$\displaystyle\mathcal{L}^{\text{mass}}_{G\sigma_{N}\sigma_{S}}=\mathcal{L}_{\text{dil}}-\frac{m_{0}^{2}}{2}\frac{G^{2}}{G_{0}^{2}}(\sigma_{N}^{2}+\sigma_{S}^{2})-\lambda_{1}(\frac{\sigma_{N}^{2}+\sigma_{S}^{2}}{2})^{2}-\frac{\lambda_{2}}{4}(\frac{\sigma_{N}^{4}}{4}+\sigma_{S}^{4})+h_{\sigma_{N}}\sigma_{N}+h_{\sigma_{S}}\sigma_{S}-\frac{\epsilon_{S}\sigma_{S}^{2}}{2}\text{
.}$ (6.2)

An important remark concerns the $\epsilon_{S}$-term (the one on the very
right), that arises as an additional contribution to the eLSM of the type
$-2\text{Tr}[\hat{\epsilon}\Phi\Phi^{\dagger}]$, with
$\hat{\epsilon}=\text{diag}(\epsilon_{N},\epsilon_{N},\epsilon_{S})$, and
$\epsilon_{N,S}\propto m_{n,s}^{2}$. This is a standard mass term for
(pseudo)scalar mesons, is subleading w.r.t. the term proportional to the
matrix $H$, but is nevertheless important in the scalar-isoscalar sector. As
usual without loss of generality we set $\epsilon_{N}=0$. Also, the parameter
$\lambda_{1}$, that could not be univocally fixed in Sec. 3.4, is here taken
into account. Upon minimizing the potential as in Sec. 3.3, the bare glueball
mass reads

$\displaystyle
m_{G}^{2}=\Big{(}\frac{m_{0}}{G_{0}}\Big{)}^{2}\Big{(}\phi_{N}^{2}+\phi_{S}^{2}\Big{)}+\frac{m_{G}^{2}G_{0}^{2}}{\Lambda^{2}}\Big{(}1+3\ln{\Big{|}\frac{G_{0}}{\Lambda_{G}}\Big{|}}\Big{)}\,.$
(6.3)

In total, 5 parameters ($\Lambda$, $m_{G}$, $\lambda_{1}$, $h_{1}$ and
$\epsilon_{S}$) are relevant in the scalar isoscalar sector. They were fitted
to the experimental quantities of Table 6.2.

Resonances | PDG Mass | eLSM Mass | $\Gamma^{\text{PDG}}_{f_{0}\rightarrow\pi\pi}$ | $\Gamma^{\text{PDG}}_{f_{0}\rightarrow K\overline{K}}$ | $\Gamma^{\text{eLSM}}_{f_{0}\rightarrow\pi\pi}$ | $\Gamma^{\text{eLSM}}_{f_{0}\rightarrow K\overline{K}}$
---|---|---|---|---|---|---
$f_{0}(1370)$ | 1200-1500 | 1444 | $-$ | $-$ | $423.6$ | $117.5$
$f_{0}(1500)$ | $1505\pm 6$ | 1534 | $38.04\pm 4.95$ | $9.37\pm 1.69$ | 39.2 | 9.1
$f_{0}(1710)$ | $1720\pm 6$ | 1750 | $29.3\pm 6.5$ | $71.4\pm 29.1$ | 28.3 | 73.4
Table 6.2: Fitting data for scalar mesons and glueball.

The admixtures as calculated in Ref. [7] read (see also Eq. (2.123)):

$\left(\begin{array}[c]{c}f_{0}(1370)\\\ f_{0}(1500)\\\
f_{0}(1710)\end{array}\right)=\left(\begin{array}[c]{ccc}-0.91&0.24&-0.33\\\
0.30&0.94&-0.17\\\
-0.27&0.26&0.93\end{array}\right)\left(\begin{array}[c]{c}\sigma_{N}\\\
\sigma_{S}\\\ G\end{array}\right)\text{ ,}$ (6.4)

that leads to the probabilities glueball contents of $f_{0}$ resonances in
Table 6.3. Interestingly, the parameter $\Lambda_{G}$ turns out to be large,
of the order of 3 GeV. As in Sec. 3.4, various local minima exist, yet all of
those that generate a satisfactory scalar phenomenology are such that
$\Lambda_{G}\gtrsim 1$ GeV, thus larger than the right edge of the interval
quoted in Sec. 2.2. This outcome is in agreement with the study in Ref. [157]:
a small scale $\Lambda_{G}$ generates a very broad scalar glueball. Of course,
a viable explanation would be to admit that the scalar glueball is too broad
to be detected experimentally (similarly to the axial-tensor mesons in Sec.
4.2). Yet such a broad glueball would be also in disagreement with
large-$N_{c}$ expectations. Here, we conclude by noticing that this specific
point of the dilaton potential, the value of the dimensionful parameter
$\Lambda_{G}$, is not yet fully clarified. A possible way to determine it
would be to measure the scattering of two scalar glueballs in lattice YM
theory, see the discussion in Ref. [163].

Resonances | $\sigma_{N}$ | $\sigma_{S}$ | G
---|---|---|---
$f_{0}(1370)$ | 83% | 6% | 11%
$f_{0}(150)$ | 9% | 88% | 3%
$f_{0}(1710)$ | 8% | 6% | 86%
Table 6.3: Results for the $\bar{q}q$-meson and glueball mixtures of the $f_{0}$-states.  Parameter | Value | Parameter | Value
---|---|---|---
$\Lambda_{G}$ |  $3297\,[$MeV$]$ | $m_{G}$ |  $1525\,[$MeV$]$
$\lambda_{1}$ | $6.25$ | $h_{1}$ | $-3.22$
$\epsilon_{S}$ |  $0.4212\times 10^{6}\,[$MeV${}^{2}]$ |  |
Table 6.4: Parameter values for the $J^{\mathcal{P}\mathcal{C}}=0^{++}$
sector.

Further predictions of the approach are reported in Table 6.5.

Decay Channel | eLSM [MeV] | PDG [MeV] | Decay Channel | eLSM [MeV] | PDG [MeV]
---|---|---|---|---|---
$f_{0}(1370)\rightarrow K\bar{K}$ | $117.5$ | - | $f_{0}(1370)\rightarrow\eta\eta$ | $43.3$ | -
$f_{0}(1370)\rightarrow\rho\rho\rightarrow 4\pi$ | $13.8$ | - | $f_{0}(1500)\rightarrow\eta\eta$ | $4.7$ | $5.56\pm 1.34$
$f_{0}(1500)\rightarrow\rho\rho\rightarrow 4\pi$ | $0.2$ | $>54.0\pm 7.1$ | $f_{0}(1710)\rightarrow\eta\eta$ | $57.9$ | $34.3\pm 17.6$
$f_{0}(1710)\rightarrow\rho\rho\rightarrow 4\pi$ | $0.5$ | - |  |  |
Table 6.5: Results of the eLSM compared to the PDG.

The eLSM implies that $f_{0}(1370)$ decays predominantly into two pions with a
decay width of about $400$ MeV, in agreement with its interpretation as
predominantly nonstrange $\bar{q}q$ state [121, 122]. This is in qualitative
agreement with the experimental analysis of Ref. [248], where
$\Gamma_{f_{0}(1370)\rightarrow\pi\pi}=325$ MeV,
$\Gamma_{f_{0}(1370)\rightarrow 4\pi}\approx 50$ MeV, and
$\Gamma_{f_{0}(1370)\rightarrow\eta\eta}/\Gamma_{f_{0}(1370)\rightarrow\pi\pi}=0.19\pm
0.07$. Moreover, the decay channel $f_{0}(1500)\rightarrow\eta\eta$ is in good
agreement with the PDG, while the decay channel
$f_{0}(1710)\rightarrow\eta\eta$ is slightly larger than but compatible with
the PDG value, see Table 6.5.

### 6.3 The tensor glueball

The second lightest glueball is the tensor glueball
($J^{\mathcal{P}\mathcal{C}}=2^{++}$), see Table 6.1. As in the scalar sector,
there are various isoscalar $J^{\mathcal{P}\mathcal{C}}=2^{++}$ resonances
listed in PDG. We show them in Table 6.6.

Resonances | Masses (MeV) | Decay Widths (MeV) | Decay Channels
---|---|---|---
$f_{2}(1910)$ | $1900\pm 9$ | $167\pm 21$ |  $\pi\pi$, $KK$, $\eta\eta$, $\omega\omega$, $\eta\eta^{\prime}$, $\eta^{\prime}\eta^{\prime}$, $\rho\rho$, $a_{2}(1320)\pi$, $f_{2}(1270)\eta$
$f_{2}(1950)$ | $1936\pm 12$ | $464\pm 24$ |  $\pi\pi$, $KK$, $\eta\eta$, $K^{*}K^{*}$, $4\pi$
$f_{2}(2010)$ | $2011^{+60}_{-80}$ | $202\pm 60$ |  $KK$, $\phi\phi$
$f_{2}(2150)$ | $2157\pm 12$ | $152\pm 30$ |  $\pi\pi$, $\eta\eta$, $KK$, $a_{2}(1320)\pi$, $f_{2}(1270)\eta$
$f_{J}(2220)$ | $2231.1\pm 3.5$ | $23^{+8}_{-7}$ | $\eta\eta^{\prime}$
$f_{2}(2300)$ | $2297\pm 28$ | $149\pm 41$ |  $KK$, $\phi\phi$
$f_{2}(2340)$ | $2345^{+50}_{-40}$ | $322^{+70}_{-60}$ |  $\eta\eta$, $\phi\phi$
Table 6.6: Spin-2 resonances heavier than 1.9 GeV listed in PDG [13].

The chirally invariant eLSM interaction Lagrangian describing the decay of the
tensor glueball reads:

$\displaystyle\mathcal{L}_{T_{G}}^{\text{int}}=\lambda_{T_{G}}^{(1)}T_{G\,,\mu\nu}\Big{(}\text{Tr}\Big{[}\\{L^{\mu},L^{\nu}\\}\Big{]}+\text{Tr}\Big{[}\\{R^{\mu},R^{\nu}\\}\Big{]}\Big{)}+\lambda_{T_{G}}^{(2)}T_{G\,,\mu\nu}\Big{(}\text{Tr}\Big{[}\Phi\textbf{R}^{\mu\nu}\Phi^{\dagger}\Big{]}+\text{Tr}\Big{[}\Phi^{\dagger}\textbf{L}^{\mu\nu}\Phi\Big{]}\Big{)}\,.$
(6.5)

The corresponding decay ratios are listed in Table 6.7, where the mass of the
tensor glueball was set to 2.2 GeV, in agreement with the analysis of Ref.
[249]. We observe that the dominant decay channels are the vector-vector decay
products, especially $\rho\rho$ and $K^{\star}\overline{K}^{\star}$.
Large-$N_{c}$ analyses of the tensor glueball decaying into $\pi\pi$
($\Gamma_{G_{2}\rightarrow\pi\pi}\simeq 15$ MeV) is comparable to the one
obtained within the so-called Witten-Sakai-Sugimoto model [81] (
$\Gamma_{G_{2}\rightarrow\pi\pi}\simeq 28$ MeV).

Decay Ratio | eLSM | Decay Ratio | eLSM | Decay Ratio | eLSM
---|---|---|---|---|---
$\frac{T_{G}\rightarrow\bar{K}\,K}{T_{G}\rightarrow\pi\,\pi}$ | $0.4$ | $\frac{T_{G}\rightarrow\rho\,\rho}{T_{G}\rightarrow\pi\,\pi}$ | $55$ | $\frac{T_{G}\rightarrow a_{1}(1260)\,\pi}{T_{G}\rightarrow\pi\,\pi}$ | $0.24$
$\frac{T_{G}\rightarrow\eta\,\eta}{T_{G}\rightarrow\pi\,\pi}$ | $0.1$ | $\frac{T_{G}\rightarrow\bar{K}^{\ast}\,\bar{K}^{\ast}}{T_{G}\rightarrow\pi\,\pi}$ | $46$ | $\frac{T_{G}\rightarrow K_{1\,,A}\,K}{T_{G}\rightarrow\pi\,\pi}$ | $0.08$
$\frac{T_{G}\rightarrow\eta\,\eta^{\prime}}{T_{G}\rightarrow\pi\,\pi}$ | $0.004$ | $\frac{T_{G}\rightarrow\omega\,\omega}{T_{G}\rightarrow\pi\,\pi}$ | $18$ | $\frac{T_{G}\rightarrow f_{1}(1285)\,\eta}{T_{G}\rightarrow\pi\,\pi}$ | $0.02$
$\frac{T_{G}\rightarrow\eta^{\prime}\,\eta^{\prime}}{T_{G}\rightarrow\pi\,\pi}$ | $0.006$ | $\frac{T_{G}\rightarrow\phi\,\phi}{T_{G}\rightarrow\pi\,\pi}$ | $6$ | $\frac{T_{G}\rightarrow f_{1}(1285)\,\eta}{T_{G}\rightarrow\pi\,\pi}$ | $0.02$
|  |  |  | $\frac{T_{G}\rightarrow f_{1}(1420)\,\eta}{T_{G}\rightarrow\pi\,\pi}$ | $0.01$
Table 6.7: Decay ratios of the tensor glueball within eLSM.

In Table 6.8 we also compared the theoretical predictions to the experimental
data for some tensor $2^{++}$ resonances listed in PDG. It shows that the
relatively broad resonance $f_{2}(1950)$ is the favored tensor glueball
candidate despite having a mass slightly smaller than the quenched LQCD
predictions in Table 6.1. Namely, unquenching effects may lead to lowering the
masses of the tensor glueball.

Resonances | Branching Ratios | PDG | eLSM
---|---|---|---
$f_{2}(1910)$ | $\rho(770)\rho(770)/\omega(782)\omega(782)$ | $2.6\pm 0.4$ | $3.1$
$f_{2}(1910)$ | $f_{2}(1270)\eta/a_{2}(1320)\pi$ | $0.09\pm 0.05$ | $0.07$
$f_{2}(1910)$ | $\eta\eta/\eta\eta^{\prime}(958)$ | $<0.05$ | $\sim 8$
$f_{2}(1910)$ | $\omega(782)\omega(782)/\eta\eta\prime(958)$ | $2.6\pm 0.6$ | $\sim 200$
$f_{2}(1950)$ | $\eta\eta/\pi\pi$ | $0.14\pm 0.05$ | $0.081$
$f_{2}(1950)$ | $K\overline{K}/\pi\pi$ | $\sim 0.8$ | $0.32$
$f_{2}(1950)$ | $4\pi/\eta\eta$ | $>200$ | $>700$
$f_{2}(2150)$ | $f_{2}(1270)\eta/a_{2}(1320)\pi$ | $0.79\pm 0.11$ | $0.1$
$f_{2}(2150)$ | $K\overline{K}/\eta\eta$ | $1.28\pm 0.23$ | $\sim 4$
$f_{2}(2150)$ | $\pi\pi/\eta\eta$ | $<0.33$ | $\sim 10$
$f_{J}(2220)$ | $\pi\pi/K\overline{K}$ | $1.0\pm 0.5$ | $\sim 2.5$
Table 6.8: Test of spin-2 resonances as the tensor glueball.

### 6.4 The pseudoscalar glueball

The pseudoscalar glueball is the third-lightest glueball [250]. Recently,
BESIII measured the pseudoscalar ($0^{-+}$) resonances $X(2370)$ [251, 252]
and $X(2600)$ [253]. The pseudoscalar glueball field $P_{G}$ couples in a
chirally invariant but $U(1)_{A}$ breaking way to light mesons [127]:

$\mathcal{L}_{P_{G}}^{\text{int}}=ic_{P_{G}}P_{G}\left(\text{{det}}\Phi-\text{{det}}\Phi^{{\dagger}}\right)\text{
.}$ (6.6)

The branching ratios of $P_{G}$ relative to the total decay width of the
pseudoscalar glueball $\Gamma_{P_{G}}^{tot}$ are reported in Table 6.9 for two
choices of the pseudoscalar masses: $M_{P_{G}}=2.6$ GeV predicted by lattice
QCD [158, 105] and in agreement with $X(2600)$, and $M_{P_{G}}=2.37$ GeV, in
agreement with the $X(2370)$.

Quite remarkably, the eLSM + DIG approach of Ref. [49] could estimate the
coupling constant entering in Eq. (6.6): $c_{P_{G}}\approx 11$. Namely, while
chiral symmetry and its anomalous breaking dictate the form of the Lagrangian
that was already presented in Ref. [127] (and in a similar form even before in
Ref. [110]), the microscopic DGI method allows to estimate the actual value of
the coupling constants. This is, evidently, an important advance.

As a consequence, one gets $\Gamma(P_{G}\rightarrow K\bar{K}\pi)\approx 0.24$
GeV and $\Gamma(P_{G}\rightarrow\pi\pi\eta^{\prime})\approx 0.05$ GeV.
Notably, both resonances $X(2370)$ and $X(2600)$ have been seen by BESIII in
the $\pi\pi\eta^{\prime}$ channel. Our results support the interpretation that
these resonances may contain a large gluonic amount, with a decay controlled
(and in some channels enhanced) by the chiral anomaly. We note that the mass
found in lattice QCD [105] agrees with the identification of $X(2600)$ as a
predominantly gluonic, but further studies are needed.

Quantity |  $M_{P_{G}}=2.6$ GeV |  $M_{P_{G}}=2.37$ GeV
---|---|---
$\Gamma_{P_{G}\rightarrow KK\eta}/\Gamma_{P_{G}}^{tot}$ | $0.049$ | $0.043$
$\Gamma_{P_{G}\rightarrow KK\eta^{\prime}}/\Gamma_{P_{G}}^{tot}$ | $0.019$ | $0.011$
$\Gamma_{P_{G}\rightarrow\eta\eta\eta}/\Gamma_{P_{G}}^{tot}$ | $0.016$ | $0.013$
$\Gamma_{P_{G}\rightarrow\eta\eta\eta^{\prime}}/\Gamma_{P_{G}}^{tot}$ | $0.0017$ | $0.00082$
$\Gamma_{P_{G}\rightarrow\eta\eta^{\prime}\eta^{\prime}}/\Gamma_{\tilde{G}}^{tot}$ | $0.00013$ | $0$
$\Gamma_{P_{G}\rightarrow KK\pi}/\Gamma_{P_{G}}^{tot}$ | $0.47$ | $0.47$
$\Gamma_{P_{G}\rightarrow\eta\pi\pi}/\Gamma_{P_{G}}^{tot}$ | $0.16$ | $0.17$
$\Gamma_{P_{G}\rightarrow\eta^{\prime}\pi\pi}/\Gamma_{P_{G}}^{tot}$ | $0.095$ | $0.090$
Table 6.9: Branching ratios for the pseudoscalar glueball within eLSM.

In conclusion, we predict that $KK\pi$ is the dominant decay channel, followed
by sizable $\eta\pi\pi$ and $\eta^{\prime}\pi\pi$ decay channels (16% and 10%
respectively). These are simple and testable theoretical predictions which can
be helpful in future experimental searches at the PANDA experiment [254],
where glueballs can directly form in the proton-antiproton fusion process. In
particular, the expected mass of the pseudoscalar glueball corresponds to the
low range for particle formation in the PANDA experiment. The pseudoscalar
glueball and its radial excitation were further studied within eLSM in Refs.
[255, 256].

### 6.5 The vector glueball

The vector glueball is an example of a glueball with $\mathcal{C}=-1$, which
therefore contains at least three constituent gluons. As a consequence, its
mass is larger and lies close to 3.8 GeV, see Table 6.1.

The vector glueball has been coupled to the eLSM in Ref. [129]. Following that
work, here we present the chirally invariant interaction terms describing the
coupling of the vector glueball field $V_{G}$ to various quark-antiquark
multiplets introduced in Sec. 2.4:

$\displaystyle\mathcal{L}^{int}_{V_{G}}=\lambda_{V_{G}}^{(1)}G\,V_{G\,,\mu}\text{Tr}\left[\Phi^{\dagger}\Phi^{\mu}+\Phi^{\mu{\dagger}}\Phi\right]+\lambda_{V_{G}}^{(2)}V_{G\,,\mu}\text{Tr}\left[L^{\mu}\Phi\Phi^{\dagger}+R^{\mu}\Phi^{\dagger}\Phi\right]+\lambda_{V_{G}}^{(3)}\varepsilon_{\mu\nu\rho\sigma}\partial^{\rho}V_{G}^{\sigma}\text{Tr}\left[L^{\mu}\Phi
R^{\nu}\Phi^{\dagger}\right]$ (6.7)

In the first two interaction terms the main decay modes are $V_{G}\rightarrow
b_{1}\pi\rightarrow\omega\pi\pi$, $V_{G}\rightarrow\omega\pi\pi$, and
$V_{G}\rightarrow\pi KK^{\star}(892)$. Also the decays involving kaons and
$K_{1}(1270)$ and $K_{1}(1400)$ are sizable. An interesting small but peculiar
decay emerging from the first term involves the scalar glueball: $G\equiv
f_{0}(1710)$, leading to $\frac{\Gamma_{V_{G}\rightarrow
f_{0}(1710)b_{1}\pi}}{\Gamma_{V_{G}\rightarrow b_{1}\pi}}=3.9\cdot 10^{-6}$.
Thus, two glueballs are present in this process.

The third interaction term, which violates dilatation invariance, predicts
decays into vector-pseudoscalar pairs. The dominant decay modes are into
$\rho\pi$, $KK^{\ast}(892)$, and $\rho a_{1}(1230)$ (with increasing
strength).

Unfortunately, we cannot determine the coupling constants in the present
framework, but we calculate various decay ratios for the three interaction
terms. The results are presented in Table 6.10.

Extending this analyses to the charmed meson sector we may estimate the
following ratio $\frac{\Gamma_{V_{G}\rightarrow
DD}}{\Gamma_{V_{G}\rightarrow\omega\pi\pi}}\approx 0.029$ shows that the
$\bar{D}D$ mode is also expected to be small. This result is important because
it shows that the vector glueball, even if according to lattice QCD has a mass
above the $\bar{D}D$ threshold, decays quite weakly into charmed mesons.

Decay Ratio | Value | Decay Ratio | Value | Decay Ratio | Value
---|---|---|---|---|---
$\frac{{}_{V_{G}\rightarrow\eta h_{1}(1170)}}{V_{G}\rightarrow b_{1}\pi}$ | 0.17 | $\frac{{}_{V_{G}\rightarrow KK^{\ast}(892)}}{{}_{V_{G}\rightarrow\rho\pi}}$ | 1.3 | $\frac{{}_{V_{G}\rightarrow KK\rho}}{{}_{V_{G}\rightarrow\omega\pi\pi}}$ | 0.50
$\frac{{}_{V_{G}\rightarrow\eta h_{1}(1380)}}{{}_{V_{G}\rightarrow b_{1}\pi}}$ | 0.11 | $\frac{{}_{V_{G}\rightarrow\eta\omega}}{{}_{V_{G}\rightarrow\rho\pi}}$ | 0.16 | $\frac{{}_{V_{G}\rightarrow KK\omega}}{{}_{V_{G}\rightarrow\omega\pi\pi}}$ | 0.17
$\frac{{}_{V_{G}\rightarrow\eta^{\prime}h_{1}(1170)}}{{}_{V_{G}\rightarrow b_{1}\pi}}$ | 0.15 | $\frac{{}_{V_{G}\rightarrow\eta^{\prime}\omega}}{{}_{V_{G}\rightarrow\rho\pi}}$ | 0.13 | $\frac{{}_{V_{G}\rightarrow KK\phi}}{{}_{V_{G}\rightarrow\omega\pi\pi}}$ | 0.21
$\frac{{}_{V_{G}\rightarrow\eta^{\prime}h_{1}(1380)}}{{}_{V_{G}\rightarrow b_{1}\pi}}$ | 0.10 | $\frac{{}_{V_{G}\rightarrow\eta\phi}}{{}_{V_{G}\rightarrow\rho\pi}}$ | 0.21 | $\frac{V_{G}\rightarrow\pi KK^{\ast}(892)}{{}_{V_{G}\rightarrow\omega\pi\pi}}$ | 1.2
$\frac{{}_{V_{G}\rightarrow KK_{1}(1270)}}{{}_{V_{G}\rightarrow b_{1}\pi}}$ | 0.75 | $\frac{{}_{V_{G}\rightarrow\eta^{\prime}\phi}}{{}_{V_{G}\rightarrow\rho\pi}}$ | 0.18 | $\frac{{}_{V_{G}\rightarrow\eta\eta\omega}}{{}_{V_{G}\rightarrow\omega\pi\pi}}$ | 0.064
$\frac{{}_{V_{G}\rightarrow KK_{1}(1400)}}{{}_{V_{G}\rightarrow b_{1}\pi}}$ | 0.30 | $\frac{{}_{V_{G}\rightarrow\rho a_{1}(1230)}}{{}_{V_{G}\rightarrow\rho\pi}}$ | 1.8 | $\frac{{}_{V_{G}\rightarrow\eta\eta^{\prime}\omega}}{{}_{V_{G}\rightarrow\omega\pi\pi}}$ | 0.019
$\frac{{}_{V_{G}\rightarrow K_{0}^{\ast}(1430)K^{\ast}(1680)}}{{}_{V_{G}\rightarrow b_{1}\pi}}$ | 0.20 | $\frac{{}_{V_{G}\rightarrow}\omega f_{1}(1285)}{{}_{V_{G}\rightarrow\rho\pi}}$ | 0.55 | $\frac{{}_{V_{G}\rightarrow\eta\eta\phi}}{{}_{V_{G}\rightarrow\omega\pi\pi}}$ | 0.039
$\frac{{}_{V_{G}\rightarrow a_{0}(1450)\rho(1700)}}{{}_{V_{G}\rightarrow b_{1}\pi}}$ | 0.14 | $\frac{{}_{V_{G}\rightarrow}\omega f_{1}(1420)}{{}_{V_{G}\rightarrow\rho\pi}}$ | 0.82 | $\frac{{}_{V_{G}\rightarrow\eta\eta^{\prime}\phi}}{{}_{V_{G}\rightarrow\omega\pi\pi}}$ | 0.011
Table 6.10: Decay ratios of the vector glueball within eLSM.

## 7 Other topics

In this section, we enlarge/modify the eLSM along 3 different directions: (i)
the inclusion of isospin breaking effects; (ii) an example of a radially
excited multiplet of (pseudo)scalar mesons; (iii) the extension to the four-
flavor case.

### 7.1 Isospin breaking

Isospin symmetry is broken in Nature, i.e. the masses of the up and down
quarks are different. Since this effect is of the order of a few MeV, it is
neglected in many cases, as in the previous sections. As a consequence of the
$m_{u}-m_{d}$ mass difference, the masses of the charged and neutral mesons of
a given isospin multiplet are slightly different. In addition, there is
another non-QCD isospin breaking effect, the electromagnetic interaction,
which due to the different quark charges also causes a difference between the
charged and neutral masses. This has been studied first in a vector meson
extended effective model in Ref. [257] and very recently in eLSM in Ref.
[131]. In the latter, besides the modification of the particle masses, the
decay widths, the PCAC relation, the role of the electromagnetic mass
contributions and the violation of Dashen’s theorem [258, 259] are also
investigated. Technically, the isospin violation can be incorporated into the
eLSM by changing the external fields (see (3.19) and (3.26)) to

$\displaystyle H$
$\displaystyle=H_{0}t_{0}+H_{3}t_{3}+H_{8}t_{8}=\frac{1}{2}\mathop{}\\!\mathrm{diag}(h_{0N}+h_{03},h_{0N}h_{03},\sqrt{2}h_{0S})\;,$
(7.1) $\displaystyle\Delta$
$\displaystyle=\Delta_{0}t_{0}+\Delta_{3}t_{3}+\Delta_{8}t_{8}=\mathop{}\\!\mathrm{diag}(\delta_{u},\delta_{d},\delta_{S})\text{
,}$ (7.2)

and assume that $h_{03}\neq 0$ and $\delta_{3}\equiv\delta_{u}-\delta_{d}\neq
0$. In this case, besides $\sigma_{N}$ and $\sigma_{S}$, $a_{0}^{0}$ can also
have non-zero vacuum expectation value, denoted as
$\phi_{3}\equiv\braket{a_{0}^{0}}$, for a total of 3 scalar condensates. The
value of $\phi_{3}$ in the vacuum is determined by minimizing the modified
classical potential $V_{G\Phi}(G=G_{0},\phi_{N},\phi_{S},\phi_{3},0,...,0)$
after also shifting the $a_{0}^{0}$ field by its expectation value, i.e.
$a_{0}^{0}\rightarrow\phi_{3}+a_{0}^{0}$. The classical potential is given by

$\displaystyle
V_{G\Phi}(G=G_{0},\sigma_{N}=\phi_{N},\sigma_{S}=\phi_{S},a_{0}^{0}=\phi_{3},0,...,0)$
$\displaystyle=\frac{m_{0}^{2}}{2}\left[\phi_{N}^{2}+\phi_{S}^{2}+(\phi_{3})^{2}\right]+\frac{\lambda_{1}}{4}\left[\phi_{N}^{2}+\phi_{S}^{2}+(\phi_{3})^{2}\right]^{2}$
(7.3)
$\displaystyle+\frac{\lambda_{2}}{4}\left[\frac{\phi_{N}^{4}}{2}+3\phi_{N}^{2}(\phi_{3})^{2}+\frac{(\phi_{3})^{4}}{2}+\phi_{S}^{4}\right]-h_{0N}\phi_{N}-h_{0S}\phi_{S}-h_{03}\phi_{3}\;,$

while the field equations – given by $\partial
V_{G\Phi}/\partial\phi_{N/S/3}=0$ – are

$\displaystyle h_{0N}$
$\displaystyle=\phi_{N}\left\\{m_{0}^{2}+\lambda_{1}\left[\phi_{N}^{2}+\phi_{S}^{2}+(\phi_{3})^{2}\right]+\frac{\lambda_{2}}{2}\left[\phi_{N}^{2}+3(\phi_{3})^{2}\right]\right\\}\;,$
(7.4) $\displaystyle h_{0S}$
$\displaystyle=\phi_{S}\left\\{m_{0}^{2}+\lambda_{1}\left[\phi_{S}^{2}+\phi_{N}^{2}+(\phi_{3})^{2}\right]+\lambda_{2}\phi_{S}^{2}\right\\}\;,$
(7.5) $\displaystyle h_{03}$
$\displaystyle=\phi_{3}\left\\{m_{0}^{2}+\lambda_{1}\left[\phi_{N}^{2}+\phi_{S}^{2}+(\phi_{3})^{2}\right]+\frac{\lambda_{2}}{2}\left[3\phi_{N}^{2}+(\phi_{3})^{2}\right]\right\\}\;.$
(7.6)

The tree-level masses and decay widths can be calculated similarly to the
isospin-symmetric case, i.e. from the quadratic and triple coupling terms of
the Lagrangian written with the shifted fields. In the isospin violating case,
the expressions for the masses and decay widths are generally quite long due
to the various mixings (see their explicit form in Ref. [131]). For the sake
of illustration, we show here only the mass expressions for the neutral and
charged kaons,

$\displaystyle m_{K^{\pm}}^{2}$
$\displaystyle=Z_{K^{\pm}}^{2}\left(m_{0}^{2}+\lambda_{1}(\phi_{N}^{2}+\phi_{3}^{2}+\phi_{S}^{2})+\lambda_{2}\left[-\frac{1}{\sqrt{2}}(\phi_{N}+\phi_{3})\phi_{S}+\frac{(\phi_{N}+\phi_{3})^{2}}{2}+\phi_{S}^{2}\right]\right)+\bar{m}^{2}_{\text{em}}\;,$
(7.7) $\displaystyle m_{K^{0}}^{2}$
$\displaystyle=Z_{K^{0}}^{2}\left(m_{0}^{2}+\lambda_{1}(\phi_{N}^{2}+\phi_{3}^{2}+\phi_{S}^{2})+\lambda_{2}\left[-\frac{1}{\sqrt{2}}(\phi_{N}-\phi_{3})\phi_{S}+\frac{(\phi_{N}-\phi_{3})^{2}}{2}+\phi_{S}^{2}\right]\right)\;,$
(7.8)

where

$Z_{K^{\pm}}=\frac{2m_{K_{1}^{\pm}}}{\sqrt{4m_{K_{1}^{\pm}}^{2}-g_{1}^{2}(\phi_{N}+\phi_{3}+\sqrt{2}\phi_{S})^{2}}}\text{
,
}Z_{K^{0}}=\frac{2m_{K_{1}^{0}}}{\sqrt{4m_{K_{1}^{0}}^{2}-g_{1}^{2}(\phi_{N}-\phi_{3}+\sqrt{2}\phi_{S})^{2}}}\text{
.}$ (7.9)

The quantity $\bar{m}^{2}_{\text{em}}$ represents a phenomenological
electromagnetic mass contribution to the charged kaon mass. As can be seen in
the above expressions, the difference is also due to the differing $Z$
factors. Regarding the decay widths, similar to the isospin symmetric case, we
show here the explicit form of the $\rho\to\pi\pi$ decays,

$\Gamma_{\rho^{0}\to\pi^{+}\pi^{-}}=\frac{k_{\rho^{0}}^{3}}{6\pi
M_{\rho^{0}}^{2}}\left|B^{\rho}_{1}+\frac{1}{2}C^{\rho}_{1}M_{\rho^{0}}^{2}\right|^{2}\text{
, }\Gamma_{\rho^{\pm}\to\pi^{\pm}\pi^{0}}=\frac{k_{\rho^{\pm}}^{3}}{24\pi
M_{\rho^{\pm}}^{2}}\left|D^{\rho}_{1}-D^{\rho}_{2}+F^{\rho}_{1}M_{\rho^{\pm}}^{2}\right|^{2}\text{
,}$ (7.10)

where

$\displaystyle B^{\rho}_{1}$ $\displaystyle=$ $\displaystyle
Z_{\pi^{\pm}}^{2}\left\\{\left[g_{1}+\phi_{N}\left(h_{3}-g_{1}^{2}\right)w_{a_{1}^{\pm}}\right]\cos\vartheta_{V}-\frac{2}{3}\phi_{3}(h_{2}+h_{3})w_{a_{1}^{\pm}}\sin\vartheta_{V}\right\\}\;,$
(7.11) $\displaystyle C^{\rho}_{1}$ $\displaystyle=$ $\displaystyle-
g_{2}Z_{\pi^{\pm}}^{2}w_{a_{1}^{\pm}}^{2}\cos\vartheta_{V}$ (7.12)
$\displaystyle D^{\rho}_{1}$ $\displaystyle=$ $\displaystyle
Z_{\pi^{\pm}}\left\\{\left[g_{1}+\phi_{N}\left(h_{3}-g_{1}^{2}\right)w_{a_{1}^{\pm}}\right]{\mathds{O}_{P}}_{21}-\phi_{3}\big{(}h_{3}-g_{1}^{2}\big{)}w_{a_{1}^{\pm}}{\mathds{O}_{P}}_{11}\right\\}\;,$
(7.13) $\displaystyle D^{\rho}_{2}$ $\displaystyle=$ $\displaystyle-
Z_{\pi^{\pm}}\left[g_{1}{\mathds{O}_{P}}_{21}+\phi_{N}\big{(}h_{3}-g_{1}^{2}\big{)}\big{(}w_{\eta}^{a}{\mathds{O}_{P}}_{11}+w_{\pi}^{a}{\mathds{O}_{P}}_{21}\big{)}-\phi_{3}\big{(}h_{2}-h_{3}+2g_{1}^{2}\big{)}\big{(}w_{\eta}^{f}{\mathds{O}_{P}}_{11}+w_{\pi}^{f}{\mathds{O}_{P}}_{21}\big{)}\right]\;,$
(7.14) $\displaystyle F^{\rho}_{1}$ $\displaystyle=$ $\displaystyle-
g_{2}Z_{\pi^{\pm}}w_{a_{1}^{\pm}}\big{(}w_{\eta}^{a}{\mathds{O}_{P}}_{11}+w_{\pi}^{a}{\mathds{O}_{P}}_{21}\big{)},\quad
F^{\rho}_{2}=-F^{\rho}_{1}\;,$ (7.15)

and

$k_{\rho^{0}}=\frac{1}{2}\sqrt{M_{\rho^{0}}^{2}-4M_{\pi^{\pm}}^{2}}\text{ ,
}k_{\rho^{\pm}}=\frac{\sqrt{(M_{\rho^{\pm}}^{2}-M_{\pi^{\pm}}^{2}-M_{\pi^{0}}^{2})^{2}-4M_{\pi^{\pm}}^{2}M_{\pi^{0}}^{2}}}{2M_{\rho^{\pm}}}\text{
,}$ (7.16)

where $\vartheta_{V}$ is the mixing angle between $\rho^{0}$ and $\omega_{N}$
(not to be confused with the previous strange-nonstrange mixing angle
$\beta_{v}$ defined in Eq. 2.129). The various coefficients $w_{X}$ refer to
the (axial-)vector field shifts given in section II of [131], $\mathds{O}_{P}$
is the transformation matrix between the pseudoscalar physical $\pi^{0}$,
$\eta$, $\eta^{\prime}$ fields and the original $\eta_{N}$, $\pi^{0}$,
$\eta_{S}$ fields of the Lagrangian, while the various $M_{X}$ are the
physical masses. The expressions for the charged and neutral $\rho\to\pi\pi$
decay widths look more complicated than their isospin symmetric counterparts
of Eq. (3.45), but it can be shown that as $\phi_{3}\to 0$ and $\delta_{3}\to
0$ the charged and neutral expressions coincide with each other and with Eq.
(3.45). By using the tree-level expressions for the charged and neutral
masses, decay widths and pion and kaon decay constants, a global fit can be
performed similarly as in the isospin symmetric case. An illustrative fit
result is shown in Table 7.1, while the corresponding parameter values are
given in Table 7.2.

Observable | Exp. val. [MeV] |  $\text{Fit}_{\text{DS},\text{no-}\omega}$ [MeV] | $\chi^{2}$ | Observable | Exp. val. [MeV] |  $\text{Fit}_{\text{DS},\text{no-}\omega}$ [MeV] | $\chi^{2}$
---|---|---|---|---|---|---|---
$f_{\pi^{+}}$ | $92.06\pm 4.60$ | $96.72$ | $1.0$ | $f_{K^{+}}$ | $110.10\pm 5.51$ | $110.45$ | $0.0$
$\bar{M}_{\pi}$ | $138.04\pm 6.90$ | $140.20$ | $0.1$ | $\Delta M_{\pi}$ | $-4.59\pm 0.92$ | $-4.56$ | $0.0$
$M_{\eta}$ | $547.86\pm 27.39$ | $547.39$ | $0.0$ | $M_{\eta^{\prime}}$ | $957.78\pm 47.89$ | $952.44$ | $0.0$
$\bar{M}_{K}$ | $495.64\pm 24.78$ | $482.47$ | $0.3$ | $\Delta M_{K}$ | $3.93\pm 0.79$ | $3.93$ | $0.0$
$\bar{M}_{\rho}$ | $775.16\pm 38.76$ | $761.61$ | $0.1$ | $\Delta M_{\rho}$ | $0.15\pm 0.57$ | $0.13$ | $0.0$
$M_{\omega}$ | $782.66\pm 39.13$ | $761.86$ | $0.3$ | $M_{\phi}$ | $1019.46\pm 50.97$ | $986.41$ | $0.4$
$\bar{M}_{K^{\star}}$ | $895.50\pm 44.78$ | $882.52$ | $0.1$ | $\Delta M_{K^{\star}}$ | $0.08\pm 0.90$ | $0.19$ | $0.0$
$\bar{M}_{a_{1}}$ | $1230.00\pm 246.00$ | $1115.71$ | $0.2$ | $M_{f_{1}^{L}}$ | $1281.90\pm 256.38$ | $1222.40$ | $0.1$
$M_{f_{1}^{H}}$ | $1426.30\pm 285.26$ | $1367.72$ | $0.0$ | $\bar{M}_{K_{1}}$ | $1253.00\pm 250.60$ | $1260.84$ | $0.0$
$\bar{M}_{a_{0}}$ | $1474.00\pm 294.80$ | $1140.38$ | $1.0$ | $\bar{M}_{K_{0}^{\star}}$ | $1425.00\pm 285.00$ | $1237.23$ | $0.4$
$M_{f_{0}^{L}}$ | $1350.00\pm 675.00$ | $1136.72$ | $0.1$ | $M_{f_{0}^{H}}$ | $1733.00\pm 866.50$ | $1326.58$ | $0.2$
$\bar{\Gamma}_{\rho\rightarrow\pi\pi}$ | $148.53\pm 7.43$ | $154.85$ | $0.7$ | $\Delta\Gamma_{\rho\rightarrow\pi\pi}$ | $-1.70\pm 1.60$ | $-1.89$ | $0.0$
$\bar{\Gamma}_{\phi\rightarrow\bar{K}K}$ | $1.76\pm 0.09$ | $1.10$ | $0.3$ | $\Delta\Gamma_{\phi\rightarrow\bar{K}K}$ | $-0.65\pm 0.13$ | $-0.58$ | $0.2$
$\bar{\Gamma}_{K^{\star}\rightarrow K\pi}$ | $46.75\pm 2.34$ | $46.20$ | $0.0$ | $\Delta\Gamma_{K^{\star}\rightarrow K\pi}$ | $1.10\pm 1.80$ | $0.40$ | $0.2$
$\bar{\Gamma}_{a_{1}\rightarrow\rho\pi}$ | $425.00\pm 175.00$ | $428.34$ | $0.0$ | $\Gamma_{a_{1}\rightarrow\pi\gamma}$ | $0.64\pm 0.25$ | $0.68$ | $0.0$
$\Gamma_{f_{1}^{H}\rightarrow K^{\star}K}$ | $43.60\pm 8.72$ | $43.84$ | $0.0$ | $\Gamma_{a_{0}}$ | $265.00\pm 53.00$ | $239.03$ | $0.2$
$\Gamma_{K_{0}^{\star}\rightarrow K\pi}$ | $270.00\pm 80.00$ | $333.96$ | $0.6$ |  |  |  |
Table 7.1: Detailed fit results in case of isospin violation. Experimental data, fit result in the Dashen case (DS) without fitting the $\omega\to\pi\pi$ decay and the corresponding chi square values. Parameter | DS, no-$\omega$ | Parameter | DS, no-$\omega$
---|---|---|---
$\phi_{N}$ [MeV] | $163.93$ |  $\phi_{S}$ [MeV] | $133.40$
$\phi_{3}$ [MeV] | $-4.72\times 10^{-3}$ |  $m_{0}^{2}$ [MeV2] | $-6.39\times 10^{+5}$
$\tilde{m}_{1}^{2}$ [MeV2] | $8.00\times 10^{+5}$ | $\lambda_{1}$ | $0.09$
$\lambda_{2}$ | $44.79$ | $h_{1}$ | $26.24$
$h_{2}$ | $23.82$ | $h_{3}$ | $5.41$
$g_{1}$ | $5.53$ | $g_{2}$ | $3.01$
$c_{1}$ [$\text{MeV}^{-2}$] | $2.68\times 10^{-4}$ |  $\tilde{\delta}_{S}$ [MeV2] | $1.46\times 10^{+5}$
$\delta_{3}$ [MeV2] | $3.75$ |  $\delta m^{2}_{V}$ [MeV2] | $-1.26\times 10^{+2}$
$\delta m^{2}_{A}$ [MeV2] | $-1.91\times 10^{+5}$ |  $m^{2}_{\text{em},S}$ [MeV2] | $9.93\times 10^{+3}$
$m^{2}_{\text{em},P}$ [MeV2] | $-3.69\times 10^{+3}$ |  $m^{2}_{\text{em},V}$ [MeV2] | $-3.20\times 10^{+2}$
$m^{2}_{\text{em},A}$ [MeV2] | $9.91\times 10^{+3}$ |  |
Table 7.2: Parameter set for the fit result of Table 7.1.

In the fit, the Dashen case refers to the fulfillment of Dashen’s theorem,
i.e. the electromagnetic contribution within a nonet is the same for the pion-
like and kaon-like particles. In this fit, the $\omega\to\pi\pi$ decay width
was not used, which is the only isospin violating decay that cannot be fit
within the eLSM at tree-level. The reduced chi-squared of the fit is
$\chi^{2}_{\text{red}}=0.6$, showing a very good agreement with data.

As a concluding remark, we notice that the eLSM, when properly equipped, is
capable of describing also relatively small effects. In the future, further
improvements are possible by adding small additional large-$N_{c}$ suppressed
terms, such as the already mentioned nonstrange-strange mixing for homochiral
mesons (such as (axial-)vector ones).

### 7.2 Radially excited (pseudo)scalar mesons

Up to now we solely considered $\bar{q}q$ nonets which are ground-states in
the radial quantum numbers $n=1$. There are however quite well-established
nonets of radially excited vector and pseudoscalar mesons [13, 50]. For what
concerns radially excited vector states, we refer to [197] for a detailed
treatment in the framework of a flavor-symmetry based approach.

Here, following Ref. [130], we concentrate on the chiral multiplet of radially
excited (pseudo)vector states made out of excited pseudoscalar and scalar
nonets. The nonet of radially excited pseudoscalar states
($J^{\mathcal{P}\mathcal{C}}=0^{-+}$ with $L=S=0,$ spectroscopic notation $2$
${}^{1}S_{0},$ denoted as $P_{E}$) contains the resonances {$\pi(1300),$
$K(1460),$ $\eta(1295)$, $\eta(1405)/\eta(1475)$} and shares the same
transformation properties of the ground-state pseudoscalar nonet $P$ (note,
the two resonances $\eta(1405)$ and $\eta(1475)$ can be interpreted as a
single resonances denoted as $\eta(1440)$). In fact, the radial excitation is
formally associated to the same local current.

Similarly, the nonet of radially excited scalar states
($J^{\mathcal{P}\mathcal{C}}=0^{++}$ with $L=S=1$, spectroscopic notation $2$
${}^{1}S_{0},$ denoted as $S_{E}$) contains the resonances $a_{0}(1950),$
$K_{0}^{\ast}(1950),$ $f_{0}(1790)$, $f_{0}(2020)/f_{0}(2100)$. This nonet is
formally analogous to $S.$ Their matrix expressions read

$P_{E}=\frac{1}{\sqrt{2}}\begin{pmatrix}\frac{\eta_{N,E}+\pi_{E}^{0}}{\sqrt{2}}&\pi_{E}^{+}&K_{E}^{+}\\\
\pi_{E}^{-}&\frac{\eta_{N,E}-\pi_{E}^{0}}{\sqrt{2}}&K_{E}^{0}\\\
K_{E}^{-}&\bar{K}_{E}^{0}&\eta_{S,E}\end{pmatrix}\,\text{,
}S_{E}=\frac{1}{\sqrt{2}}\begin{pmatrix}\frac{\sigma_{N,E}+a_{0,E}^{0}}{\sqrt{2}}&a_{0,E}^{+}&K_{0,E}^{\ast+}\\\
a_{0,E}^{-}&\frac{\sigma_{N,E}-a_{0,E}^{0}}{\sqrt{2}}&K_{0,E}^{\ast 0}\\\
K_{0,E}^{\ast-}&\bar{K}_{0,E}^{\ast 0}&\sigma_{S,E}\text{ ,}\end{pmatrix}$
(7.17)

out of which we construct the heterochiral multiplet of radially excited
(pseudo)scalar states (in short, excited heteroscalars) as
$\Phi_{E}=S_{E}+iP_{E}$, which transforms just as $\Phi$ under chiral and
$\mathcal{C}$ and $\mathcal{P}$ transformations. This is a general feature for
radially excited chiral multiplets: for what concerns transformation
properties, they are formally identical to the ground-state ones.

The Lagrangian describing the masses of radially excited (pseudo)scalars is
given by

$\displaystyle\mathcal{L}_{\Phi_{E}}^{\text{mass}}=\text{Tr}\Big{[}\Big{(}\frac{m_{0}^{\ast\,2}\,G^{2}}{2\,G_{0}^{2}}+\Delta_{E}\Big{)}\Big{(}\Phi_{E}^{\dagger}\Phi_{E}\Big{)}\Big{]}+\frac{\lambda_{\Phi_{E}}}{2}\text{Tr}\Big{(}\Phi^{\dagger}\Phi\Big{)}\text{Tr}\Big{(}\Phi_{E}^{\dagger}\Phi_{E}\Big{)}+\lambda_{2}^{\ast}\text{Tr}\Big{(}\Phi_{E}^{\dagger}\Phi_{E}\Phi^{\dagger}\Phi\Big{)}+\xi_{2}\text{Tr}\Big{(}\Phi_{E}^{\dagger}\Phi\Phi_{E}^{\dagger}\Phi\Big{)}$
(7.18)

where $\Delta_{E}=\text{diag}(0,0,\epsilon_{S}^{E})$ describes, as usual, the
mass contribution arising from nonzero quark masses for this multiplet. The
resulting expressions for the masses are reported in Table 7.3.

Mass squares | Analytical expressions | Estimates of masses (MeV)
---|---|---
$m_{\sigma_{N}^{E}}^{2}$ | $(m_{0}^{\ast})^{2}+\frac{\lambda_{2}^{\ast}+\xi_{2}}{2}\phi_{N}^{2}$ | $1790\pm 35$
$m_{a_{0}^{E}}^{2}$ | $(m_{0}^{\ast})^{2}+\frac{\lambda_{2}^{\ast}+\xi_{2}}{2}\phi_{N}^{2}=m_{\sigma_{N}^{E}}^{2}$ | $1790\pm 35$
$m_{\eta_{N}^{E}}^{2}$ | $(m_{0}^{\ast})^{2}+\frac{\lambda_{2}^{\ast}-\xi_{2}}{2}\phi_{N}^{2}$ | $1294\pm 4$
$m_{\pi^{E}}^{2}$ | $(m_{0}^{\ast})^{2}+\frac{\lambda_{2}^{\ast}-\xi_{2}}{2}\phi_{N}^{2}=m_{\eta_{N}^{E}}^{2}$ | $1294\pm 4$
$m_{\eta_{S}^{E}}^{2}$ | $(m_{0}^{\ast})^{2}-2\epsilon_{S}^{E}+\left(\lambda_{2}^{\ast}-\xi_{2}\right)\phi_{S}^{2}$ | $1432\pm 10$
$m_{\sigma_{S}^{E}}^{2}$ | $(m_{0}^{\ast})^{2}-2\epsilon_{S}^{E}+\left(\lambda_{2}^{\ast}+\xi_{2}\right)\phi_{S}^{2}$ | $1961\pm 38$
$m_{K^{E}}^{2}$ | $(m_{0}^{\ast})^{2}-\epsilon_{S}^{E}+\frac{\lambda_{2}^{\ast}}{4}\phi_{N}^{2}-\frac{\xi_{2}}{\sqrt{2}}\phi_{N}\phi_{S}+\frac{\lambda_{2}^{\ast}}{2}\phi_{S}^{2}$ | $1366\pm 6$
$m_{K_{0}^{\star E}}^{2}$ | $(m_{0}^{\ast})^{2}-\epsilon_{S}^{E}+\frac{\lambda_{2}^{\ast}}{4}\phi_{N}^{2}+\frac{\xi_{2}}{\sqrt{2}}\phi_{N}\phi_{S}+\frac{\lambda_{2}^{\ast}}{2}\phi_{S}^{2}$ | $1877\pm 36$
Table 7.3:  Masses of radially excited (pseudo)scalar mesons within eLSM (we
assume $\sigma_{N}^{E}\equiv f_{0}(1790)$, $\eta_{N}^{E}\equiv\eta(1295)$ and
$\eta_{S}^{E}\equiv\eta(1440)$).

The chiral interaction Lagrangian for the decays of radially excited
(pseudo)scalar mesons reads

$\displaystyle\mathcal{L}_{\Phi_{E}LR}$
$\displaystyle=h_{2}^{\ast}\mathop{\mathrm{Tr}}\Big{(}\Phi_{E}^{\dagger}L_{\mu}L^{\mu}\Phi+\Phi^{\dagger}L_{\mu}L^{\mu}\Phi_{E}+R_{\mu}\Phi_{E}^{\dagger}\Phi
R^{\mu}+R_{\mu}\Phi^{\dagger}\Phi_{E}R^{\mu}\Big{)}$ (7.19)
$\displaystyle+2h_{3}^{\ast}\mathop{\mathrm{Tr}}\Big{(}L_{\mu}\Phi_{E}R^{\mu}\Phi^{\dagger}+L_{\mu}\Phi
R^{\mu}\Phi_{E}^{\dagger}\Big{)}+\cdots(\text{large-$N_{c}$ suppressed
terms})\text{ .}$

As a result, we obtain the decay channels given in Table 7.4. By using
$\Gamma_{f_{0}(1790)\rightarrow\pi\pi}=270\pm 45\,\text{MeV}$ and
$\Gamma_{f_{0}(1790)\rightarrow KK}=70\pm 40\,\text{MeV}$ in order to fit the
parameters of Eq.(7.19) , one obtains $h_{2}^{\ast}=67\pm 63$ and
$h_{3}^{\ast}=79\pm 63$.

Decay Rates | Estimates (MeV) | Decay Rates | Estimates (MeV)
---|---|---|---
$\sigma_{N}^{E}\rightarrow a_{1}(1260)\pi$ | $47\pm 8$ | $\sigma_{N}^{E}\rightarrow\eta\eta^{\prime}$ | $10\pm 2$
$\sigma_{N}^{E}\rightarrow\eta\eta$ | $7\pm 1$ | $\sigma_{N}^{E}\rightarrow f_{1}(1285)\eta$ | $1\pm 0$
$\eta_{N}^{E}\rightarrow\eta\pi\pi+\eta^{\prime}\pi\pi+\pi KK$ | $7\pm 3$ | $\eta_{S}^{E}\rightarrow K^{\star}K$ | $128^{+204}_{-128}$
$\eta_{S}^{E}\rightarrow\eta\pi\pi$ and $\eta^{\prime}\pi\pi$ | $156^{+245}_{-156}$ | $\sigma_{S}^{E}\rightarrow KK$ | $21^{+39}_{-21}$
$\sigma_{S}^{E}\rightarrow\eta\eta^{\prime}$ | $12\pm 2$ | $\sigma_{S}^{E}\rightarrow\eta\eta$ | $6\pm 1$
$\sigma_{S}^{E}\rightarrow K_{1}K$ | $2^{+5}_{-2}$ | $\sigma_{S}^{E}\rightarrow\eta^{\prime}\eta^{\prime}$ | $1\pm 0$
$a_{0}^{E}\rightarrow\eta\pi$ | $72\pm 12$ | $a_{0}^{E}\rightarrow KK$ | $70\pm 40$
$a_{0}^{E}\rightarrow\eta^{\prime}\pi$ | $\;\,32\pm\;\,5$ | $a_{0}^{E}\rightarrow f_{1}(1285)\pi$ | $\;\,16\pm\;\,3$
$K_{0}^{\star E}\rightarrow K\pi$ | $\;\;51\pm 35$ | $K_{0}^{\star E}\rightarrow\eta^{\prime}K$ | $\;24\pm\,4$
$K_{0}^{\star E}\rightarrow K_{1}\pi$ | $\;\;\,6\pm\,4$ | $K_{0}^{\star E}\rightarrow\eta K$ | $\;\;\,4^{+7}_{-4}$
$K_{0}^{\star E}\rightarrow a_{1}(1260)K$ | $\;\;\;\;3\pm\;\,2$ | $K_{0}^{\star E}\rightarrow f_{1}(1285)K$ | $\;\;\;\;1\pm\;\,1$
Table 7.4: Decay rates of excited scalar mesons. Decay Rates | Estimates (MeV) | Decay Rates | Estimates (MeV)
---|---|---|---
$\eta_{S}^{E}\rightarrow KK\pi$ | $3\pm 0$ | $\pi^{E}\rightarrow\rho\pi$ | $368\pm 37$
$\pi^{E}\rightarrow 3\pi$ | $204\pm 15$ | $\pi^{E}\rightarrow KK\pi$ | $\;\;\;\,2\pm\;\,0$
$K^{E}\rightarrow K^{\star}\pi$ | $112\pm 11$ | $K^{E}\rightarrow K\pi\pi$ | $\;\;\,35\pm\;\;\,4$
$K^{E}\rightarrow\rho K$ | $\;\;\,20\pm\;\;\,2$ | $K^{E}\rightarrow\omega K$ | $\;\;\;\;\,7\pm\;\;\,1$
Table 7.5: Decay rates of excited pseudoscalar mesons.

While the assignment fo the radially excited pseudoscalars is rather
established (ses the quark model review of the PDG [14]), this is not the case
for the scalars. According to the model results, the resonances $f_{0}(1790)$
(with decay width $405\pm 96\,\text{MeV}$) and $a_{0}(1950)$ (with decay width
$271\pm 40$ MeV [260]) are predicted to be the excited quarkonium states (the
latter one is the isotriplet member). The state $K_{0}^{\star}(1950)$ is a
strong candidate for an excited scalar kaon. We also predict an excited
strange-antistrange isoscalar state $\sigma_{S}^{E}\equiv\overline{s}s$ with a
mass $m_{\sigma_{S}^{E}}=2038\pm 24\,\text{MeV}$ positioned between the masses
of $f_{0}(2020)$ and $f_{0}(2100)$ and with a relatively small decay width (
$\Gamma_{\sigma_{S}^{E}}\leq 110\,\text{MeV}$).

The extension to other radially excited chiral multiplets is possible, e.g. to
(axial-)vector mesons forming homochiral fields ($L_{\mu}^{E}$ and
$R_{\mu}^{E}$). Based on recent PDG 2024 data [14], one has the vector states
$\rho(1450)$, $K^{\ast}(1410)$, $\phi(1680)$, $\omega(1420)$ and the axial-
vector ones $a_{1}(1640)$, $K_{1}(1650)$. One may also extend the procedure to
tensor and axial-tensor homochiral multiplets ($\textbf{L}_{\mu\nu}^{E}$ and
$\textbf{R}_{\mu\nu}^{E}$), using e.g. the resonances $a_{2}(1700)$,
$K^{\ast}_{2}(1980)$, $f_{2}(1640)$, and so on.

### 7.3 Extension of the eLSM to $N_{f}=4$

In this subsection, following Ref. [123], we present the results for the four-
flavor extended linear sigma model. In this case, all nonets are enlarged to
$4\times 4$ matrices, hence 16-th multiples.

To this end, we introduce new mesons including the charm quark. For the case
of pseudoscalars, these are the charmed states $D^{0,\pm}$, the open strange-
charmed states $D^{\pm}_{s}$, and a hidden charmed ground state
$\eta_{c}(1S)$, which are collected within the matrix $P$ given below:

$P=\frac{1}{\sqrt{2}}\left(\begin{array}[c]{cccc}\bar{u}i\gamma^{5}u&\bar{d}i\gamma^{5}u&\bar{s}i\gamma^{5}u&\bar{c}i\gamma^{5}u\\\
\bar{u}i\gamma^{5}d&\bar{d}i\gamma^{5}d&\bar{s}i\gamma^{5}d&\bar{c}i\gamma^{5}d\\\
\bar{u}i\gamma^{5}s&\bar{d}i\gamma^{5}s&\bar{s}i\gamma^{5}s&\bar{c}i\gamma^{5}s\\\
\bar{u}i\gamma^{5}c&\bar{d}i\gamma^{5}c&\bar{s}i\gamma^{5}c&\bar{c}i\gamma^{5}c\end{array}\right)=\frac{1}{\sqrt{2}}\left(\begin{array}[c]{cccc}\frac{1}{\sqrt{2}}(\eta_{N}+\pi^{0})&\pi^{+}&K^{+}&D^{0}\\\
\pi^{-}&\frac{1}{\sqrt{2}}(\eta_{N}-\pi^{0})&K^{0}&D^{-}\\\
K^{-}&\overline{K}^{0}&\eta_{S}&D_{s}^{-}\\\
\overline{D}^{0}&D^{+}&D_{s}^{+}&\eta_{c}\end{array}\right)\text{ .}$ (7.20)

For scalar fields, we include the open charmed $D^{\ast\,0,\pm}_{0}$, the
strange-charmed meson $D^{\ast\,\pm}_{s0}$, and the hidden charmed resonance
$\chi_{c0}\equiv\chi_{c0}(1P)$, which are assigned to
$D^{\ast}_{0}(2400)^{0,\pm}$, $D_{s0}^{\ast}(2317)^{\pm}$, and to the
$\eta_{c}$ meson. The matrix has the following form:

$S=\frac{1}{\sqrt{2}}\left(\begin{array}[c]{cccc}\frac{1}{\sqrt{2}}(\sigma_{N}+a_{0}^{0})&a_{0}^{+}&K_{0}^{\ast+}&D_{0}^{\ast
0}\\\ a_{0}^{-}&\frac{1}{\sqrt{2}}(\sigma_{N}-a_{0}^{0})&K_{0}^{\ast
0}&D_{0}^{\ast-}\\\ K_{0}^{\ast-}&\overline{K}_{0}^{\ast
0}&\sigma_{S}&D_{s0}^{\ast-}\\\ \overline{D}_{0}^{\ast
0}&D_{0}^{\ast+}&D_{s0}^{\ast+}&\chi_{c0}\end{array}\right)\text{ ,}$ (7.21)

where also the explicit quark content is displayed. As for the three-flavor
case, we construct a chiral field containing both of the above-mentioned
nonets $\Phi=S+iP$, which has analogous transformation rules under parity,
charge conjugation, and chiral symmetry (here under $SU(4)_{L}\times
SU(4)_{R}$) presented in Table 2.6. Of course, a large explicit breaking is
present due to the large $c$-quark mass, but the idea is that this breaking
can be relegated to the linear and quadratic symmetry-breaking mass terms.

For vector mesons, one includes the nonstrange-charmed fields $D^{\ast
0},\,D^{\ast\pm}$, that corresponds to $D^{\ast}(2007)^{0}$ and
$D^{\ast}(2010)^{\pm}$, the strange-charmed $D_{0}^{\ast\pm}$ is assigned to
the resonance $D_{0}^{\ast\pm}$ (with mass $m_{D_{0}^{\ast\pm}}$), and the
well-known charmonium state $J/\psi(1S)$. All of them are part of the matrix
$V^{\mu}$:

$V^{\mu}=\frac{1}{\sqrt{2}}\left(\begin{array}[c]{cccc}\frac{1}{\sqrt{2}}(\omega_{N}+\rho^{0})&\rho^{+}&K^{\ast}(892)^{+}&D^{\ast
0}\\\
\rho^{-}&\frac{1}{\sqrt{2}}(\omega_{N}-\rho^{0})&K^{\ast}(892)^{0}&D^{\ast-}\\\
K^{\ast}(892)^{-}&\bar{K}^{\ast}(892)^{0}&\omega_{S}&D_{s}^{\ast-}\\\
\overline{D}^{\ast
0}&D^{\ast+}&D_{s}^{\ast+}&J/\psi\end{array}\right)^{\mu}\text{.}$ (7.22)

For axial-vector fields, one adds the open charmed mesons $D_{1}$ and
$D_{s1}$, assigned to $D_{1}(2420)$ and $D_{s1}(2536)$, and the
$c\overline{c}$ state $\chi_{c1}$ that corresponds to the well-known resonance
$\chi_{c1}(1P)$. The matrix is:

$A_{1}^{\mu}=\frac{1}{\sqrt{2}}\left(\begin{array}[c]{cccc}\frac{1}{\sqrt{2}}(f_{1,N}+a_{1}^{0})&a_{1}^{+}&K_{1}^{+}&D_{1}^{0}\\\
a_{1}^{-}&\frac{1}{\sqrt{2}}(f_{1,N}-a_{1}^{0})&K_{1}^{0}&D_{1}^{-}\\\
K_{1}^{-}&\bar{K}_{1}^{0}&f_{1,S}&D_{s1}^{-}\\\
\bar{D}_{1}^{0}&D_{1}^{+}&D_{s1}^{+}&\chi_{c1}\end{array}\right)^{\mu}\,.$
(7.23)

Homochiral (axial-)vector fields are $L^{\mu}=V^{\mu}+A_{1}^{\mu}$ and
$R^{\mu}=V^{\mu}-A_{1}^{\mu}$ and follow the corresponding transformation
rules given in Table 2.6 for three-flavor fields.

Field | PDG | Quark content | $I$ | $J^{\mathcal{P}\mathcal{C}}$ | Mass (MeV)
---|---|---|---|---|---
$\bar{D}^{0}$, $D^{0}$, $D^{\pm}$ | $D$ |  $u\bar{c}$, $\bar{u}c$,$d\bar{c}$, $\bar{d}c$ | $\frac{1}{2}$ | $0^{-+}$ | $1864.86\pm 0.13$
$D_{S}^{\pm}$ | $D_{S}$ |  $s\bar{c}$, $\bar{s}c$ | $0$ | $0^{-+}$ | $1968.50\pm 0.32$
$\eta_{c}$ | $\eta_{c}(1s)$ | $c\bar{c}$ | $0$ | $0^{-+}$ | $2673\pm 118$
$\bar{D}_{0}^{\ast}$, $D_{0}^{\ast}$ , $D_{0}^{\ast\pm}$ | $D_{0}^{\ast}(2400)^{0}$ |  $u\bar{c}$, $\bar{u}c$,$d\bar{c}$, $\bar{d}c$ | $\frac{1}{2}$ | $0^{++}$ | $2414\pm 77$
$D_{s0}^{\ast\pm}$ | $D_{s0}^{\ast}(2317)^{\pm}$ |  $s\bar{c}$, $\bar{s}c$ | 0 | $0^{++}$ | $2467\pm 76$
$\chi_{c0}$ | $\chi_{c0}(1P)$ | $c\bar{c}$ | 0 | $0^{++}$ | $3144\pm 128$
$\bar{D}^{\ast 0}$, $D^{\ast 0}$, $D^{\ast\pm}$ | $D^{\ast}(2007)^{0}$ |  $u\bar{c}$, $\bar{u}c$,$d\bar{c}$, $\bar{d}c$ | $\frac{1}{2}$ | $1^{--}$ | $2168\pm 70$
$D_{s}^{\ast\pm}$ | $D^{\ast\pm}_{S}(2010)$ |  $s\bar{c}$, $\bar{s}c$ | 0 | $1^{--}$ | $2203\pm 69$
$J/\psi$ | $J/\psi(1s)$ | $c\bar{c}$ | 0 | $1^{--}$ | $2947\pm 109$
$\bar{D}_{1}^{0}$, $D_{1}^{0}$, $D_{1}^{\pm}$ | $D_{1}(2420)^{0}$ |  $u\bar{c}$, $\bar{u}c$,$d\bar{c}$, $\bar{d}c$ | $\frac{1}{2}$ | $1^{++}$ | $2429\pm 63$
$D_{s1}^{\pm}$ | $D_{s1}^{\pm}$ |  $s\bar{c}$, $\bar{s}c$ | 0 | $1^{++}$ | $2480\pm 63$
$\chi_{c1}$ | $\chi_{c1}(1P)$ | $c\bar{c}$ | 0 | $1^{++}$ | $3239\pm 101$
Table 7.6: Charmed meson fields within eLSM, PDG correspondence, quark
content, quantum numbers, and PDG mass values from [261].

The Lagrangian of the eLSM presented in Sec. 3.2 retains its form when the
matrices are $4\times 4$. A special attention is devoted to the symmetry
breaking terms, which are a straightforward extensions of those in 3.2 but are
now important (and not only a correction) to describe the large charm mass:

$\displaystyle\mathcal{L}_{\text{sym-
brk}}^{N_{f}=4}=\tilde{c}\Big{(}\det{\Phi}-\det{\Phi}^{\dagger}\Big{)}^{2}+\mathrm{Tr}[H(\Phi^{{\dagger}}+\Phi)]-2\,\mathrm{Tr}[\hat{\epsilon}\Phi^{\dagger}\Phi]+\mathrm{Tr}[\Delta(L_{\mu\nu}^{2}+R_{\mu\nu}^{2})]$
(7.24)

where $\tilde{c}\equiv 2c_{2}/\phi_{C}^{2}$ and explicit forms of the matrices
(only diagonal elements are non-zero) are

$H=\text{diag}\Big{(}\frac{h_{0N}}{2},\frac{h_{0N}}{2},\frac{h_{0S}}{2},\frac{h_{0C}}{\sqrt{2}}\Big{)}\,,\,\Delta=\text{diag}\Big{(}\delta_{N},\delta_{N},\delta_{S},\delta_{C}\sim
m_{c}^{2}\Big{)}\text{ ,
}\hat{\epsilon}=\text{diag}\Big{(}\varepsilon_{N},\varepsilon_{N},\varepsilon_{S},\varepsilon_{C}\Big{)}\,.$
(7.25)

Above: $h_{0,N,S,C}\propto m_{n,s,c}$, $\delta_{N,S,C}\propto m_{n,s,c}^{2}$,
and $\epsilon_{N,C,S}\propto m_{n,s,c}^{2}$. As previously, we set
$\delta_{N}=\epsilon_{N}=0$. SSB implies that also the scalar charmed meson
condenses, hence we need to perform a shift on the field $\chi_{C0}$ similar
to Eq. (3.27) by its vacuum expectation value $\phi_{C}$ as

$\displaystyle\chi_{c0}\rightarrow\chi_{c0}+\phi_{C}\,.$ (7.26)

We present the expression for the masses of (pseudo)scalar and (axial-)vector
charmed mesons within the eLSM in Tables 7.7 and 7.8. In particular, for
chiral partners one has:

$\displaystyle m_{D_{1}}^{2}$
$\displaystyle=m_{D^{\star}}^{2}+\sqrt{2}(g_{1}^{2}-h_{3})\phi_{N}\phi_{C}\,,$
(7.27) $\displaystyle m_{\chi_{c1}}^{2}$
$\displaystyle=m_{J/\psi}^{2}+2(g_{1}^{2}-h_{3})\phi_{C}^{2}\,,$ (7.28)
$\displaystyle m_{D_{s1}}^{2}$
$\displaystyle=m_{D_{S}^{\star}}^{2}+2(g_{1}^{2}-h_{3})\phi_{S}\phi_{C}\,\text{
,}$ (7.29)

thus the mass differences do not depend on the charm mass, but only on the
chiral condensates, as in the light sector. For the coupling constants of the
model ($g_{1}$, $\lambda_{1,2}$ and $h_{1,2,3}$), we use the values determined
in the $N_{f}=3$ case, see Table 3.5. Using the PDG masses for 12 mesonic
states given in Table 7.6, we determine the following three parameters:

$\displaystyle\phi_{C}=178\pm 28\,\text{ MeV}\,,\quad\delta_{C}=(3.91\pm
0.36)\cdot 10^{6}\,\text{ MeV}^{2}\,,\quad\varepsilon_{C}=(2.23\pm 0.71)\cdot
10^{6}\,\text{ MeV}^{2}\,.$

It is worth noting that the charm condensate is comparable to the non-strange
and strange quark condensates. The masses of open charmed mesons closely align
with PDG values within theoretical errors, while the masses of charmonia,
excluding $J/\psi$, are underestimated by about $10\%$.

Mass | Equations of Mass Square | eLSM (MeV)
---|---|---
$m_{\eta_{C}}$ | $Z_{\eta_{C}}^{2}\left[m_{0}^{2}+\lambda_{1}(\phi_{N}^{2}+\phi_{S}^{2})+(\lambda_{1}+\lambda_{2})\phi_{C}^{2}+\frac{c}{8}\phi_{N}^{4}\phi_{S}^{2}\right]$ | $2673\pm 118$
$m_{D}$ | $Z_{D}^{2}\left[m_{0}^{2}+\left(\lambda_{1}+\frac{\lambda_{2}}{2}\right)\phi_{N}^{2}+\lambda_{1}\phi_{S}^{2}-\frac{\lambda_{2}}{\sqrt{2}}\phi_{N}\phi_{C}+(\lambda_{1}+\lambda_{2})\phi_{C}^{2}\right]$ | $1981\pm 73$
$m_{D_{s}}$ | $Z_{D_{S}}^{2}\left[m_{0}^{2}+\lambda_{1}\phi_{N}^{2}+\left(\lambda_{1}+\lambda_{2}\right)\phi_{S}^{2}-\lambda_{2}\phi_{S}\phi_{C}+(\lambda_{1}+\lambda_{2})\phi_{C}^{2}\right]$ | $2004\pm 74$
$m_{\chi_{C0}}$ | $m_{0}^{2}+\lambda_{1}(\phi_{N}^{2}+\phi_{S}^{2})+3(\lambda_{1}+\lambda_{2})\phi_{C}^{2}$ | $3144\pm 128$
$m_{D_{0}^{\star}}$ | $Z_{D_{0}^{\star}}^{2}\left[m_{0}^{2}+\left(\lambda_{1}+\frac{\lambda_{2}}{2}\right)\phi_{N}^{2}+\lambda_{1}\phi_{S}^{2}+\frac{\lambda_{2}}{\sqrt{2}}\phi_{N}\phi_{C}+(\lambda_{1}+\lambda_{2})\phi_{C}^{2}\right]$ | $2414\pm 77$
$m_{D_{s0}^{\star}}$ | $Z_{D_{s0}}^{2}\left[m_{0}^{2}+\lambda_{1}\phi_{N}^{2}+\left(\lambda_{1}+\lambda_{2}\right)\phi_{S}^{2}+\lambda_{2}\phi_{S}\phi_{C}+(\lambda_{1}+\lambda_{2})\phi_{C}^{2}\right]$ | $2467\pm 76$
Table 7.7:  Masses of (pseudo)scalar charmed mesons within the eLSM.  Mass | Equations of Mass Square | eLSM (MeV)
---|---|---
$m_{D^{\star}}$ | $m_{1}^{2}+\delta_{N}+\delta_{C}+\frac{\phi_{N}^{2}}{2}(\frac{g_{1}^{2}}{2}+h_{1}+\frac{h_{2}}{2})+\frac{\phi_{N}\phi_{C}}{\sqrt{2}}(h_{3}-g_{1}^{2})+\frac{\phi_{C}^{2}}{2}(g_{1}^{2}+h_{1}+h_{2})+\frac{h_{1}}{2}\phi_{S}^{2}$ | $2168\pm 70$
$m_{J/\psi}$ | $m_{1}^{2}+2\delta_{C}+\frac{h_{1}}{2}(\phi_{N}^{2}+\phi_{S}^{2})+(\frac{h_{1}}{2}+h_{2}+h_{3})\phi_{C}^{2}$ | $2947\pm 109$
$m_{D_{s}^{\star}}$ | $m_{1}^{2}+\delta_{S}+\delta_{C}+\frac{\phi_{S}^{2}}{2}(g_{1}^{2}+h_{1}+h_{2})+\phi_{S}\phi_{C}(h_{3}-g_{1}^{2})+\frac{\phi_{C}^{2}}{2}(g_{1}^{2}+h_{1}+h_{2})+\frac{h_{1}}{2}\phi_{N}^{2}$ | $2203\pm 69$
$m_{D_{s1}}$ | $m_{1}^{2}+\delta_{S}+\delta_{C}+\frac{\phi_{S}^{2}}{2}(g_{1}^{2}+h_{1}+h_{2})+\phi_{S}\phi_{C}(g_{1}-h_{3}^{2})+\frac{\phi_{C}^{2}}{2}(g_{1}^{2}+h_{1}+h_{2})+\frac{h_{1}}{2}\phi_{N}^{2}$ | $2480\pm 63$
$m_{D_{1}}$ | $m_{1}^{2}+\delta_{N}+\delta_{C}+\frac{\phi_{N}^{2}}{2}(\frac{g_{1}^{2}}{2}+h_{1}+\frac{h_{2}}{2})+\frac{\phi_{N}\phi_{C}}{\sqrt{2}}(g_{1}^{2}-h_{3})+\frac{\phi_{C}^{2}}{2}(g_{1}^{2}+h_{1}+h_{2})+\frac{h_{1}}{2}\phi_{S}^{2}$ | $2429\pm 63$
$m_{\chi_{C1}}$ | $m_{1}^{2}+2\delta_{C}+\frac{h_{1}}{2}(\phi_{N}^{2}+\phi_{S}^{2})+2g_{1}\phi_{C}^{2}+\phi_{C}^{2}(\frac{h_{1}}{2}+h_{2}-h_{3})$ | $3239\pm 101$
Table 7.8:  Masses of (axial-)vector charmed mesons within the eLSM.

Table 7.9 illustrates decay channels in eLSM versus the PDG, see Ref. [123]
for details. Of course, a chiral model, being intrinsically low-energy QCD
approach, cannot be as precise as approaches constructed for heavy quarks
(e.g.based on heavy quark symmetry [262]). Yet the very fact that many decay
widths can be correctly reproduced (especially the ones that are allowed by
large-$N_{c}$ dominant terms such as the first entry in Table 7.9) by using
the parameters that were determined in the low-energy sector, implies that a
‘remnant’ of chiral symmetry is still active for charmed mesons as well, even
if the explicit breaking is large. The extension of the eLSM to $N_{f}=4$ also
predicts the interesting decay channel of $\eta_{c}$ into pseudoscalar
glueball $\Gamma_{\eta_{c}\rightarrow P_{G}\pi\pi}=0.124\,\text{MeV}$ (wit
$2.6$ GeV for the pseudoscalar glueball mass). Other decay channels of
$\eta_{c}$ within eLSM can be found in Ref. [263].

The somewhat unexpected result that some interactions still fulfill chiral
symmetry opens also interesting outlooks, such as the inclusion of
(axial-)tensor chiral multiples ($\mathbf{L}^{\mu\nu}$ and
$\mathbf{R}^{\mu\nu}$) by (including the resonances $\chi_{c2}(1P)$ and
$\psi_{2}(3823)$) and the study of the charm condensate at nonzero
temperature.

Decay Channel | eLSM [MeV] | PDG [MeV]
---|---|---
$D_{0}^{\ast}(2400)^{0}\rightarrow D\pi$ | $139_{-114}^{+243}$ | $\Gamma_{\text{tot}}=267\pm 40$
$D_{0}^{\ast}(2400)^{+}\rightarrow D\pi$ | $51_{-51}^{+182}$ | $\Gamma_{\text{tot}}=283\pm 24\pm 34$
$D^{\ast}(2007)^{0}\rightarrow D^{0}\pi^{0}$ | $0.025\pm 0.003$ | $<1.3$
$D^{\ast}(2010)^{+}\rightarrow D^{+}\pi^{0}$ | $0.018_{-0.003}^{+0.002}$ | $0.029\pm 0.008$
$D^{\ast}(2010)^{+}\rightarrow D^{0}\pi^{+}$ | $0.038_{-0.004}^{+0.005}$ | $0.065\pm 0.017$
$D_{1}(2420)^{0}\rightarrow D^{\ast}\pi$ | $65_{-37}^{+51}$ | $\Gamma_{\text{tot}}=27.4\pm 2.5$
$D_{1}(2420)^{0}\rightarrow D^{0}\pi\pi$ | $0.59\pm 0.02$ | seen
$D_{1}(2420)^{0}\rightarrow D^{+}\pi^{-}\pi^{0}$ | $0.21_{-0.015}^{+0.01}$ | seen
$D_{1}(2420)^{+}\rightarrow D^{\ast}\pi$ | $65_{-36}^{+51}$ | $\Gamma_{\text{tot}}=25\pm 6$
$D_{1}(2420)^{+}\rightarrow D^{+}\pi\pi$ | $0.56\pm 0.02$ | seen
$D_{1}(2420)^{+}\rightarrow D^{0}\pi^{0}\pi^{+}$ | $0.22\pm 0.01$ | seen
Table 7.9: Decay widths of charmed mesons.

## 8 The eLSM at nonzero temperature and density

### 8.1 General features and the coupling of the eLSM to quark d.o.f.

The study of the strong interaction in the medium represents one of the most
significant research areas in high-energy physics in recent decades. This is
due to the fact that matter produced in hadron colliders (CERN, RHIC) heats up
and becomes very dense during the collision. Subsequently, as it expands and
cools, it undergoes a trajectory in the $T$-$\rho_{B}$ or $T$-$\mu_{B}$ plane,
where $T$ is the temperature, while $\rho_{B}$ and $\mu_{B}$ are the baryonic
density and chemical potential, respectively. The envisioned image of the
phase diagram of the strongly interacting matter can be seen in Fig. 8.1.

Figure 8.1: The envisioned phase diagram of strongly interacting matter, as
presented in [264], is shown along with the regions of exploration for various
experiments.

According to the figure, it is commonly believed that at large temperature
and/or baryon chemical potential/density, the normal hadronic matter
transforms into the quasi-free gas of quarks and gluons. There is a phase
boundary between the two phases. The transition itself is the deconfinement
phase transition. When the matter is deconfined, the chiral symmetry is also
restored, which is called the chiral phase transition. The two phase
boundaries, i.e. deconfinement and chiral, go hand in hand on some part of the
$T-\mu_{B}$ plane, but there can be regions where they are separated. From
lattice QCD it is known that at $\mu_{B}=0$ the chiral phase transition is
continuous or crossover type [265], while other parts of the phase diagram are
less known, at least either directly from QCD or experimentally. Lattice QCD
cannot be used directly at finite $\mu_{B}$, due to the notorious sign problem
[266]. However, most effective models of QCD – with which finite $\mu_{B}$
regions can be reached – predict that the phase transition as a function of
$\mu_{B}$ at $T=0$ is of first order. Accordingly, there should be a so-called
critical end point (CEP) along the chiral phase boundary separating the
crossover region from the first-order one. Different models predict the
location of the CEP in a wide range of $T$ and $\mu_{B}$. On the other hand,
current and future experiments address different parts of the phase diagram,
but the CEP has not yet been found. Another interesting part of the phase
diagram is the small temperature $T\approx 0$ and large $\mu_{B}$ region where
the so-called neutron stars or more generally compact stars live. Neutron
stars, which are extremely dense stable astrophysical objects, cool down to
$T\sim 100$ eV within a few years after their formation due to neutrino
cooling, while their core density is estimated to be around $2-5\rho_{0}$,
where $\rho_{0}=0.16\text{\,fm}^{-3}$ is the nuclear saturation density. In
the last decade, neutron star observations have regained momentum with new
experiments such as the NICER experiment [267], which can simultaneously
measure the radii and masses of neutron stars. The advantage of effective
models of QCD such as the eLSM is that all parts of the phase diagram can be
explored. Thus, the eLSM can be used to determine the chiral phase boundary,
or even to compute mass-radius curves for compact stars that can be directly
compared with experimental astrophysical data. We follow here the treatment
presented in Ref. [8] (for additional eLSM works in the medium, see Refs.
[139, 268, 269, 270, 271]). It is also worth noting that the phase structure
is thought to be more complicated in the low temperature, high density regime,
where a color superconducting phase is expected [272, 273]. However, as far as
we know, this phase has not been studied in the eLSM.

Theoretically, the in-medium properties of strongly interacting matter can be
described within the framework of finite temperature field theory [274, 275,
276]. To go to finite temperature, the standard procedure is to perform an
analytic continuation to imaginary times, i.e. $t\to-i\tau$. Then the grand
potential $\Omega(T,\mu_{q})$ can be calculated from the partition function
$\mathcal{Z}$ for a spatially uniform system of volume $V$ in thermal
equilibrium for temperature $T=1/\beta$. Any thermodynamic quantity can be
derived from the grand potential. So far we have only considered mesons, but
to get to finite baryon density/baryon chemical potential, particles carrying
baryon number should be included. In practice, instead of baryons, the so-
called constituent quarks ($q_{i},\;i\in(u,d,s)$) are included in the eLSM by
adding a Yukawa term to the Lagrangian as follows,

$\bar{q}\left[i\gamma_{\mu}D^{\mu}-\mathcal{M}\right]q\,,$ (8.1)

with

$q=\left(\begin{array}[c]{c}q_{u}\\\ q_{d}\\\ q_{s}\end{array}\right)\;\text{
, }D^{\mu}q=\partial^{\mu}q-iG^{\mu}q\text{ ,
}\mathcal{M}=g_{F}\left(\mathds{1}_{4\times 4}S+i\gamma_{5}P\right)\;.$ (8.2)

Here $S$ and $P$ are the scalar and pseudoscalar nonets (see Eq. (2.181)) and
as before $G^{\mu}$ is the gluon field. Since the eLSM is an effective
approach where the fields are physical meson fields, the gluons are used to
generate an effective, so-called Polyakov loop variable, which mimics some
properties of confinement (see below). Returning to the grand potential and
the partition function, $\mathcal{Z}$ can be written according to [274],

$\displaystyle{\mathcal{Z}}$ $\displaystyle=e^{-\beta
V\Omega(T,\mu_{q})}=\textnormal{Tr}\exp[-\beta\bigg{(}\hat{\mathcal{H}}-\sum_{i=u,d,s}\mu_{i}\hat{\mathcal{Q}}_{i}\bigg{)}\bigg{]}$
$\displaystyle=\int\limits_{\textnormal{PBC}}\prod_{a}{\mathcal{D}}\varphi_{a}\int\limits_{\textnormal{APBC}}\prod_{i}{\mathcal{D}q_{i}}{\mathcal{D}q^{\dagger}_{i}}\exp[-\int_{0}^{\beta}d\tau\int_{V}d^{3}x\bigg{(}{\mathcal{L}}+\mu_{q}\sum_{i}q^{\dagger}_{i}q_{i}\bigg{)}\bigg{]},$
(8.3)

where PBC and APBC refer to the periodic and antiperiodic boundary conditions,
$\hat{\mathcal{Q}}_{i}$ is the conserved charge operator, and $\varphi_{a}$
denotes all the mesonic fields. There is no integration over the gluons, since
they are only used as a constant background field to generate the mean-field
level Polyakov loop variable. Moreover, here we assumed symmetric quark
matter, i.e. $\mu_{u}=\mu_{d}=\mu_{s}\equiv\mu_{q}=\mu_{B}/3$, where $\mu_{q}$
is the quark and $\mu_{B}$ is the baryon chemical potential. Unfortunately,
even in the case of eLSM, the partition function in Eq. (8.3) cannot be
calculated exactly, so some approximation is needed. In [8] a so-called hybrid
approach was used, where the meson potential is at the tree-level, the fermion
determinant obtained after performing the functional integration over the
quark fields is evaluated for vanishing mesonic fluctuating fields (but the
integration itself is exact), and the lightest mesonic thermal fluctuations
($\pi$, $K$, $f_{0}^{L}$) are taken into account in the pressure. This
approach considers fermion vacuum and thermal fluctuations, neglects meson
vacuum fluctuations, and considers only the lightest meson fluctuations in the
pressure. Considering the mesons, this approach is not self-consistent. Self-
consistent approaches at different loop levels in the eLSM are outlined in
[236]. Before the explicit form of the grand potential can be given in this
hybrid approach, the above mentioned Polyakov loop variables should be
introduced.

### 8.2 Polyakov loop and Polyakov loop variables

The Polyakov loop operator, which is a special Wilson loop of the gauge field
in the temporal direction is defined after going to imaginary times
($G_{0}(t,\mathbf{x})\to-iG_{4}(\tau,\mathbf{x})$) as

$L=\mathcal{P}\exp\left(i\int_{0}^{\beta}d\tau G_{4}(\tau,\mathbf{x})\right),$
(8.4)

where we assumed that the spatial components of the gluon field vanish (for
more details see e.g. [277, 278, 279, 280]). $L$ and $L^{\dagger}$ are
matrices in the fundamental representation of the color gauge group $SU(3)$.
Next we define the color traced Polyakov loop as,

$\Phi_{P}(\mathbf{x})=\frac{1}{3}\Tr_{c}L(\mathbf{x}),\quad\bar{\Phi}_{P}(\mathbf{x})=\frac{1}{3}\Tr_{c}L^{\dagger}(\mathbf{x})\,,$
(8.5)

while the Polyakov loop variables are the thermal expectation values of
$\Phi_{P}(\mathbf{x})$ and $\bar{\Phi}_{P}(\mathbf{x})$,

$\braket{\Phi_{P}}_{T}=\frac{1}{3}\braket{\Tr_{c}L(\mathbf{x})}_{T},\;\braket{\bar{\Phi}_{P}}_{T}=\frac{1}{3}\braket{\Tr_{c}L^{\dagger}(\mathbf{x})}_{T}\;.$
(8.6)

In the pure gauge case, $\braket{\Phi_{P}}_{T}$ and
$\braket{\bar{\Phi}_{P}}_{T}$ are related to the free energy of the infinitely
heavy static quark and antiquark, respectively. As a further simplification,
we go to the Polyakov gauge in which
$G_{4}(\tau,\mathbf{x})=G_{4}(\mathbf{x})$ is time-independent and diagonal in
color space, and we also assume that $G_{4}(\mathbf{x})=G_{4}$ is homogeneous,
so it can be written as

$G_{4}=\varphi_{3}\lambda_{3}+\varphi_{8}\lambda_{8}\;,\text{ with
}\varphi_{3},\varphi_{8}\in\mathbb{R}\;.$ (8.7)

This can be substituted into the covariant derivative expression for the
quarks (8.2) and calculate the propagation of the constituent quarks on a
constant gluon background. The role of the gluons in this approximation will
be a color dependent imaginary chemical potential [8].

The Polyakov loop variables also have a Polyakov loop potential that drives
the deconfinement phase transition with increasing temperature. The potential
comes from pure gauge theory, and its form is constructed to reproduce some
thermodynamic properties calculated on the lattice. There are still several
possibilities for the form of the potential, but a commonly used one can be
given as

$\beta^{4}U_{\text{Pol}}(\Phi_{P},\bar{\Phi}_{P})=-\frac{1}{2}a(T)\Phi_{P}\bar{\Phi}_{P}+b(T)\ln\left(1-6\Phi_{P}\bar{\Phi}_{P}+4(\Phi_{P}^{3}+\bar{\Phi}_{P}^{3})-3(\Phi_{P}\bar{\Phi}_{P})^{2}\right),$
(8.8)

with

$a(T)=a_{0}+a_{1}\left(\frac{T_{0}}{T}\right)+a_{2}\left(\frac{T_{0}}{T}\right)^{2},\quad
b(T)=b_{3}\left(\frac{T_{0}}{T}\right)^{3},$ (8.9)

where for simplicity we used
$\Phi_{P}\equiv\braket{\Phi_{P}}_{T},\bar{\Phi}_{P}\equiv\braket{\bar{\Phi}_{P}}_{T}$
and the values of the constants are $a_{0}=3.51$, $a_{1}=-2.47$,
$a_{2}=15.22$, and $b_{3}=-1.75$.

Returning to the grand potential in the hybrid approximation outlined above,
it can be written as

$\Omega_{\textnormal{H}}(T,\mu_{q})=U(\left<\Phi\right>)+U_{\text{Pol}}(\Phi_{P},\bar{\Phi}_{P})+\Omega_{\bar{q}q}^{(0)}(T,\mu_{q}),$
(8.10)

where $U(\left<\Phi\right>)$ represents the tree-level meson potential,
$U(\Phi_{P},\bar{\Phi}_{P})$ is the Polyakov loop potential, and
$\Omega_{\bar{q}q}^{(0)}$ is the fermion contribution for non-vanishing scalar
backgrounds $\phi_{N}$ and $\phi_{S}$ and vanishing mesonic fluctuating
fields. The tree-level classical potential is given by

$U(\left<\Phi\right>)=\frac{m_{0}^{2}}{2}\big{(}\phi_{N}^{2}+\phi_{S}^{2}\big{)}-\frac{c_{1}}{2\sqrt{2}}\phi_{N}^{2}\phi_{S}-h_{S}\phi_{S}-h_{N}\phi_{N}+\frac{\lambda_{1}}{4}\big{(}\phi_{N}^{2}+\phi_{S}^{2}\big{)}^{2}+\frac{\lambda_{2}}{8}\big{(}\phi_{N}^{4}+2\phi_{S}^{4}\big{)}\;,$
(8.11)

which, a part from the anomaly term, coincides with Eq.
$V_{G\Phi}(G_{0},\phi_{N},\phi_{S})$ discussed in Eq. (3.31). Skipping some
details of the calculation of the fermion part – see [8] – one can arrive at

$\displaystyle\Omega_{\bar{q}q}^{(0)}(T,\mu_{q})$
$\displaystyle=\Omega_{\bar{q}q}^{(0)\textnormal{v}}+\Omega_{\bar{q}q}^{(0)\textnormal{T}}(T,\mu_{q}),$
(8.12) $\displaystyle\Omega_{\bar{q}q;R}^{(0)\textnormal{v}}$
$\displaystyle=-\frac{3}{8\pi^{2}}\sum_{i=u,d,s}m_{i}^{4}\ln\frac{m_{i}}{M_{0}}$
(8.13) $\displaystyle\Omega_{\bar{q}q}^{(0)\textnormal{T}}(T,\mu_{q})$
$\displaystyle=-2T\sum_{i}\int\frac{d^{3}p}{(2\pi)^{3}}\big{[}\ln
g_{i}^{+}(p)+\ln g_{i}^{-}(p)\big{]}\;,$ (8.14)

where

$\displaystyle g_{i}^{+}(p)$ $\displaystyle=1+3\bar{\Phi}_{P}\,e^{-\beta
E_{i}^{+}(p)}+3\Phi_{P}\,e^{-2\beta E_{i}^{+}(p)}+e^{-3\beta E_{i}^{+}(p)}\;,$
(8.15) $\displaystyle g_{i}^{-}(p)$ $\displaystyle=1+3\Phi_{P}\,e^{-\beta
E_{i}^{-}(p)}+3\bar{\Phi}_{P}\,e^{-2\beta E_{i}^{-}(p)}+e^{-3\beta
E_{i}^{-}(p)}\;,$ (8.16) $\displaystyle E_{i}^{\pm}(p)$
$\displaystyle=E_{i}(p)\mp\mu_{i},\;E_{i}(p)=\sqrt{\mathbf{p}^{2}+m_{i}^{2}}\;,$
(8.17) $\displaystyle m_{u,d}$
$\displaystyle=\frac{g_{F}}{2}\phi_{N}\quad\textrm{and}\quad
m_{s}=\frac{g_{F}}{\sqrt{2}}\phi_{S}\;.$ (8.18)

In (8.12) the contribution consists of two parts, a vacuum and a thermal part.
The vacuum part in (8.13) is already renormalized. In the thermal part (8.14)
the appearing $\ln g_{f}^{\pm}$ factors contain the Polyakov loop variables
and are related to modified Fermi-Dirac factors (see next section). Eq (8.18)
are the tree-level constituent quark masses. It is worth noting that if we
consider the PCAC relations, then the $\phi_{N/S}$ condensates are fixed by
the pion and kaon decay constants. Consequently, if one wanted to use baryons
instead of the constituent quarks, then $m_{p}=g_{F}\phi_{N}/2$ would be the
proton mass, which would lead to a very large value for the $g_{F}$ Yukawa
coupling. However, since $g_{F}$ enters the expressions for the meson masses
through the fermion vacuum fluctuations, this would lead to an unacceptable
meson spectrum. Therefore, the baryons cannot be used instead of the
constituent quarks in the way described above.

### 8.3 Field equations in the eLSM

The field equations, which determines the temperature and baryon chemical
potential dependence of the order parameters of the model, which are the
$\phi_{N}$, $\phi_{S}$ scalar condensates and the $\Phi_{P}$ and
$\bar{\Phi}_{P}$ Polyakov loop variables, are given by the stationary point of
the grand potential,

$\frac{\partial\Omega_{\textnormal{H}}}{\partial\phi_{N}}=\frac{\partial\Omega_{\textnormal{H}}}{\partial\phi_{S}}=\frac{\partial\Omega_{\textnormal{H}}}{\partial\Phi}=\frac{\partial\Omega_{\textnormal{H}}}{\partial\bar{\Phi}}=0,$
(8.19)

which leads in the current hybrid approximation to

$\displaystyle-\frac{d}{d\Phi_{P}}\left(\frac{U_{\text{Pol}}(\Phi_{P},\bar{\Phi}_{P})}{T^{4}}\right)+\frac{6}{T^{3}}\sum_{i=u,d,s}\int\frac{d^{3}p}{(2\pi)^{3}}\left(\frac{e^{-\beta
E_{i}^{-}(p)}}{g_{i}^{-}(p)}+\frac{e^{-2\beta
E_{i}^{+}(p)}}{g_{i}^{+}(p)}\right)=0,$ (8.20)
$\displaystyle-\frac{d}{d\bar{\Phi}_{P}}\left(\frac{U_{\text{Pol}}(\Phi_{P},\bar{\Phi}_{P})}{T^{4}}\right)+\frac{6}{T^{3}}\sum_{i=u,d,s}\int\frac{d^{3}p}{(2\pi)^{3}}\left(\frac{e^{-\beta
E_{i}^{+}(p)}}{g_{i}^{+}(p)}+\frac{e^{-2\beta
E_{i}^{-}(p)}}{g_{f}^{-}(p)}\right)=0,$ (8.21) $\displaystyle
m_{0}^{2}\phi_{N}+\left(\lambda_{1}+\frac{1}{2}\lambda_{2}\right)\phi_{N}^{3}+\lambda_{1}\phi_{N}\phi_{S}^{2}-\frac{1}{\sqrt{2}}c_{1}\phi_{N}\phi_{S}-h_{0N}+\frac{3}{2}g_{F}\left(\langle\bar{q}_{u}q_{u}\rangle_{{}_{T}}+\langle\bar{q}_{d}q_{d}\rangle_{{}_{T}}\right)=0,$
(8.22) $\displaystyle
m_{0}^{2}\phi_{S}+\left(\lambda_{1}+\lambda_{2}\right)\phi_{S}^{3}+\lambda_{1}\phi_{N}^{2}\phi_{S}-\frac{\sqrt{2}}{4}c_{1}\phi_{N}^{2}-h_{0S}+\frac{3}{\sqrt{2}}g_{F}\langle\bar{q}_{s}q_{s}\rangle_{{}_{T}}=0,$
(8.23)

where the renormalized quark tadpole integral is given by

$\displaystyle\langle\bar{q}_{i}q_{i}\rangle_{{}_{T}}$
$\displaystyle=4m_{i}\left[-\frac{m_{i}^{2}}{16\pi^{2}}\left(\frac{1}{2}+\ln\frac{m_{i}^{2}}{M_{0}^{2}}\right)+T_{i}\right],$
(8.24) $\displaystyle T_{i}$
$\displaystyle=\int\frac{d^{3}p}{(2\pi)^{3}}\frac{1}{2E_{i}(p)}\big{(}f^{-}_{i}(p)+f^{+}_{i}(p)\big{)}\,,$
(8.25) $\displaystyle f^{+}_{i}(p)$
$\displaystyle=\frac{\bar{\Phi}_{P}e^{-\beta E_{i}^{+}(p)}+2\Phi_{P}e^{-2\beta
E_{i}^{+}(p)}+e^{-3\beta E_{i}^{+}p)}}{g^{+}_{i}(p)},$ (8.26) $\displaystyle
f^{-}_{i}(p)$ $\displaystyle=\frac{\Phi_{P}e^{-\beta
E_{i}^{-}(p)}+2\bar{\Phi}_{P}e^{-2\beta E_{i}^{-}(p)}+e^{-3\beta
E_{i}^{-}(p)}}{g^{-}_{i}(p)}\;.$ (8.27)

Here $f^{\pm}_{i}(p)$ are the modified Fermi-Dirac distribution functions. To
solve the set of field equations Eqs. (8.20)-(8.23) a set of parameters is
needed, which are usually determined at $T=\mu_{B}=0$. The parameterization
procedure is described in Section 3.4. One could choose as an example the
parameters presented in Section 3.4, but at finite temperature there are some
new aspects to consider. The most important one is that from the lattice it is
known that the pseudocritical temperature $T_{c}$ at $\mu_{B}=0$ is
$T_{c}=151$ MeV [265]. However, once we have fixed the parameters, the set of
field equations can be solved for a given value of $T,\mu_{B}$. In practice,
one usually starts with a solution at $T=\mu_{B}=0$ and increases either $T$
or $\mu_{B}$, using the previous solution as the initial condition. Since the
set of equations is highly nonlinear, it is very difficult to find a solution
without a good initial condition. In this way we can determine the order
parameters ($\phi_{N}$, $\phi_{S}$, $\Phi_{P}$, $\bar{\Phi}_{P}$) either as a
function of $T$ for a fixed $\mu_{B}=\mu_{B,\text{fix}}$ or as a function of
$\mu_{B}$ for a fixed $T=T_{\text{fix}}$. As an example a parameterization
result is shown from [8] in Table 8.1, together with the values for the
parameters in Table 8.2.

Observable | Fit $\left[\text{MeV}\right]$ | Experiment $\left[\text{MeV}\right]$ | Observable | Fit $\left[\text{MeV}\right]$ | Experiment $\left[\text{MeV}\right]$
---|---|---|---|---|---
$f_{\pi}$ | $95.5$ | $92.2\pm 0.2$ | $f_{K}$ | $109.4\pm 0.6$ | $110.5\pm 0.8$
$m_{\pi}$ | $140.5$ | $138\pm 3.0$ | $m_{K}$ | $499.5$ | $495.6\pm 2.0$
$m_{\eta}$ | $542.1$ | $547.9\pm 0.02$ | $m_{\eta^{\prime}}$ | $964.3$ | $957.8\pm 0.06$
$m_{\rho}$ | $806.4$ | $775.3\pm 0.3$ | $m_{K^{\star}}$ | $915.2$ | $894.7\pm 0.3$
$m_{\phi}$ | $990.1$ | $1019.5\pm 0.02$ | $m_{a_{1}}$ | $1076.6$ | $1230\pm 40$
$m_{f_{1}\left(1420\right)}$ | $1416.0$ | $1426.4\pm 0.9$ | $m_{a_{0}(980)}$ | $720.8$ | $980\pm 20$
$m_{K_{0}^{\star}\left(800\right)}$ | $752.9$ | $682\pm 29$ | $m_{f_{0}^{L}(500)}$ | $283.7$ | $475\pm 75$
$m_{f_{0}^{H}(980)}$ | $737.6$ | $990\pm 20$ | $m_{u,d}$ | $322.4$ | $308\pm 31$
$m_{s}$ | $457.7$ | $483\pm 49$ | $\Gamma_{\rho\rightarrow\pi\pi}$ | $151.5$ | $149.1\pm 1.1$
$\Gamma_{K^{\star}\rightarrow K\pi}$ | $47.8$ | $48\pm 1.3$ | $\Gamma_{\phi\rightarrow\bar{K}K}$ | $3.53$ | $3.54\pm 0.02$
$\Gamma_{a_{1}\rightarrow\rho\pi}$ | $199.4$ | $425\pm 175$ | $\Gamma_{a_{1}\rightarrow\pi\gamma}$ | $0.37$ | $0.64\pm 0.25$
$\Gamma_{f_{1}\left(1420\right)\rightarrow K^{\star}K}$ | $44.5$ | $44.5\pm 2.1$ | $\Gamma_{a_{0}(980)}$ | $68.3$ | $75\pm 25$
$\Gamma_{K_{0}^{\star}(800)\rightarrow K\pi}$ | $600.1$ | $547\pm 24$ | $\Gamma_{f_{0}^{L}(500)\rightarrow\pi\pi}$ | $554.2$ | $550\pm 150$
$\Gamma_{f_{0}^{L}(500)\rightarrow KK}$ | $0.0$ | $0.0\pm 100$ | $\Gamma_{f_{0}^{H}(980)\rightarrow\pi\pi}$ | $81.7$ | $70\pm 30$
$\Gamma_{f_{0}^{H}(980)\rightarrow KK}$ | $0.0$ | $0.0\pm 20$ | $T_{c}(\mu_{B}=0)$ | $170.4$ | $151\pm 15.1$
Table 8.1: An example of fit results from [8], together with the experimental values taken from PDG (2012) [261].  Parameter | Value | Parameter | Value
---|---|---|---
$\phi_{N}$ [GeV] | $0.141$ | $g_{1}$ | $5.62$
$\phi_{S}$ [GeV] | $0.142$ | $g_{2}$ | $3.05$
$m_{0}^{2}$ [GeV2] | $2.39\times 10^{-4}$ | $h_{1}$ | $27.46$
$m_{1}^{2}$ [GeV2] | $6.33\times 10^{-8}$ | $h_{2}$ | $4.23$
$\lambda_{1}$ | $-1.67$ | $h_{3}$ | $5.98$
$\lambda_{2}$ | $23.51$ | $g_{F}$ | $4.57$
$c_{1}$ [GeV] | $1.31$ |  $M_{0}$ [GeV] | $0.35$
$\delta_{S}$ [GeV2] | $0.11$ |  |
Table 8.2: Parameter values for the fit shown in Table 8.1.

Some notes are in order, in this particular parameterization the scalar-
isoscalar $f_{0}^{L}$ and $f_{0}^{H}$ masses and their decay widths are also
fitted. In addition, to be compatible with lattice results, the pseudocritical
temperature $T_{c}$ at $\mu_{B}=0$ is also part of the fit. Note that scalar
states below $1$ GeV are used here simply because they gave better chi-squared
values than other particle assignments. In this respect, a tension between the
vacuum’s results and the present ones is visible. Namely, the role of the
$f_{0}^{L}$ (or $\sigma$) mass is paramount, since if its value is too high
the value of $T_{c}$ is dramatically increased, e.g. for $m_{f_{0}^{L}}>1$
GeV, $T_{c}>500$ MeV and the phase transition along the $\mu_{B}$ axis would
also be crossover instead of first order, so there would be no CEP. The
existence of the CEP is a common belief, but in principle it is not excluded
that the chiral or deconfinement phase boundary is a crossover all along.

### 8.4 Selected results

In this section we show some results using the solution of Eqs. (8.20)-(8.23)
with the parameter set of Table 8.2. First, the left figure of Fig. 8.2 shows
the temperature dependence of the $\phi_{N}(T)$ and $\phi_{S}(T)$ non-strange
and strange scalar condensates together with the Polyakov loop variable
$\Phi_{P}(T)$ at $\mu_{B}=0$.

Figure 8.2:  Temperature (left) and baryon chemical potential (right)
dependence of the order parameters ($\phi_{N}$, $phi_{S}$, $\Phi_{P}$,
$\bar{\Phi}_{P}$). On the left $\mu_{B}=0$, on the right $T=0$.

At $\mu_{B}$ the two Polyakov loop variables are the same
$\Phi_{P}(T,\mu_{B}=0)=\bar{\Phi}_{P}(T,\mu_{B}=0)$. It can be seen that both
$\phi_{N}$ and $\phi_{S}$ decrease smoothly from some initial value at $T=0$
and tend to zero at $T\to\infty$. There are various definitions for the
pseudocritical temperature $T_{c}$ in the case of a crossover transition. One
possibility is the value of the inflection point of $\phi_{n}(T)$, i.e.
$d\phi_{N}/dT=0$ and $d^{2}\phi_{N}/dT^{2}=0$ for some $T=T_{c}$. Another
definition can be the value of $T$ for which $\phi_{N}(T)/phi_{N}(T=0)=0.5$,
i.e. when the non/strange condensates drop to half of their original value.
While one definition, which is also used on the [281] lattice, is based on the
so-called subtracted condensate, which is defined as

$\Delta_{\text{sub}}(T)=\frac{(\phi_{N}-\frac{h_{0N}}{h_{0S}}\phi_{S})|_{T}}{(\phi_{N}-\frac{h_{0N}}{h_{0S}}\phi_{S})|_{T=0}},$
(8.28)

and which takes values between $1$ and $0$, while the definition of $T_{c}$ is
$\Delta_{\text{sub}}(T_{c})=0.5$. In this particular case, both the inflection
point of $\phi_{N}$ and the half value of the subtracted condensate give
$T_{c}=172$ MeV. The right figure in Fig. 8.2 shows the $\phi_{N/S}$
condensates as a function of $\mu_{B}$ at $T=0$. At $T=0$ the Polyakov loop
variables are identically zero,
$\Phi_{P}(\mu_{B},T=0)=\bar{\Phi}_{P}(\mu_{B},T=0)=0$. The phase transition is
first order along the $\mu_{B}$ axis, the transition temperature can again be
defined as the infection point in the unstable region, while the two spinodals
are given by the condition $d\phi_{N}/dT=\infty$ (or $dT/d\phi_{N}=0$).

The so-called curvature masses are defined as the second derivative of the
grand potential (or effective potential) at the minimum, which is the solution
of the field equations,

$m^{2}_{ab}=\frac{\partial^{2}\Omega_{H}(T,\mu_{q})}{\partial\varphi_{a}\partial\varphi_{b}}\bigg{|}_{\textnormal{min}}=m^{2}_{\text{tree},ab}+\delta
m^{2}_{\text{vac},ab}+\delta m^{2}_{T,ab},$ (8.29)

where $\varphi_{a}$ can be any scalar or pseudoscalar field. The curvature
mass has a tree-level part, a fermion vacuum part, and a fermion thermal part
coming from the zero and finite temperature fermion loop corrections to the
grand potential. The details and the explicit expression for the scalar and
pseudoscalar curvature masses can be found in [8]. The temperature dependence
of the masses can be seen in Fig. 8.3.

Figure 8.3: Temperature dependence of the scalar and pseudoscalar meson
masses.

It can be seen that the masses degenerate around $T_{c}$ as expected, but not
all of them. This is due to the fact that the $U(1)_{A}$ axial symmetry is not
fully restored, at least up to temperatures of $350-400$ MeV, where the model
is no longer reliable. In the case of mesons heavier than the pion or kaon,
there is a drop down around $T_{c}$ to reach the degenerate state with their
lighter partners. After this region, the thermal contributions become dominant
in all masses and all masses increase with increasing $T$. Recently, the
temperature dependence of the curvature masses of the vector and axial vectors
in [236] has also been investigated within the eLSM.

After solving the field equations and substituting the solution at each $T$
and $\mu_{B}$ back into the grand potential, it can be used to determine any
thermodynamic quantity. The pressure is given by

$p(T,\mu_{q})=\Omega_{\textnormal{H}}(T=0,\mu_{q})-\Omega_{\textnormal{H}}(T,\mu_{q}),$
(8.30)

while the entropy, baryon number density, baryon number susceptibility and
energy density are given by

$\displaystyle s$ $\displaystyle=\frac{\partial p}{\partial
T}\;,\quad\rho_{B}=\frac{\partial p}{\partial\mu_{B}}\;,$ (8.31)
$\displaystyle\chi_{B}$
$\displaystyle=\frac{\partial^{2}p}{\partial\mu^{2}_{B}}\;,\quad\varepsilon=-p+Ts+\mu_{B}\rho_{B}\;.$
(8.32)

The normalized pressure $p/T^{4}$ is shown in Fig. 8.4 together with a
continuum lattice result taken from [282] for comparison as a function of the
reduced temperature $t=(T-T_{c})/T_{c}$.

Figure 8.4:  The normalized pressure as a function of the reduced temperature
($t=(T-T_{c})/T_{c}$). A continuum lattice result is also shown. Thermal meson
fluctuations are also added for the lowest mass mesons ($\pi$, $K$,
$f_{0}^{L}$). The arrow on the right indicates the Stefan-Boltzmann limit of
QCD: $p_{SB}/T^{4}=5.2$.

As already mentioned, meson fluctuations are only considered in the pressure
with the following contribution

$\Delta p_{x}(T)=-N_{x}T\int\frac{d^{3}p}{(2\pi)^{3}}\ln(1-e^{-\beta
E_{x}(p)}),\quad E_{x}(p)=\sqrt{p^{2}+m_{x}^{2}},\quad x\in(\pi,K,f_{0}^{L}),$
(8.33)

where $N_{x}$ is the meson multiplicity factor with $N_{\pi}=3$, $N_{K}=4$ and
$N_{f_{0}^{L}}=1$. This treatment of the meson thermal fluctuation is not
self-consistent, since it is not introduced at the level of the grand
potential, so there is no meson fluctuation in the field equation. The
introduction of a self-consistent meson contribution beside the fermion one is
more complicated, as described in [236]. It can be seen that the meson
contributions, especially for the pion, have a huge effect at lower
temperatures and with their help the pressure curve is much closer to the
lattice result. On the other hand, as $T$ increases, the additional thermal
meson fluctuations overshoot the lattice results more and more. However, it
should be noted again that the validity of the model decreases at higher
temperatures.

Determining the pseudocritical temperatures for increasing values of $\mu_{B}$
draws the line of the phase boundary on the $T$-$\mu_{B}$ plane. A particular
example is shown in Fig. 8.5 for the parameter set presented in the previous
section.

Figure 8.5:  Chiral phase boundary in eLSM for a given set of parameters from
Table 8.2. The dashed curve shows the crossover, while the solid curve shows
the first order phase transition. The dashed-dotted curve is the chemical
freeze-out curve from [283]. The coordinates of the CEP are
$(T,\mu_{B})=(885,52.7)$ MeV.

As $\mu_{B}$ increases, the phase transition as a function of $T$ becomes
stronger and stronger, while for a certain value the derivative of
$d\phi_{N}/dT$ at the infection point becomes infinite. At this point, the
phase transition is second order, which we call the critical endpoint or CEP.
As $\mu_{B}$ continues to increase, the phase transition becomes first order
and there is an unstable region bounded by the two spinodal lines as shown in
the figure. In this particular example the coordinates of the CEP are
$(T,\mu_{B})=(885,52.7)$ MeV. With different parameterizations and different
approximations the location of the CEP can change, it can also happen,
especially for parameterizations with large $f_{0}^{L}$ mass
($m_{f_{0}^{L}}\gtrapprox 300$ MeV), that the CEP vanishes and the phase
boundary is only crossover. In many other models, the CEP also exists, but it
could be in a very different place, so finding the CEP experimentally is not
easy because there is no clear indication of where to find it.

As a final remark, if we introduce in addition to the scalar condensates also
condensates of different vector fields in the temporal direction, one can
calculate $p(\mu_{B})$ and $\varepsilon(\mu_{B})$ at $T=0$ and determine the
so-called equation of state, i.e. the $p(\varepsilon)$ curve. This can then be
used to study hybrid stars in the eLSM (for more details see [284, 285]).

## 9 Summary and outlook

In this review, the genesis, motivation, and main results of the extended
Linear Sigma Model (eLSM) have been presented [6, 7, 8, 9, 10]. The eLSM is a
chiral model for low-energy hadrons that contains on the same level chiral
partners, which are further classified as heterochiral and homochiral
multiplets [48, 49]. We have shown that the eLSM for $N_{f}=3$ can be used to
successfully describe the most salient features of mesonic phenomenology up to
2.5 GeV. Below, we summarize some of them and then we discuss possible future
tasks.

  * 1.

Besides (pseudo)scalar fields, which are present in every LSM, the eLSM
contains (axial) vector mesons from the beginning. For instance, the broad
resonance $a_{1}(1230)$ is regarded as the chiral partner of the $\rho$ meson
[6, 121].

  * 2.

The chiral partners of pseudoscalar fields, the famous scalar mesons, turn out
to be heavier than $1$ GeV. In this respect, the chiral partner of the pion is
the resonance $f_{0}(1370)$ [6, 7]. However, when the eLSM is studied in the
medium [8, 139], light scalar states are required tot correctly reproduce the
chiral phase transition. A way out of this conundrum could be the inclusion at
the same time of a nonet of light four-quark states (scalar resonances below 1
GeV) into the eLSM.

  * 3.

Other $J^{\mathcal{P}\mathcal{C}}$ quark-antiquark mesons have been coupled to
the eLSM as well. An interesting example is the inclusion of tensor and their
chiral partners, the axial-tensor mesons [10]. The former states are very well
known experimentally, the latter are still mostly unknown. The eLSM could make
prediction for the not-yet discovered axial-tensor states, finding that they
are broad.

  * 4.

Within the eLSM, hybrid mesons can be easily added, since they form chiral
multiplets which, under chiral transformations, behave just as regular quark-
antiquark multiplets [9]. The lowest hybrid nonet involves the exotic quantum
numbers $J^{\mathcal{P}\mathcal{C}}=1^{-+}$. In particular, the state
$\pi_{1}(1600)$ and the recently discovered resonance $\eta_{1}(1855)$ fit
well in this picture [224, 225]. Two additional predicted states,
$\eta_{1}(1660)$ and $K_{1}(1750)$, are still missing.

  * 5.

The eLSM contains the dilaton/glueball from its very onset, since the model is
built (in the chiral limit and neglecting the chiral anomaly) as being
dilatation and chirally invariant, the former being broken by one dimensionful
parameter present in the dilaton potential. The decays and mixing patterns of
the scalar glueballs are then naturally described within the eLSM. The
resonance $f_{0}(1710)$ turns out to be mostly gluonic [7, 122].

  * 6.

The tensor glueball can be easily coupled to the eLSM [128]: the main result
is a strong coupling to (axial-)vector mesons, thus the channel
$T_{G}\rightarrow\rho\rho\rightarrow 4\pi$ is expected to dominate. At
present, the resonance $f_{2}(1950)$ can be regarded as a good glueball
candidate, but further studies are needed.

  * 7.

The pseudoscalar glueball is coupled to regular $\bar{q}q$ states of the eLSM
via an $U(1)_{A}$ anomalous term [127]. The main decay channels are $KK\pi$ as
well as $\eta\pi\pi$ and $\eta^{\prime}\pi\pi$. The recently confirmed
pseudoscalar states $X(2370)$ and $X(2600)$ (both measured by BESIII in the
channel $\eta^{\prime}\pi\pi$ [251, 252, 253]) are good candidates for
containing a sizable pseudoscalar glueball amount in their wave function.
Note, as discussed in Ref. [49] also the intensity of the interaction as
driven by an instanton gas agrees with the experimentally measured widths of
these states. An important task for the future is the exact allocation of the
glueball amount in the two resonances mentioned above.

  * 8.

The vector glueball has been also coupled to the eLSM [129]. This is an
example in which a relatively heavy glueball (about 3.8 GeV) is considered:
the main outcome is that the decay channel $\omega\pi\pi$ is dominant.

  * 9.

Further topics have been considered: (i) a successful description of isospin
breaking is achieved, that is able to describe e.g. the small mass difference
of the neutral vs the charged pions [131]; (ii) the inclusion of radially
excited (pseudo)scalar fields allow to describe various resonances such as
$\pi(1300)$ and $f_{0}(1790)$ [130]; (iii) the extension to the four-flavor
case, $N_{f}=4$, leads to the correct description of certain decay widths,
such as $D^{*}(2010)^{+}\rightarrow D^{+}\pi^{0}$ [123].

  * 10.

The eLSM, together with Polyakov loop and quarks d.o.f., allows to study at
the same time the chiral and the deconfinement phase transitions and offers a
phenomenological description in agreement with other models and lattice QCD.
Moreover, it offers a consistent framework to investigate the location of the
CEP [8].

Besides what it has been achieved up to now, it is important to mention
interesting topics that can be studied in the future. The list below is not
intended to be complete, but it offers an idea of the many tasks that can be
addressed with the eLSM.

  1. 1.

The addition and/or a study in depth of quark-antiquark multiplets is a
possible straightforward application of the eLSM. For instance, the
heterochiral multiplet involving the pseudovector mesons and the orbitally
excited vector mesons has already been introduced in the eLSM but has not yet
been studied in detail. Also, the novel inclusion of the pseudotensor mesons
and the mesons with $J^{\mathcal{P}\mathcal{C}}=3^{--}$ [205] together with
their -yet unknown- chiral partners is doable, eventually shading light on the
latter states. Moreover, further radially excited states can be added, such as
the rather well known radially excited vector mesons together with their
chiral partners, the radially excited axial-vector mesons (for which
candidates do exist in the PDG).

  2. 2.

The tension linked to the scalar meson assignment shows that a nonet of light
scalar mesons in form of four-quark states should be added to the eLSM, thus
completing the approaches of Refs. [112, 114, 115]. In particular, in the
scalar-isoscalar sector, five scalar states need to be present: two mainly
four-quark objects, two quarkonia, and one glueball. Preliminary two-flavor
works that involve light four-quark mesonic field(s)have been discussed in
Refs. [286, 34, 134].

  3. 3.

A particularly interesting topic is the systematic study of terms that break
the chiral anomaly and involve other multiplets than the (pseudo)scalar one.
In this respect, a novel and quite promising mathematical object (denoted as
‘GPKJ $\epsilon$-product’, see Eq. (4.4) and Refs. [48, 49]) has been
introduced for this purpose and affects all the heterochiral multiplets.

  4. 4.

A related topic to the previous one is a systematic study of isoscalar mixing
in each homochiral nonet (such as the (axial-)vector one), for which there are
no axial anomalous terms. To this end, some terms that were at first neglected
in the eLSM because large-$N_{c}$ suppressed can be added and have a small but
significant contribution. Such terms may play also an important role when
isospin breaking is taken into account.

  5. 5.

The behavior of the model when changing certain parameters can be studied.
While the quark masses are fixed in Nature, it is quite easy to modify them in
models and on the lattice. The behavior of masses and decays as function of
the bare quark masses is in this respect promising.

  6. 6.

Two further and specular directions are possible: one is the study of the
$N_{f}=1$ case. In this limit, most mesons are stable and, quite
interestingly, it is possible to study such a system on the lattice [287,
288]. An eLSM study could help to understand the physics of the dilaton and
the lightest scalar meson. The other limit consists in analyzing the case
$N_{f}=5$: even if the explicit breaking of chiral symmetry is huge, some
specific decay terms could still show that some specific interactions retain
chiral symmetry, even in such a different energy domain.

  7. 7.

There is a full tower of glueballs predicted by lattice QCD and other
approaches. Up to now, only four of them (the 3 lightest ones and the vector
glueball) have been coupled to the eLSM. On the one hand, open questions
concerning the glueballs already coupled to the eLSM still exist. On the other
hand, in view of the interest of the community on such states, and also in the
perspective of future experimental searches, their complete study within the
eLSM can be useful to isolate their dominant decay widths and to predict
various decay ratios.

  8. 8.

Another important direction is the development of the model in the baryonic
sector, in particular the addition of the decuplet and its chiral partners,
extending the work of Ref. [137].

  9. 9.

Novel studies of the QCD phase diagrams are interesting, such as the
investigation of neutron stars, in connection with processes emitting
gravitational waves. Also the detailed investigation of quarkyonic matter
belongs to the outlook of the eLSM approach.

  10. 10.

The application of deep learning techniques in the study of parameterization
could be useful, since this problem is similar to finding a global minimum of
the loss function in deep learning.

In conclusion, due to its versatility and generality, the eLSM was applied in
various areas of low-energy QCD phenomenology, but at the same time many
potential studies -both in the vacuum and in the medium- can be realized in
the future. It is also expected that many experimental data from ongoing and
planned experiments will enlarge and complete the PDG tables of mesonic (as
well as baryonic) states, requiring for a useful, general, and adequate
modelling. Together with other approaches and with computer simulations, the
eLSM offers its valuable contribution to the understanding of old and new
resonances, in particular relating chiral partners and isolating states which
cannot be described as (predominantly) quark-antiquark objects.

## Acknowledgements

It is a great pleasure to thank our numerous collaborators and colleagues with
whom we extensively discussed about the eLSM over the past 15 years: F.
Divotgey, W. Eshraim, C. Fischer, A. Heinz, S. Janowski, A. Koenigstein, Gy.
Kovács, P. Lakaschus, L. Olbrich, G. Pagliara, D. Parganlija, M. Piotrowska,
R. Pisarski, C. Reisinger, D. H. Rischke, J. Sammet, J. Schaffner-Bielich, V.
Shastry, S. Strueber, Zs. Szép, J. Takátsy, A. Vereijken, Gy. Wolf, N.
Weickgenannt, Th. Wolkanowski-Gans, and M. Zetenyi. F. G. acknowledges support
from the Polish National Science Centre (NCN) through the OPUS 440 Project No.
2019/33/B/ST2/00613 during the preparation of this paper (up to 16/2/2024). S.
J. has partially been supported by the Polish National Science Centre (NCN)
Grant SONATA project No. 2020/39/D/ST2/02054. P. K. acknowledges support by
the Hungarian National Research, Development and Innovation Fund under Project
numbers FK 131982 (until 31/05/2024) and K 138277.

##  Appendix A Brief recall of the $U(N)$ and $SU(N)$ groups

Elements $U$ of the unitary $U(N)$ group are (in the fundamental
representation) $N\times N$ complex matrices satisfying the following
conditions

$\displaystyle UU^{\dagger}=U^{\dagger}U=\mathbb{1}_{N\times N}\text{ .}$
(A.1)

Each matrix $U$ can be written as (see e.g. Ref. [289]):

$\displaystyle U=\exp{-i\sum_{a=0}^{N^{2}-1}\theta_{a}t_{a}}\,\text{ ,}$ (A.2)

where the $t^{a}$ form a basis of Hermitian matrices. As usual, we set
$t_{0}=\frac{1}{\sqrt{2N}}\mathbb{1}_{N\times N}$ and for
$(a,b)\in[0,\cdots,N^{2}-1]$:

$\displaystyle\mathrm{Tr}\Big{(}t_{a}t_{b}\Big{)}=\frac{1}{2}\delta^{ab}\,\text{
.}$ (A.3)

Then, for $a\in[1,\cdots,N^{2}-1]$ one has $\mathrm{Tr}(t_{a})=0$.

Adding the additional condition $\det U=1$ to the elements of $U(N)$ we obtain
the special unitary group $SU(N)$. It can be expressed as in Eq. (A.2) upon
setting $\theta_{0}=0$, thus there are $N^{2}-1$ generators that satisfy the
following commutation relation

$\displaystyle[t_{a},t_{b}]=if_{abc}t_{c}\,,\text{ where
}(a,b,c)\in[1,\cdots,N^{2}-1]\,,$ (A.4)

where $f_{abc}$ are the antisymmetric structure constants of $SU(N)$. The
Jacobi identity is also fulfilled

$\displaystyle[[t_{a},t_{b}],t_{c}]+[[t_{b},t_{c}],t_{a}]+[[t_{c},t_{a}],t_{b}]=0\,\text{
.}$ (A.5)

Moreover, the symmetric structure constants $d_{abc}$ are given by

$\displaystyle\\{t_{a},t_{b}\\}=\frac{\delta_{ab}}{N}+d_{abc}t_{c}\text{ .}$
(A.6)

Another important subgroup of $SU(N)$ is the discrete center group $Z_{N}$,
defined as the diagonal elements $Z_{n}=e^{2\pi n/N}\mathbb{1}_{N\times N}$
for $n=0,1,...,N-1$. The symmetry under $Z_{N}$ is spontaneously broken at
high $T$, since one of the $N$ possible choice is picked up, but is fulfilled
in the QCD vacuum. This is correctly implemented by the Polyakov loop
variable, see chapter 8 for the eLSM application to the case $N=M_{c}=3$ and
Ref. [139] for the arbitrary case $N=N_{c}$.

The dimensions of the generators, as well as the structure constants, are
dependent on the form of the representation. Throughout the paper, we are
interested in the fundamental and the adjoint representation. For the generic
representation $\mathcal{R}$, $SU(N)$ generators satisfy

$\displaystyle\mathrm{Tr}\Big{(}t^{\mathcal{R}}_{a}t_{b}^{\mathcal{R}}\Big{)}=C_{1}^{\mathcal{R}}\delta^{ab}\,,\qquad\text{and}\,,\qquad\sum_{a=1}^{N^{2}-1}\Big{(}t^{\mathcal{R}}_{a}\Big{)}_{ij}\Big{(}t^{\mathcal{R}}_{a}\Big{)}_{jk}=C_{2}^{\mathcal{R}}\delta_{ik}$
(A.7)

Denoting the dimension of the representation $d^{\mathcal{R}}$ and the number
of the independent parameters $n_{G}$ of the group $G$, one has the general
relation $d^{\mathcal{R}}C_{2}^{\mathcal{R}}=n_{G}C_{1}^{\mathcal{R}}$. In the
case of the fundamental representation $\mathcal{F}$, its dimension
$d^{\mathcal{F}}=N$, $n_{SU(N)}=N^{2}-1$, $C_{1}^{\mathcal{F}}=\frac{1}{2}$
and $C_{2}^{\mathcal{F}}=\frac{N^{2}-1}{2N}$. For the adjoint representation
$\mathcal{A}$, the dimension of the representation is equal to the independent
parameters of the group: $d^{\mathcal{A}}=n_{G}=N^{2}-1$ and consequently,
$C_{1}^{\mathcal{A}}=C_{2}^{\mathcal{A}}=N$. In the later, a generic
transformation of $\phi^{a}$ under the adjoint representation reads
$\phi^{a}\rightarrow(U_{\mathcal{A}})_{a,b}\phi^{b}$. Upon introducing the
$N\times N$ matrix $\phi=\sum_{a=1}^{N^{2}-1}\phi^{a}t^{a}$, the matrix $\phi$
transforms as $\phi\rightarrow U\phi U^{\dagger}$, where $U$ belongs to the
fundamental representation. The latter equation is applied allover the
manuscript.

In the case $N=2$, one sets $t^{a}=\sigma^{a}/2$, where $\sigma^{a}$ are the
Pauli matrices defined in Eq. (2.3). In this case, $f_{abc}=\epsilon_{abc}$,
$d_{abc}=0$, and the $SU(2)$ group elements take the simple form

$\exp{\Big{(}i\vec{\sigma}\cdot\vec{n}\theta\Big{)}}=\cos{\theta}+i\vec{\sigma}\cdot\vec{n}\sin{\theta}\text{
.}$ (A.8)

For $N=3$, the generators are represented in terms of the 8 Gell-Mann matrices
$\lambda_{a}$ as $t_{a}=\frac{\lambda_{a}}{2}$ given in Eq. (A.9).

$\displaystyle\lambda^{1}=\begin{pmatrix}{0}&{1}&{0}&\\\ {1}&{0}&{0}\\\
{0}&{0}&{0}\end{pmatrix},\qquad\lambda^{2}=\begin{pmatrix}{0}&{-i}&{0}\\\
{i}&{0}&{0}\\\
{0}&{0}&{0}\end{pmatrix},\qquad\lambda^{3}=\begin{pmatrix}{1}&{0}&{0}\\\
{0}&{-1}&{0}\\\
{0}&{0}&{0}\end{pmatrix},\qquad\lambda^{4}=\begin{pmatrix}{0}&{0}&{1}\\\
{0}&{0}&{0}\\\ {1}&{0}&{0}\end{pmatrix}$ (A.9)
$\displaystyle\lambda^{5}=\begin{pmatrix}{0}&{0}&{-i}\\\ {0}&{0}&{0}\\\
{i}&{0}&{0}\end{pmatrix},\quad\lambda^{6}=\begin{pmatrix}{0}&{0}&{0}\\\
{0}&{0}&{1}\\\
{0}&{1}&{0}\end{pmatrix},\qquad\lambda^{7}=\begin{pmatrix}{0}&{0}&{0}\\\
{0}&{0}&{-i}\\\
{0}&{i}&{0}\end{pmatrix},\quad\lambda^{8}=\frac{1}{\sqrt{3}}\begin{pmatrix}{1}&{0}&{0}\\\
{0}&{1}&{0}\\\ {0}&{0}&{-2}\end{pmatrix}\,\text{ ,}$

and $\lambda_{0}=\sqrt{2/3}\mathbb{1}_{3\times 3}$. The structure constants
$f^{abc}$ read [289]

$\displaystyle f_{abc}=-2i\text{Tr}\Big{(}[t_{a},t_{b}]t_{c}\Big{)}\quad$
$\displaystyle\rightarrow\quad
f^{147}=f^{165}=f^{246}=f^{257}=f^{345}=f^{376}={\frac{1}{2}}\,,$ (A.10)
$\displaystyle f^{123}=1\ ,\quad f^{458}=f^{678}={\frac{\sqrt{3}}{2}}\,,$

while $d^{abc}$ are:

$\displaystyle
d_{abc}=2\text{Tr}\Big{(}\\{t_{a},t_{b}\\}t_{c}\Big{)}\,\quad\rightarrow$
$\displaystyle\,\qquad
d^{146}=d^{157}=d^{256}=d^{344}=d^{355}=-d^{247}=-d^{366}=-d^{377}=\frac{1}{2}\,,$
(A.11) $\displaystyle d^{118}$
$\displaystyle=d^{228}=d^{338}=-d^{888}=\frac{1}{\sqrt{3}}\,,\qquad
d^{448}=d^{558}=d^{668}=d^{778}=-\frac{1}{2\sqrt{3}}\,.$

##  Appendix B Brief recall of the large-$N_{c}$ limit in QCD

As well known, QCD is nonperturbative in its low-energy domain since the
coupling constant is large. Yet, a different kind of expansion as function of
the inverse of the color numbers $1/N_{c}$ is possible [15]. This expansion,
analogous to the semi-classical expansion, allows to explain many non-
perturbative features of QCD [16]. To this end, one needs to assume the
scaling $g_{\text{QCD}}\propto N_{c}^{-1/2}$, which is a natural consequence
of $\Lambda_{\text{QCD}}$ being taken as $N_{c}$-independent, see Sec. 2.1.
Moreover, it is convenient to introduce the double-line representation, since
it greatly simplifies the $1/N_{c}$ counting. In this notation, gluons can be
seen as color-anticolor objects, as it is shown in Fig. B.1.

According to this counting, the dominant Feynman graphs are the planar ones
with a minimal number of quark loops (diagrams on the plane without any
crossing lines), see details in Ref. [16]. Also, gluon loops dominate w.r.t.
quark loops, see for instance Fig. B.2.

Figure B.1: The normal (left) and the double line (right) notation for the
propagators and vertices of QCD. The indices on the right are color indices.

Figure B.2: The one-loop gluon (upper part) and fermion (lower part)
contribution to the gluon self-energy in the usual (left) and the double line
notation (right). The upper diagram scales as $g^{2}N_{c}\propto N_{c}^{0}$,
while the lower one as $g^{2}\propto N_{c}^{-1}$.

Following the recent review of Ref. [19], the consequences of large-$N_{c}$
phenomenology are summarized below:

  1. 1.

The masses of $\bar{q}q$ states $Q$, glueballs $G\equiv gg$, and hybrids
$H\equiv\overline{q}qg$ are $N_{c}$-independent at large $N_{c}$:

$M_{Q}\propto N_{c}^{0}\text{ , }M_{G}\propto N_{c}^{0}\text{ , }M_{H}\propto
N_{c}^{0}\text{ .}$ (B.1)

  2. 2.

The interaction between $n_{Q}$ quarkonia scales as

$A_{n_{Q}Q}\propto\frac{N_{c}}{N_{c}^{n_{Q}/2}}\text{ for }n_{Q}\geq 1\text{
,}$ (B.2)

implying that the scattering amplitudes vanish at large-$N_{c}$. In
particular, the decay process corresponds to $n_{Q}=3$, thus
$A_{\text{decay}}\propto N_{c}^{-1/2}$. Then, a generic decay width scales as
$\Gamma\propto 1/N_{c}$. Conventional quarkonia are stable at large $N_{c}.$

  3. 3.

The scattering amplitude between $n_{G}$ glueballs goes as

$A_{n_{G}G}\propto\frac{N_{c}^{2}}{N_{c}^{n_{G}}}\text{ for }n_{G}\geq 1\text{
,}$ (B.3)

even smaller than the previous case.

  4. 4.

The scattering amplitude between $n_{Q}$ quarkonia and $n_{G}$ glueballs
scales as

$A_{\left(n_{Q}Q\right)\left(n_{G}G\right)}\propto\frac{N_{c}}{N_{c}^{n_{Q}/2}N_{c}^{n_{G}}}\text{
for }n_{Q}\geq 1\text{ .}$ (B.4)

The glueball-quarkonium mixing corresponds to $n_{G}=n_{Q}=1$ and scales as
$A_{\text{mixing}}\propto N_{c}^{-1/2}$, hence suppressed as well.

  5. 5.

The scattering amplitude for $n_{Q}$ quarkonia and $n_{H}$ hybrids behaves as

$A_{\left(n_{Q}Q\right)\left(n_{H}H\right)}\propto\frac{N_{c}}{N_{c}^{n_{Q}/2}N_{c}^{n_{H}/2}}\text{
for }n_{Q}+n_{H}\geq 1\text{ .}$ (B.5)

For $n_{Q}=n_{H}=1$ one sees that the quarkonium-hybrid mixing scales as
$N_{c}^{0}$, implying that quarkonia and hybrids are indistinguishable at
large-$N_{c}$.

  6. 6.

The general scattering amplitude with $n_{H}$ quarkonia, $n_{G}$ glueballs,
and $n_{H}$ hybrids reads:

$A_{\left(n_{Q}Q\right)\left(n_{G}G\right)\left(n_{H}H\right)}\propto\frac{N_{c}}{N_{c}^{n_{Q}/2}N_{c}^{n_{G}}N_{c}^{n_{H}/2}}\text{
for }n_{Q}+n_{H}\geq 1\text{ .}$ (B.6)

  7. 7.

Four-quark states (both molecular and diquark-anti-diquark) tend to disappear
at large $N_{c}$, see however the discussion in Refs. [18, 19].

##  Appendix C The non-relativistic limit as a justification for the currents

As illustrative examples, we consider the cases of pseudoscalar, scalar, and
vector mesons with quark content $u\bar{d}$, that correspond to the mesons
$\pi^{+}$, $a_{0}^{+},$ and $\rho^{+}$, respectively. All the other cases can
be derived similarly.

The quark field ‘$i$’ (with $i=u,d,s$) can be expressed as

$q_{i}(x)=\sum_{s=\pm
1/2}\int\frac{d^{3}p}{(2\pi)^{3/2}}\sqrt{\frac{m_{i}}{E_{i}}}\left(b_{i}^{(s)}(\mathbf{p})u_{i}^{(s)}(\mathbf{p})e^{-ipx}+d_{i}^{(s)\dagger}(\mathbf{p})v_{i}^{(s)}(\mathbf{p})e^{ipx}\right)$
(C.1)

where $u^{(s)}(\mathbf{p})$ and $v^{(s)}(\mathbf{p})$ are spinors, while
$b_{i}^{(s)}$ and $d_{i}^{(s)\dagger}$ are the annihilation and creation
operators of quarks and antiquarks. Choosing the Dirac representation, suited
for the non-relativistic limit, one has:

$u_{i}^{(s)}(\mathbf{p})=\sqrt{\frac{E_{i}+m_{i}}{2m_{i}}}\left(\begin{array}[c]{l}\chi^{(s)}\\\
\frac{\mathbf{\sigma\cdot p}}{E_{i}+m_{i}}\chi^{(s)}\end{array}\right)\text{ ,
}v_{i}^{(s)}(\mathbf{p})=\sqrt{\frac{E_{i}+m_{i}}{2m_{i}}}\left(\begin{array}[c]{l}\frac{\mathbf{\sigma\cdot
p}}{E_{i}+m_{i}}\varepsilon\chi^{(s)}\\\
\varepsilon\chi^{(s)}\end{array}\right)\text{ ,}$ (C.2)

where the matrix $\varepsilon$ is such that
$\left(\varepsilon\right)^{ij}=\varepsilon^{ij}$ (Levi-Civita), and with

$\chi^{(1/2)}=\left(\begin{array}[c]{c}1\\\ 0\end{array}\right)\text{ ,
}\chi^{(-1/2)}=\left(\begin{array}[c]{c}0\\\ 1\end{array}\right)\text{ ,
}p\cdot x=E_{i}t-\mathbf{p\cdot x}\text{ ,
}E_{i}=\sqrt{\mathbf{p}^{2}+m_{i}^{2}}\text{ .}$ (C.3)

The vector $\chi^{(s)}$ describes the spin of the quark. The anti-commutation
relations read

$\left\\{b_{i}^{(s)}(\mathbf{p}_{1}),b_{j}^{(r)\dagger}(\mathbf{p}_{2})\right\\}=\delta_{ij}\delta_{rs}\delta(\mathbf{p}_{1}-\mathbf{p}_{2})\text{
;
}\left\\{d_{i}^{(s)}(\mathbf{p}_{1}),d_{j}^{(r)\dagger}(\mathbf{p}_{2})\right\\}=\delta_{ij}\delta_{rs}\delta(\mathbf{p}_{1}-\mathbf{p}_{2})\text{
.}$ (C.4)

The generic pseudoscalar current is
$P_{ij}\sim\bar{q}_{j}(x)i\gamma^{5}q_{i}(x),$ hence the state
$\left|\pi^{+}\right\rangle$

$\left|\pi^{+}\right\rangle\sim\int
d^{4}xe^{-ipx}\bar{q}_{u}(x)i\gamma^{5}q_{d}(x)\left|0\right\rangle$ (C.5)

corresponds to a pion-like structure with four-momentum
$p=(p^{0},\mathbf{p}).$ Namely, when acting on the vacuum $\bar{q}_{u}$
creates a quark $u$ and $q_{d}$ an antiquark $\bar{d}$. A straightforward
calculation (by plugging Eq. (C.2) into the one above), upon choosing
$\mathbf{p}=\mathbf{0}$, leads to (neglecting an overall constant):

$\left|\pi^{+}\right\rangle\sim\sum_{s_{1},s_{2}=\pm 1/2}\int
d^{3}q\sqrt{\frac{m_{u}}{E_{u}}}\sqrt{\frac{m_{d}}{E_{d}}}\bar{u}_{u}^{(s_{1})}(\mathbf{q})i\gamma^{5}v_{d}^{(s_{2})}(-\mathbf{q})b_{u}^{(s_{1})\dagger}(\mathbf{q})d_{d}^{(s_{1})\dagger}(-\mathbf{q})\left|0\right\rangle\text{
.}$ (C.6)

Thus, upon using

$\bar{u}_{u}^{(s_{1})}(\mathbf{q})i\gamma^{5}v_{d}^{(s_{2})}(-\mathbf{q})=-i\sqrt{\frac{E_{u}+m_{u}}{2m_{u}}}\sqrt{\frac{E_{d}+m_{d}}{2m_{d}}}\left(1+\frac{\mathbf{q}^{2}}{\left(E_{u}+m_{u}\right)\left(E_{d}+m_{d}\right)}\right)\varepsilon^{s_{1}s_{2}}\sim\varepsilon^{s_{1}s_{2}}\text{
,}$ (C.7)

one arrives at:

$\left|\pi^{+}\right\rangle\sim\int
d^{3}qA(\mathbf{q}^{2})\left(b_{u}^{(1/2)\dagger}(\mathbf{q})d_{d}^{(-1/2)\dagger}(-\mathbf{q})-b_{u}^{(-1/2)\dagger}(\mathbf{q})d_{d}^{(1/2)\dagger}(-\mathbf{q})\right)\left|0\right\rangle\text{
,}$ (C.8)

where the spin singlet
$\left|\uparrow\downarrow\right\rangle-\left|\downarrow\uparrow\right\rangle$
emerges automatically, showing that $S=0$. In general, the function
$A(\mathbf{q}^{2})$ is the momentum wave function of the pion, which here
cannot be determined since we used local currents (for nonlocal extension, see
e.g. Refs. [164, 64]). Yet, the generic expression $A(\mathbf{q}^{2})$, being
rotational invariant, assures that $L=0.$ Finally, a study of parity and
charge conjugation confirms the previously shown assignment. Under parity,

$U_{\mathcal{P}}\left|\pi^{+}\right\rangle=-\left|\pi^{+}\right\rangle$ (C.9)

follows from
$U_{\mathcal{P}}b_{u}^{(s)\dagger}(\mathbf{q})U_{\mathcal{P}}^{\dagger}=b_{u}^{(s)\dagger}(-\mathbf{q})$
and
$U_{\mathcal{P}}d_{d}^{(1/2)\dagger}(-\mathbf{q})U_{\mathcal{P}}^{\dagger}=-d_{d}^{(1/2)\dagger}(\mathbf{q}),$
while under charge conjugation

$U_{\mathcal{C}}\left|\pi^{+}\right\rangle=\left|\pi^{-}\right\rangle$ (C.10)

is a consequence of
$U_{\mathcal{C}}b_{u}^{(s)\dagger}(\mathbf{q})U_{\mathcal{C}}^{\dagger}=d_{u}^{(s)\dagger}(\mathbf{q})$
and
$U_{\mathcal{C}}d_{d}^{(1/2)\dagger}(\mathbf{q})U_{\mathcal{C}}^{\dagger}=u_{d}^{(1/2)\dagger}(\mathbf{q})$.

The scalar case is, at first glance simple, but indeed the result is quite
instructive. Upon using the scalar current $S_{ij}\sim\bar{q}_{j}(x)q_{i}(x),$
for the $\bar{d}u$ object one studies

$\left|a_{0}^{+}\right\rangle\sim\sum_{s_{1},s_{2}=\pm 1/2}\int
d^{3}q\sqrt{\frac{m_{u}}{E_{u}}}\sqrt{\frac{m_{d}}{E_{d}}}\bar{u}_{u}^{(s_{1})}(\mathbf{q})v_{d}^{(s_{2})}(-\mathbf{q})b_{u}^{(s_{1})\dagger}(\mathbf{q})d_{d}^{(s_{1})\dagger}(-\mathbf{q})\left|0\right\rangle$
(C.11)

where $\gamma^{5}$ has been replaced by the identity matrix. Hence:

$\bar{u}_{u}^{(s_{1})}(\mathbf{q})v_{d}^{(s_{2})}(-\mathbf{q})\sim\chi^{(s_{1})\dagger}\mathbf{\sigma}\cdot\mathbf{q}\chi^{(s_{2})\dagger}\text{
.}$ (C.12)

Let us look, for instance, at the specific direction $q_{z},$ leading to the
following component of the wave function:

$\left|a_{0}^{+}\right\rangle\sim\int
d^{3}qq_{z}A(\mathbf{q}^{2})\left(b_{u}^{(1/2)\dagger}(\mathbf{q})d_{d}^{(1/2)\dagger}(-\mathbf{q})-b_{u}^{(-1/2)\dagger}(\mathbf{q})d_{d}^{(-1/2)\dagger}(-\mathbf{q})\right)\left|0\right\rangle+...$
(C.13)

where dots refer to the terms proportional to $q_{x}$ and $q_{y},$
respectively. It is evident that the spin is $S=1,$ since the structure
$\left|\uparrow\uparrow\right\rangle-\left|\downarrow\downarrow\right\rangle$
arises. Moreover, the momentum wave function $q_{z}A(\mathbf{q}^{2})$ belongs
to the $L=1$ multiplet. Thus, this exercise shows that scalar current
corresponds to $L=S=1$, as expected. Moreover, it is a scalar $J=0$ since the
original current is such. A direct application of parity and charge
conjugation confirms that

$U_{\mathcal{P}}\left|a_{0}^{+}\right\rangle=\left|a_{0}^{+}\right\rangle\text{
},\text{
}U_{\mathcal{C}}\left|a_{0}^{+}\right\rangle=\left|a_{0}^{-}\right\rangle\text{
}.$ (C.14)

As a last example we may consider vector mesons with currents
$V_{ij}^{\mu}\sim\bar{q}_{j}(x)\gamma^{\mu}q_{i}(x).$ Thus, upon choosing
$\mu=3$ for simplicity, the meson $\rho^{+}$ corresponds to

$\left|\rho^{+}\right\rangle\sim\int
d^{4}xe^{-ipx}\bar{q}_{u}(x)\gamma^{3}q_{d}(x)\left|0\right\rangle\text{ .}$
(C.15)

Since
$\bar{u}_{u}^{(s_{1})}(\mathbf{q})\gamma^{3}v_{d}^{(s_{2})}(-\mathbf{q})\sim\chi^{(s_{1})\dagger}\mathbf{\sigma}_{z}\varepsilon\chi^{(s_{2})\dagger}$,
one arrives to

$\left|\rho^{+}\right\rangle\sim\int
d^{3}qA(\mathbf{q}^{2})\left(b_{u}^{(1/2)\dagger}(\mathbf{q})d_{d}^{(-1/2)\dagger}(-\mathbf{q})+b_{u}^{(-1/2)\dagger}(\mathbf{q})d_{d}^{(-1/2)\dagger}({-}\mathbf{q})\right)\left|0\right\rangle\text{
,}$ (C.16)

which corresponds to $L=0$ and $S=1$ (the latter in the form
$\left|\uparrow\downarrow\right\rangle+\left|\downarrow\uparrow\right\rangle$,
referring to $S=1$ and $S_{z}=0).$

These arguments can be repeated for any current introduced in Sec. 2.3 and
show the correctness of the field assignments performed in this work in
particular and in all these types of model in general.

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