textbf{2\,1}^{+-}$ |  | 3800(60)
$\textbf{1 4}^{++}$ |  | 3690(80) | $\textbf{1\,2}^{+-}$ | 4230(50)(200) | 4240(80)
$\textbf{1\,0}^{-+}$ | 2560(35)(120) | 2561(40) | $\textbf{1\,3}^{+-}$ | 3600(40)(170) | 3530(80)
$\textbf{2\,0}^{-+}$ |  | 3540(80) | $\textbf{1\,4}^{+-}$ |  | 4380(80)
$\textbf{1\,2}^{-+}$ | 3040(40)(150) | 3070(60) | $\textbf{1\,1}^{-+}$ |  | 4120(80)
$\textbf{2\,2}^{-+}$ |  | 3970(70) | $\textbf{2\,1}^{-+}$ |  | 4160(80)
$\textbf{3\,2}^{-+}$ |  |  | $\textbf{3\,1}^{-+}$ |  | 4200(90)
Table 6.1: Masses of the glueball states taken from the lattice simulations
Refs. [158, 105].

Before introducing the interaction of glueballs with $\bar{q}q$ fields, we
mention that a composite model for the YM part of QCD is possible by
generalizing the dilaton Lagrangian of Section 2.2 as shown below:

$\displaystyle\mathcal{L}_{Glueballs}=\mathcal{L}^{\text{kin}}+V_{\text{dil}}+\frac{\alpha\,G^{2}}{2}T_{G\,,\mu\nu}T_{G}^{\mu\nu}+\frac{\beta
G^{2}P_{G}^{2}}{2}+\frac{\gamma\,G^{2}}{2}V_{G\,\mu}V_{G}^{\mu}+\mathcal{L}_{T_{G}}^{\text{int}}+\mathcal{L}_{P_{G}}^{\text{int}}+\mathcal{L}_{V_{G}}^{\text{int}}\text{
,}$ (6.1)

where $\mathcal{L}^{\text{kin}}$ contains the kinetic terms, $V_{\text{dil}}$
is the usual dilaton potential, and the fields $P_{G}$, $T_{G}$, and $V_{G}$
refer to the pseudoscalar, tensor, and vector glueballs, respectively. They
acquire a mass term by interacting with the scalar glueball that condenses to
a nonzero v.e.v. $G_{0}=\Lambda_{G}$; the remaining terms are self-
interactions. In agreement with dilatation invariance, this model is built
containing only dilatation invariant interaction terms with the usual
exception of $\Lambda_{G}$ in $V_{\text{dil}}$. As discussed in Ref. [159]
this model is suited to study e.g. the scattering of glueballs in a pure YM
context.

### 6.2 The scalar glueball

The scalar glueball is the lightest gluonic state predicted by lattice QCD
(see Table 6.1) and is one of the hadronic d.o.f of the eLSM emerging as the
excitation of the dilaton field [6, 122]. The eLSM makes predictions for the
lightest glueball state in a chiral framework, completing previous
phenomenological works on the subject [28, 29, 230, 185, 247, 184, 183]. The
results of the study of Ref. [7] shows that the scalar glueball is
predominately contained in the resonance $f_{0}(1710)$. This assignment is in
agreement with the pioneering lattice result of Ref. [247], with the more
recent lattice result of Ref. [103], with other hadronic approaches [28], and
lately also within the holographic approach of Ref. [81].

When expanding the eLSM Lagrangian in the scalar-isoscalar sector, it reduces
to the following terms including the fields ($G,\sigma_{N},\sigma_{S}$)