# Mass Spectrum of Non-Charmed and Charmed Meson States in Extended Linear-
Sigma Model

Azar I. Ahmadov  Department of Theoretical Physics, Baku State University, Z.
Khalilov st. 23, AZ1148, Baku, Azerbaijan  Institute for Physical Problems,
Baku State University, Z. Khalilov st. 23, AZ1148, Baku, Azerbaijan  Abdel
Nasser Tawfik  Future University in Egypt (FUE), Fifth Settlement, 11835 New
Cairo, Egypt<EMAIL_ADDRESS>
###### Abstract

For scalar and pseudoscalar meson fields and the up, down, strange and charm
quark flavors, we construct the effective Lagrangian of the extended linear-
sigma model (eLSM). Based on its global chiral symmetry, analytical
expressions for the masses of the scalar, pseudoscalar, vector and axial-
vector meson states are derived. The various eLSM parameters are determined,
analytically and estimated, numerically. This allows a precise calculations
for the masses of sixteen non-charmed and thirteen charmed meson states. We
confront these results with the recent compilation of the particle data group
(PDG) and concluded that the masses of the sixteen non-charmed and thirteen
charmed meson states calculated with the eLSM are in excellent agreement with
the PDG. Thus, we suggest that the eLSM with its parameters is a suitable
theoretical approach to determine the mass spectra of the various non-charmed
and charmed meson states, especially at vanishing temperature.

Chiral Lagrangians, Sigma model, Charmed mesons

###### pacs:

2.39.Fe,12.40.Yx,13.25.Ft

††preprint: ECTP-2023-11††preprint: WLCAPP-2023-11††preprint: FUE-2023-11

##  I Introduction

The extended linear-sigma model (eLSM), in which four quark flavors in
addition to the scalar and pseudoscalar meson fields are properly integrated
in an effective Lagrangian [1, 2], allows the construction of meson states
$\langle\bar{q}q\rangle=\langle\bar{q}_{\ell}q_{r}-\bar{q}_{\ell}q_{r}\rangle\neq
0$ (with at least one charm quark) [3]. With their chiral structures, the
various meson states can be classified according to the relevant quantum
numbers, orbital angular momenta $J$, parity $P$, and charge conjugates $C$
into scalar $J^{PC}=0^{++}$, pseudoscalar $J^{PC}=0^{-+}$, vector
$J^{PC}=1^{--}$, and axialvector mesons $J^{PC}=1^{++}$ [4, 5]. This work
presents a systematic estimation of different non-charmed and charmed meson
states. To this end, the effective Lagrangian i) describes various low-lying
non-charmed and charmed meson states and ii) determines their mass spectra, at
vanishing temperature.

We develop and introduce an extended version of eLSM in which four quark
flavors (up, down, strange, and charm) are included together with scalar and
pseudoscalar glueball fields (dilaton Lagrangian). These are the basic degrees
of freedom in the eLSM, which was succeeded in describing the vacuum
phenomenology of the non-strange, strange, and charm mesons [6, 7, 8, 9, 10].

In this regard, we recall the quantum choromodynamics (QCD), the a gauge
theory for color interactions of quarks and gluons, in which the Lagrangian is
invariant under local color transformations, i.e., the physical content
remains unchanged if colors of quarks and gluons are transformed, while the
interactions themselves are flavor-blind. In eLSM, the global chiral symmetry
is explicitly broken by non-vanishing quark masses and quantum effects [11],
and spontaneously broken by a non-vanishing expectation value of the quark
condensate in the QCD vacuum. The new version of eLSM is an effective
framework to study the QCD symmetries including i) the isospin symmetry which
is a global transformation referring to SU(2) rotation in flavor space of up
and down quarks, where the Lagrangian is invariant for identical or vanishing
masses [12, 13, 14], ii) the global chiral symmetry, which is exact in the
chiral limit of massless QCD, i.e., left- and right-handed pieces of the
Lagrangian are separately invariant, $\Psi=\Psi_{R}+\Psi_{L}$ and
$\Psi_{R,L}=(1/2)(1\pm\gamma^{5})\Psi$ [15], iii) spontaneous chiral symmetry
breaking. i.e., the chiral symmetry broken by properties of the QCD vacuum,
e.g., Higgs mechanism, where pions are the lightest Goldstone bosons of the
broken symmetry [16, 17], iv) discrete $C$, $P$, and $T$ symmetries [18], and
v) classical dilatation (scale) symmetry [19].

As for SU(4), we emphasize that good four decades ago, the spin-zero mass
spectrum and the leptonic decay constants were determined in the one-loop
approximation of SU(4) linear-sigma model [20]. The phenomenology of charmed
mesons was investigated in eLSM [18]. Also recently, the quark-hadron phase
structure was studied in the mean-field approximation of the SU(4) Polyakov
linear-sigma model, at finite temperatures and densities [6]

The present script is organized as follows. The whole formalism of the mesonic
part of the Lagrangian of the SU(4) extended linear-sigma model is outlined in
section II. Our results on the non-charmed and charmed meson states and their
mass spectra are discussed in section III. Section IV is devoted to the final
conclusions and outlook.

##  II Mesonic Lagrangian of extended Linear-Sigma Model

The extended linear-sigma model (eLSM) is an effective QCD-like model, in
which the chiral symmetry is realized in the hadronic approaches in a linear
representation [21]. The nonlinear representation considers only the Goldstone
bosons besides the vector mesons [22, 23]. The linear representation, on the
other hand, allows to consider the chiral partners of the Goldstone bosons,
the scalar mesons. Its extension to the vector sector allows the introduction
of vector and axialvector mesons. Therefore, the construction of an extended
linear-sigma model (eLSM), with either $N_{f}=2$ or $N_{f}=3$ or recently
$N_{f}=4$ attracted the attention of many theoreticians. As shortly discussed
in the introduction, eLSM takes into account the chiral symmetry besides many
other QCD symmetries [24].

The Lagrangian for $N_{f}=4$ with a global chiral invariance [6] has an
analogous form as that of the corresponding Lagrangian for $N_{f}=3$ [10].
Only for $N_{f}=4$ the mass term $-2\mathtt{Tr}[\epsilon\Phi^{\dagger}\Phi]$
must be added [11]. Then, the Lagrangian of the mesonic sector including
scalar, pseudo-scalar, vector, and axial-vector meson states in addition to
scalar and pseudo-scalar glueballs and the interactions together with their
corresponding anomalies becomes [4, 5]

$\displaystyle\mathcal{L}_{\mathtt{m}}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{\mathtt{ps}}+\mathcal{L}_{\mathtt{av}}+\mathcal{L}_{\mathtt{int}}+\mathcal{L}_{\mathtt{anomaly}}+\mathcal{L}_{\mathtt{dilaton}}+\mathcal{L}_{\mathtt{emass}},$
(1)

where

$\displaystyle\mathcal{L}_{\mathtt{ps}}$ $\displaystyle=$
$\displaystyle\mathtt{Tr}\left[\left(\mathcal{D}^{\mu}\Phi\right)^{\dagger}\left(\mathcal{D}^{\mu}\Phi\right)-m_{0}^{2}\left(\frac{G}{G_{0}}\right)^{2}\mathtt{Tr}\left(\Phi^{\dagger}\Phi\right)\right]-\lambda_{1}\left[\mathtt{Tr}\left(\Phi^{\dagger}\Phi\right)\right]^{2}-\lambda_{2}\mathtt{Tr}\left[\left(\Phi^{\dagger}\Phi\right)\right]^{2}$
(2) $\displaystyle+$
$\displaystyle\mathtt{Tr}\left[H\left(\Phi+\Phi^{\dagger}\right)\right],$
$\displaystyle\mathcal{L}_{\mathtt{av}}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}\mathtt{Tr}\left[\left(L^{\mu\nu}\right)^{2}+\left(R^{\mu\nu}\right)^{2}\right]+\mathtt{Tr}\left\\{\left[\left(\frac{G}{G_{0}}\right)^{2}\frac{m^{2}}{2}+\Delta\right]\left[\left(L^{\mu}\right)^{2}+\left(R^{\mu}\right)^{2}\right]\right\\}$
(3)
$\displaystyle-2ig_{2}\left\\{\mathtt{Tr}\left(L_{\mu\nu}\left[L^{\mu},L^{\nu}\right]\right)+\mathtt{Tr}\left(R_{\mu\nu}\left[R^{\mu},R^{\nu}\right]\right)\right\\},$
$\displaystyle\mathcal{L}_{\mathtt{int}}$ $\displaystyle=$
$\displaystyle\frac{h_{1}}{2}\mathtt{Tr}\left(\Phi^{\dagger}\Phi\right)\mathtt{Tr}\left[\left(L^{\mu}\right)^{2}+\left(R^{\mu}\right)^{2}\right]+h_{2}\mathtt{Tr}\left[\left(R^{\mu}\right)^{2}\Phi+\left(L^{\mu}\Phi\right)^{2}\right]+2h_{3}\mathtt{Tr}\left(L^{\mu}\Phi
R_{\mu}\Phi^{\dagger}\right),\hskip 19.91692pt$ (4)
$\displaystyle\mathcal{L}_{\mathtt{anomaly}}$ $\displaystyle=$ $\displaystyle
C\left(\det\Phi+\det\Phi^{\dagger}\right)^{2}+iC_{\tilde{G}}\left(\det\Phi+\det\Phi^{\dagger}\right),$
(5) $\displaystyle\mathcal{L}_{\mathtt{dilaton}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\mathcal{D}_{\mu}G\right)^{2}-\frac{1}{4}\frac{m^{2}_{G}}{\Lambda^{2}}\left(G^{4}\ln\frac{G^{2}}{\Lambda^{2}}-\frac{G^{4}}{4}\right),$
(6) $\displaystyle\mathcal{L}_{\mathtt{emass}}$ $\displaystyle=$
$\displaystyle-2\mathtt{Tr}\left[\epsilon\left(\Phi^{\dagger}\Phi\right)\right],$
(7)

where $G$ is the scalar glueball and $\tilde{G}$ is the pseudoscalar glueball.
$C$, $C_{\tilde{G}}$ are constants introduced for a better fits of the mesons
and glueballs, respectively. $m_{G}$ is the lowest-lying glueball mass, which
is determined in the quenched approximation, i.e., no quarks [25]. $\Lambda$
is the QCD scaling parameter. The dilaton Lagrangian is conjectured to mimic
the QCD trace anomaly. The field $\Phi$ is a complex $N_{f}\times N_{f}$
matrix for scalar $\sigma_{a}$ with $J^{PC}=0^{++}$, pseudoscalar $\pi_{a}$
with $J^{PC}=0^{-+}$, vector with $J^{PC}=0^{--}$, and axialvector mesons with
$J^{PC}=0^{++}$, Appendix B

$\displaystyle\Phi$ $\displaystyle=$
$\displaystyle\sum_{a=0}^{N_{f}^{2}-1}T_{a}\left(\sigma_{a}+i\pi_{a}\right),$
(8)

where the scalar mesons are given as

$\displaystyle T_{a}\pi_{a}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\begin{tabular}[]{c c c
c}$\frac{\sigma_{0}}{2}+\frac{\sigma_{3}}{\sqrt{2}}+\frac{\sigma_{8}}{\sqrt{6}}+\frac{\sigma_{15}}{2\sqrt{3}}$&$\frac{\sigma_{1}-i\sigma_{2}}{\sqrt{2}}$&$\frac{\sigma_{4}-i\sigma_{5}}{\sqrt{2}}$&$\frac{\sigma_{9}-i\sigma_{10}}{\sqrt{\sqrt{2}}}$\\\
$\frac{\sigma_{1}+i\sigma_{2}}{\sqrt{2}}$&$\frac{\sigma_{0}}{\sqrt{2}}-\frac{\sigma_{3}}{\sqrt{2}}+\frac{\sigma_{8}}{\sqrt{6}}+\frac{\sigma_{15}}{2\sqrt{3}}$&$\frac{\sigma_{6}-i\sigma_{7}}{\sqrt{2}}$&$\frac{\sigma_{11}-i\sigma_{12}}{\sqrt{2}}$\\\
$\frac{\sigma_{4}+i\sigma_{5}}{\sqrt{2}}$&$\frac{\sigma_{6}+i\sigma_{7}}{\sqrt{2}}$&$\frac{\sigma_{0}}{2}-\sqrt{\frac{2}{3}}\sigma_{8}+\frac{\sigma_{15}}{2\sqrt{3}}$&$\frac{\sigma_{13}-i\sigma_{14}}{\sqrt{2}}$\\\
$\frac{\sigma_{9}+i\sigma_{10}}{\sqrt{2}}$&$\frac{\sigma_{11}+i\sigma_{12}}{\sqrt{2}}$&$\frac{\sigma_{13}+i\sigma_{14}}{\sqrt{2}}$&$\frac{\sigma_{0}}{2}-\frac{\sqrt{3}}{2}\sigma_{15}$\end{tabular}\right).\hskip
19.91692pt$ (13)

Similarly, the pseudo-scalar mesons become

$\displaystyle T_{a}\pi_{a}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left(\begin{tabular}[]{c c c
c}$\frac{\pi_{0}}{2}+\frac{\pi_{3}}{\sqrt{2}}+\frac{\pi_{8}}{\sqrt{6}}+\frac{\pi_{15}}{2\sqrt{3}}$&$\frac{\pi_{1}-i\pi_{2}}{\sqrt{2}}$&$\frac{\pi_{4}-i\pi_{5}}{\sqrt{2}}$&$\frac{\pi_{9}-i\pi_{10}}{\sqrt{\sqrt{2}}}$\\\
$\frac{\pi_{1}+i\pi_{2}}{\sqrt{2}}$&$\frac{\pi_{0}}{\sqrt{2}}-\frac{\pi_{3}}{\sqrt{2}}+\frac{\pi_{8}}{\sqrt{6}}+\frac{\pi_{15}}{2\sqrt{3}}$&$\frac{\pi_{6}-i\pi_{7}}{\sqrt{2}}$&$\frac{\pi_{11}-i\pi_{12}}{\sqrt{2}}$\\\
$\frac{\pi_{4}+i\pi_{5}}{\sqrt{2}}$&$\frac{\pi_{6}+i\pi_{7}}{\sqrt{2}}$&$\frac{\pi_{0}}{2}-\sqrt{\frac{2}{3}}\pi_{8}+\frac{\pi_{15}}{2\sqrt{3}}$&$\frac{\pi_{13}-i\pi_{14}}{\sqrt{2}}$\\\
$\frac{\pi_{9}+i\pi_{10}}{\sqrt{2}}$&$\frac{\pi_{11}+i\pi_{12}}{\sqrt{2}}$&$\frac{\pi_{13}+i\pi_{14}}{\sqrt{2}}$&$\frac{\pi_{0}}{2}-\frac{\sqrt{3}}{2}\pi_{15}$\end{tabular}\right).\hskip
19.91692pt$ (18)

The chiral symmetry is explicitly broken by nonvanishing external field
matrices $H$, $\Delta$, and $\epsilon$,

$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{a=0}^{N_{f}^{2}-1}h_{a}T_{a}=h_{0}T_{0}+h_{8}T_{8}+h_{15}T_{15},$
(19) $\displaystyle\Delta$ $\displaystyle=$
$\displaystyle\sum_{a=0}^{N_{f}^{2}-1}h_{a}\delta_{a}=h_{0}\delta_{0}+h_{8}\delta_{15}+h_{a}\delta_{15},$
(20) $\displaystyle\epsilon$ $\displaystyle=$
$\displaystyle\epsilon_{c}=m_{c}^{2}=\frac{1}{2}\left[m_{\chi_{c0}}^{2}-m_{0}^{2}-\lambda_{1}\left(\sigma_{x}^{2}+\sigma_{y}^{2}\right)-3\sigma_{c}^{2}\left(\lambda_{1}+\lambda_{2}\right)\right],$
(21)

where $T_{a}=\lambda_{a}/2$ are the generators of the group U($N_{f}$) with
$\lambda_{a}$ being the Gell-Mann matrices, Appendix A.

In SU(4)${}_{\ell}\,\times$SU(4)r model, the quark condensates are given as

$\displaystyle\sigma_{x}$ $\displaystyle=$
$\displaystyle\frac{\sigma_{0}}{\sqrt{2}}+\frac{\sigma_{8}}{\sqrt{3}}+\frac{\sigma_{15}}{\sqrt{6}},$
(22) $\displaystyle\sigma_{y}$ $\displaystyle=$
$\displaystyle\frac{\sigma_{0}}{2}-\sqrt{\frac{2}{3}}\sigma_{8}+\frac{1}{2\sqrt{3}}\sigma_{15},$
(23) $\displaystyle\sigma_{c}$ $\displaystyle=$
$\displaystyle\frac{\sigma_{0}}{2}+\frac{\sqrt{3}}{2}\sigma_{15},$ (24)

where $\sigma_{x}$ counts for the light quark (up and down) condensate and
$\sigma_{y}$ counts for the strange quark condensate. $\sigma_{c}$ as the name
says is the charm quark condensates. On the other hand, the quark masses
$m_{q}$ could be related to the resulting complex $N_{f}\times N_{f}$ matrix
for scalar $J^{PC}=0^{++}$, pseudoscalar $J^{PC}=0^{-+}$, vector
$J^{PC}=0^{--}$, and axialvector mesons $J^{PC}=0^{++}$, i.e., to $\Phi$, Eq.
(127). With $m_{0}=(g/2)\Phi$, where $g$ is the Yukawa coupling, we get

$\displaystyle m_{u}$ $\displaystyle=$
$\displaystyle\frac{g}{2}\left[\frac{\sigma_{0}}{\sqrt{2}}+\frac{\sigma_{8}}{\sqrt{3}}+\frac{\sigma_{15}}{\sqrt{6}}\right]=\frac{g}{2}\sigma_{x},$
(25) $\displaystyle m_{d}$ $\displaystyle=$
$\displaystyle\frac{g}{2}\left[\frac{\sigma_{0}}{\sqrt{2}}+\frac{\sigma_{8}}{\sqrt{3}}+\frac{\sigma_{15}}{\sqrt{6}}\right]=\frac{g}{2}\sigma_{x},$
(26) $\displaystyle m_{s}$ $\displaystyle=$
$\displaystyle\frac{g}{2}\left[\frac{\sigma_{0}}{\sqrt{2}}-\frac{2\sigma_{8}}{\sqrt{3}}+\frac{\sigma_{15}}{\sqrt{6}}\right]=\frac{g}{\sqrt{2}}\sigma_{y},$
(27) $\displaystyle m_{c}$ $\displaystyle=$
$\displaystyle\frac{g}{2}\left[\frac{\sigma_{0}}{\sqrt{2}}-\sqrt{\frac{3}{2}}\sigma_{15}\right]=\frac{g}{\sqrt{2}}\sigma_{c}.$
(28)

Then, following quantities could be defined

$\displaystyle\mathtt{Tr}\left(\Phi^{\dagger}\Phi\right)$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left[\left(\sigma_{x}\right)^{2}+\left(\sigma_{y}\right)^{2}+\left(\sigma_{c}\right)^{2}\right],$
(29)
$\displaystyle\left[\mathtt{Tr}\left(\Phi^{\dagger}\Phi\right)\right]^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{4}\left[\left(\sigma_{x}\right)^{4}+\left(\sigma_{y}\right)^{4}+\left(\sigma_{c}\right)^{4}+2\left(\sigma_{x}\right)^{2}\left(\sigma_{y}\right)^{2}+2\left(\sigma_{y}\right)^{2}\left(\sigma_{c}\right)^{2}+\left(\sigma_{x}\right)^{2}\left(\sigma_{c}\right)^{2}\right],\hskip
19.91692pt$ (30)
$\displaystyle\mathtt{Tr}\left[\epsilon\Phi^{\dagger}\Phi\right]$
$\displaystyle=$
$\displaystyle\frac{1}{2}\epsilon_{c}\left(\sigma_{c}\right)^{2},$ (31)
$\displaystyle\mathtt{Tr}\left[H\left(\Phi^{\dagger}+\Phi\right)\right]$
$\displaystyle=$ $\displaystyle
h_{x}\sigma_{x}+h_{y}\sigma_{y}+h_{c}\sigma_{c},$ (32) $\displaystyle
c\left(\det\Phi+\det\Phi^{\dagger}\right)$ $\displaystyle=$
$\displaystyle\frac{c}{4}\left(\sigma_{x}\right)^{2}\sigma_{y}\sigma_{c},$
(33)

The covariant derivative of the scalar mesons is expressed as

$\displaystyle\mathcal{D}^{\mu}\Phi$ $\displaystyle=$
$\displaystyle\delta_{\mu}\Phi-ig_{1}\left(L^{\mu}\Phi-\Phi
R^{\mu}\right)-ieA^{\mu}\left[T_{3},\Phi\right],$ (34)

where $A_{\mu}=gA_{\mu}^{a}\lambda^{a}/2$ is the electromagnetic field. For
vector and axialvector meson nonets, we have

$\displaystyle L^{\mu\nu}$ $\displaystyle=$
$\displaystyle\delta_{\mu}L^{\nu}-ieA^{\mu}\left[T_{3},L^{\nu}\right]-\left\\{\delta^{\nu}\mathcal{L}^{\mu}-ieA^{\nu}\left[T_{3},L^{\mu}\right]\right\\},$
(35) $\displaystyle R^{\mu\nu}$ $\displaystyle=$
$\displaystyle\delta_{\mu}R^{\nu}-ieA^{\mu}\left[T_{3},R^{\nu}\right]-\left\\{\delta^{\nu}R^{\mu}-ieA^{\nu}\left[T_{3},R^{\mu}\right]\right\\},$
(36)

where $L^{\mu}=\sum_{a=0}^{N_{f}^{2}-1}T_{a}(V_{a}^{\mu}+A_{a}^{\mu})$ and
$R^{\mu}=\sum_{a=0}^{N_{f}^{2}-1}T_{a}(V_{a}^{\mu}-A_{a}^{\mu})$.

##  III Results

###  III.1 Meson States

From Eqs. (13) and (18), we can now determine various meson states,

$\displaystyle a_{0}^{\pm}\equiv\frac{\sigma_{1}\mp
i\sigma_{2}}{\sqrt{2}},\quad$ $\displaystyle\quad a_{0}^{0}\equiv\sigma_{3},$
(37) $\displaystyle k^{\pm}\equiv\frac{\pi_{4}\mp i\pi_{5}}{\sqrt{2}},\quad$
$\displaystyle\quad
k^{0}\equiv\pi_{6}-i\pi_{7},\qquad\bar{k^{0}}\equiv\frac{\pi_{6}+i\pi_{7}}{\sqrt{2}},$
(38)
$\displaystyle\frac{1}{\sqrt{2}}\left(\sigma_{N}+a_{0}^{0}\right)=\frac{a_{0}^{0}}{\sqrt{2}}+\frac{\sigma_{0}}{2}+\frac{\sigma_{8}}{\sqrt{6}}+\frac{\sigma_{15}}{2\sqrt{3}},\quad$
$\displaystyle\quad\frac{1}{\sqrt{2}}\left(\sigma_{N}-a_{0}^{0}\right)=-\frac{a_{0}^{0}}{\sqrt{2}}+\frac{\sigma_{0}}{2}+\frac{\sigma_{8}}{\sqrt{6}}+\frac{\sigma_{15}}{2\sqrt{3}},\hskip
19.91692pt$ (39)
$\displaystyle\sigma_{s}=\frac{\sigma_{0}}{2}-\sqrt{\frac{2}{3}}\sigma_{8}+\frac{\sigma_{15}}{2\sqrt{3}},\quad$
$\displaystyle\quad\chi_{co}=\frac{\sigma_{0}}{2}-\frac{\sqrt{3}}{2}\sigma_{15},$
(40)
$\displaystyle\frac{1}{\sqrt{2}}\left(\eta_{N}+\pi^{0}\right)=\frac{\pi^{0}}{\sqrt{2}}+\frac{\pi_{0}}{2}+\frac{\pi_{8}}{\sqrt{6}}+\frac{\pi_{15}}{2\sqrt{3}},\quad$
$\displaystyle\quad\frac{1}{\sqrt{2}}\left(\eta_{N}-\pi^{0}\right)=-\frac{\pi^{0}}{\sqrt{2}}+\frac{\pi_{0}}{2}+\frac{\pi_{8}}{\sqrt{6}}+\frac{\pi_{15}}{2\sqrt{3}},$
(41)
$\displaystyle\eta_{s}=\frac{\pi_{0}}{2}-\sqrt{\frac{2}{3}}\pi_{8}+\frac{\pi_{15}}{2\sqrt{3}},\quad$
$\displaystyle\quad\eta_{c}=\frac{\pi_{0}}{2}-\frac{\sqrt{3}}{2}\pi_{15},$
(42) $\displaystyle D^{0}\equiv\frac{\pi_{0}+i\pi_{10}}{\sqrt{2}},\quad$
$\displaystyle\quad\bar{D}^{0}\equiv\frac{\pi^{0}-i\pi_{10}}{\sqrt{2}},$ (43)
$\displaystyle D^{\pm}\equiv\frac{\pi_{11}\pm i\pi_{12}}{\sqrt{2}},\quad$
$\displaystyle\quad D^{\pm}_{s}\equiv\frac{\pi_{13}\pm i\pi_{14}}{\sqrt{2}},$
(44) $\displaystyle
D^{0}_{0}\equiv\frac{\sigma_{9}+i\sigma_{10}}{\sqrt{2}},\quad$
$\displaystyle\quad\bar{D}^{0}_{0}\equiv\frac{\sigma_{9}-i\pi_{10}}{\sqrt{2}},$
(45) $\displaystyle D^{\pm}_{0}\equiv\frac{\sigma_{11}\pm
i\sigma_{12}}{\sqrt{2}},\quad$
$\displaystyle\quad\bar{D}^{\pm}_{s,0}\equiv\frac{\sigma_{13}\pm
i\pi_{14}}{\sqrt{2}}.$ (46)

Section III.2 introduced analytical expressions for the mass spectra of non-
charmed and charmed meson states.

###  III.2 Mass Spectra of Non-Charmed and Charmed Meson States

At vanishing temperature, the mass spectra of non-charmed meson states can be
classified into:

  * •

Pseudo-scalar mesons

$\displaystyle m_{\pi}^{2}$ $\displaystyle=$ $\displaystyle
Z_{\pi}^{2}\left[m_{0}^{2}+\left(\lambda_{1}+\frac{1}{2}\lambda_{2}\right)\sigma_{x}^{2}+\lambda_{1}\sigma_{y}^{2}+\lambda_{1}\sigma_{c}^{2}\right],$
(47) $\displaystyle m_{K}^{2}$ $\displaystyle=$ $\displaystyle
Z_{K}^{2}\left[m_{0}^{2}+\left(\lambda_{1}+\frac{1}{2}\lambda_{2}\right)\sigma_{x}^{2}-\frac{1}{2}\lambda_{2}\sigma_{x}\sigma_{y}+\lambda_{1}\left[\sigma_{y}^{2}+\sigma_{c}^{2}\right]+\lambda_{2}\sigma_{y}^{2}\right],$
(48) $\displaystyle m_{{\eta}_{N}}^{2}$ $\displaystyle=$ $\displaystyle
Z_{\pi}^{2}\left[m_{0}^{2}+\left(\lambda_{1}+\frac{1}{2}\lambda_{2}\right)\sigma_{x}^{2}+\lambda_{1}\left[\sigma_{y}^{2}+\sigma_{c}^{2}\right]+\frac{c}{2}\left(\sigma_{x}^{2}+\sigma_{y}^{2}+\sigma_{c}^{2}\right)\right],$
(49) $\displaystyle m_{{\eta}_{S}}^{2}$ $\displaystyle=$ $\displaystyle
Z_{{\eta}_{S}}^{2}\left[m_{0}^{2}+\lambda_{1}\left(\sigma_{x}^{2}+\sigma_{y}^{2}+\sigma_{c}^{2}\right)+\lambda_{2}\sigma_{y}^{2}+\frac{c}{8}\left(\sigma_{x}^{2}+\sigma_{x}^{2}+\sigma_{c}^{2}\right)\right],$
(50)

where, $Z_{\pi}$, $Z_{K}$, $Z_{{\eta}_{S}}$, the various wavefunction
renormalization factors, are listed in Appendix D.

  * •

Scalar mesons

$\displaystyle m_{a_{0}}^{2}$ $\displaystyle=$ $\displaystyle
m_{0}^{2}+\lambda_{1}\left[\sigma_{x}^{2}+\sigma_{y}^{2}+\sigma_{c}^{2}\right]+\frac{3}{2}\lambda_{2}\sigma_{x}^{2},$
(51) $\displaystyle m_{k^{*}_{0}}^{2}$ $\displaystyle=$ $\displaystyle
Z_{k^{*}_{0}}^{2}\left[m_{0}^{2}+\left(\lambda_{1}+\frac{1}{2}\lambda^{2}\right)\sigma_{x}^{2}+\frac{1}{\sqrt{2}}\lambda_{2}\sigma_{x}\sigma_{y}+\lambda_{1}\left[\sigma_{y}^{2}+\sigma_{c}^{2}\right]+\lambda_{2}\sigma_{c}^{2}\right],$
(52) $\displaystyle m_{{\sigma}_{N}}^{2}$ $\displaystyle=$ $\displaystyle
m_{0}^{2}+3\left(\lambda_{1}+\frac{1}{2}\lambda_{2}\right)\sigma_{x}^{2}+\lambda_{1}\left[\sigma_{y}^{2}+\sigma_{c}^{2}\right],$
(53) $\displaystyle m_{{\sigma}_{S}}^{2}$ $\displaystyle=$ $\displaystyle
m_{0}^{2}+\lambda_{1}\left[\sigma_{x}^{2}+\sigma_{c}^{2}\right]+3\left(\lambda_{1}+\lambda_{2}\right)\sigma_{y}^{2},$
(54)

where $Z_{k^{*}_{0}}$ is another wavefunction renormalization factor.

  * •

Vector mesons

$\displaystyle m^{2}_{\omega_{N}}$ $\displaystyle=$ $\displaystyle
m_{1}^{2}-m_{0}^{2}+\frac{1}{2}\left(h_{1}+h_{2}+h_{3}\right)\sigma_{x}^{2}+\frac{1}{2}h_{1}\left[\sigma_{y}^{2}+\sigma_{c}^{2}\right]+2\delta_{x},$
(55) $\displaystyle m^{2}_{\omega_{S}}$ $\displaystyle=$ $\displaystyle
m_{1}^{2}-m_{0}^{2}+\frac{1}{2}h_{1}\left[\sigma_{x}^{2}+\sigma_{c}^{2}\right]+\frac{1}{2}\left(h_{1}+2h_{2}+2h_{3}\right)\sigma_{y}^{2}+2\delta_{x},$
(56) $\displaystyle m^{2}_{K^{*}}$ $\displaystyle=$ $\displaystyle
m_{1}^{2}-m_{0}^{2}+\frac{1}{4}\sigma_{x}^{2}\left[g_{1}^{2}+2h_{1}+h_{2}\right]+\frac{1}{\sqrt{2}}\sigma_{x}\sigma_{y}\left[h_{3}-g_{1}^{2}\right]$
(57) $\displaystyle+$
$\displaystyle\frac{1}{2}\sigma_{y}^{2}\left[g_{1}^{2}+h_{1}+h_{2}\right]+\frac{1}{2}h_{1}\sigma_{c}^{2}+\delta_{x}+\delta_{y},$
$\displaystyle m^{2}_{\rho}$ $\displaystyle=$ $\displaystyle
m^{2}_{\omega_{N}}.$ (58)

  * •

Axial-vector mesons

$\displaystyle m^{2}_{a_{1}}$ $\displaystyle=$ $\displaystyle
m_{1}^{2}-m_{0}^{2}+g_{1}^{2}\sigma_{x}^{2}+\frac{1}{2}h_{1}\left[\sigma_{y}^{2}+\sigma_{c}^{2}\right]+\frac{1}{2}\sigma_{x}^{2}\left[h_{1}+h_{2}+h_{3}\right]+2\delta_{x},$
(59) $\displaystyle m^{2}_{f_{1s}}$ $\displaystyle=$ $\displaystyle
m_{1}^{2}-m_{0}^{2}+\frac{1}{2}h_{1}\left[\sigma_{x}^{2}+\sigma_{c}^{2}\right]+2g_{1}^{2}\sigma_{c}^{2}+\frac{1}{2}\left[h_{1}+2h_{2}-2h_{3}\right]\sigma_{y}^{2}+2\delta_{y},$
(60) $\displaystyle m^{2}_{k_{1}}$ $\displaystyle=$ $\displaystyle
m_{1}^{2}-m_{0}^{2}+\frac{1}{4}\sigma_{x}^{2}\left[g_{1}^{2}+2h_{1}+h_{2}\right]+\frac{1}{2}\sigma_{y}^{2}\left[g_{1}^{2}+h_{1}+h_{2}\right]$
(61) $\displaystyle+$
$\displaystyle\frac{1}{\sqrt{2}}\sigma_{x}\sigma_{y}\left[g_{1}^{2}-h_{3}\right]+\frac{1}{2}h_{1}\sigma_{c}^{2}+\delta_{x}+\delta_{y},$
$\displaystyle m^{2}_{1N}$ $\displaystyle=$ $\displaystyle m^{2}_{a_{1}}.$
(62)

Also, we determine the mass spectra of the charmed meson states.

  * •

Pseudo-scalar charmed mesons

$\displaystyle m_{D}^{2}$ $\displaystyle=$ $\displaystyle
Z_{D}^{2}\left[m_{0}^{2}+\left(\lambda_{1}+\frac{1}{2}\lambda_{2}\right)\sigma_{x}^{2}+\lambda_{1}\sigma_{y}^{2}+\left(\lambda_{1}+\lambda_{2}\right)\sigma_{c}^{2}-\frac{1}{\sqrt{2}}\lambda_{2}\sigma_{x}\sigma_{y}+\epsilon_{c}\right],$
(63) $\displaystyle m_{\eta_{c}}^{2}$ $\displaystyle=$ $\displaystyle
Z_{\eta_{c}}^{2}\left[m_{0}^{2}+\lambda_{1}\left[\sigma_{x}^{2}+\sigma_{y}^{2}\right]+\left(\lambda_{1}+\lambda_{2}\right)\sigma_{c}^{2}-\frac{c}{8}\left(\sigma_{x}^{2}+\sigma_{y}^{2}\right)+2\epsilon_{c}\right],$
(64) $\displaystyle m_{D_{s}}^{2}$ $\displaystyle=$ $\displaystyle
Z_{D_{s}}^{2}\left[m_{0}^{2}+\lambda_{1}\sigma_{x}^{2}+\left(\lambda_{1}+\lambda_{2}\right)\sigma_{y}^{2}+\left(\lambda_{1}+\lambda_{2}\right)\sigma_{c}^{2}-\lambda_{2}\sigma_{y}\sigma_{c}+\epsilon_{c}\right],$
(65)

where $Z_{D}$, $Z_{\eta_{c}}$ and $Z_{D_{s}}$ are wavefunction renormalization
factors.

  * •

Scalar charmed mesons

$\displaystyle m_{{\chi}_{c0}}^{2}$ $\displaystyle=$ $\displaystyle
m_{0}^{2}+\lambda_{1}\left[\sigma_{x}^{2}+\sigma_{y}^{2}\right]+3\left(\lambda_{1}+\lambda_{2}\right)\sigma_{c}^{2}+2\epsilon_{c},$
(66) $\displaystyle m_{{D}_{0}^{*}}^{2}$ $\displaystyle=$ $\displaystyle
Z_{D_{0}^{*}}^{2}\left[m_{0}^{2}+\left(\lambda_{1}+\frac{1}{2}\lambda_{2}\right)\sigma_{x}^{2}+\lambda_{1}\sigma_{y}^{2}+\frac{1}{\sqrt{2}}\lambda_{2}\sigma_{x}\sigma_{c}+\left(\lambda_{1}+\lambda_{2}\right)\sigma_{c}^{2}+\epsilon_{c}\right],$
(67) $\displaystyle m_{{D}_{0}^{*0}}^{2}$ $\displaystyle=$ $\displaystyle
Z_{D_{0}^{*0}}^{2}\left[m_{0}^{2}+\left(\lambda_{1}+\frac{1}{2}\lambda_{2}\right)\sigma_{x}^{2}+\lambda_{1}\sigma_{y}^{2}+\frac{1}{\sqrt{2}}\lambda_{2}\sigma_{x}\sigma_{c}+\left(\lambda_{1}+\lambda_{2}\right)\sigma_{c}^{2}+\epsilon_{c}\right],$
(68) $\displaystyle m_{{D}_{s0}^{*}}^{2}$ $\displaystyle=$ $\displaystyle
Z_{D_{s0}^{*}}^{2}\left[m_{0}^{2}+\lambda_{1}\sigma_{x}^{2}+\left(\lambda_{1}+\lambda_{2}\right)\sigma_{y}^{2}+\lambda_{2}\sigma_{y}\sigma_{c}+\left(\lambda_{1}+\lambda_{2}\right)\sigma_{c}^{2}+\epsilon_{c}\right],$
(69)

where $Z_{D_{0}^{*}}$, $Z_{D_{0}^{*0}}$ and $Z_{D_{s0}^{*}}$ are additional
wavefunction renormalization factors.

  * •

Vector charmed mesons

$\displaystyle m_{D^{*}}^{2}$ $\displaystyle=$ $\displaystyle
m_{1}^{2}-m_{0}^{2}+\frac{1}{4}\left(g_{1}^{2}+2h_{1}+h_{2}\right)\sigma_{x}^{2}+\frac{1}{\sqrt{2}}\sigma_{x}\sigma_{y}\left[h_{3}-g_{1}^{2}\right]$
(70) $\displaystyle+$
$\displaystyle\frac{1}{2}\left(g_{1}^{2}+h_{1}+h_{2}\right)\sigma_{c}^{2}+\frac{1}{2}h_{1}\sigma_{y}^{2}+\delta_{x}+\delta_{c},$
$\displaystyle m_{J/\psi}^{2}$ $\displaystyle=$ $\displaystyle
m_{1}^{2}-m_{0}^{2}+\frac{1}{2}h_{1}\left[\sigma_{x}^{2}+\sigma_{y}^{2}\right]+\frac{1}{2}\left(h_{1}+2h_{2}+2h_{3}\right)\sigma_{c}^{2}+2\delta_{c},$
(71) $\displaystyle m_{D_{s}^{*}}^{2}$ $\displaystyle=$ $\displaystyle
m_{1}^{2}-m_{0}^{2}+\frac{1}{2}\left(g_{1}^{2}+h_{1}+h_{2}\right)\left[\sigma_{y}^{2}+\sigma_{c}^{2}\right]+\frac{1}{2}h_{1}\sigma_{x}^{2}$
(72) $\displaystyle+$
$\displaystyle\left(h_{3}-g_{1}^{2}\right)\sigma_{y}\sigma_{c}+\delta_{y}+\delta_{c}.$

  * •

Axial-vector charmed mesons

$\displaystyle m_{D_{s1}}^{2}$ $\displaystyle=$ $\displaystyle
m_{1}^{2}-m_{0}^{2}+\frac{1}{2}\left(g_{1}^{2}+h_{1}+h_{2}\right)\left[\sigma_{y}^{2}+\sigma_{c}^{2}\right]+\sigma_{y}\sigma_{c}\left(g_{1}^{2}-h_{3}\right)+\frac{1}{2}h_{1}\sigma_{x}^{2}+\delta_{y}+\delta_{c},\hskip
19.91692pt$ (73) $\displaystyle m_{D_{1}}^{2}$ $\displaystyle=$ $\displaystyle
m_{1}^{2}-m_{0}^{2}+\frac{1}{4}\left(g_{1}^{2}+2h_{1}+h_{2}\right)\sigma_{x}^{2}+\frac{1}{2}\left(g_{1}^{2}+h_{1}+h_{2}\right)\sigma_{c}^{2}+\frac{1}{\sqrt{2}}\left(g_{1}^{2}-h_{3}\right)\sigma_{x}\sigma_{c}$
(74) $\displaystyle+$
$\displaystyle\frac{1}{2}h_{1}\sigma_{y}^{2}+\delta_{y}+\delta_{c},$
$\displaystyle m_{{\chi}_{c1}}^{2}$ $\displaystyle=$ $\displaystyle
m_{1}^{2}-m_{0}^{2}+\frac{1}{2}h_{1}\left[\sigma_{x}^{2}+\sigma_{y}^{2}\right]+2g_{1}^{2}\sigma_{c}^{2}+\frac{1}{2}\left(h_{1}+2h_{2}-2h_{3}\right)\sigma_{c}^{2}+2\delta_{c}.$
(75)

The Lagrangian of the extended linear-sigma model, Eq. (1), has various
parameters including $h_{1}$, $h_{2}$, $h_{3}$, $\delta_{u}$, $\delta_{d}$,
$\delta_{s}$, $\delta_{c}$, $\delta_{x}$, $\delta_{y}$, $\epsilon_{u}$,
$\epsilon_{d}$, $\epsilon_{s}$, $\epsilon_{c}$, $C$, $\lambda_{1}$,
$\lambda_{2}$, $g_{1}$, and $g_{2}$. All these parameters are explained and
determined in the Appendix E. For large $N_{c}$, both parameters $h_{1}$ and
$\lambda_{1}$ vanish, especially for SU(3) meson masses. The other set of
parameters, $\sigma_{u}$, $\sigma_{d}$, $\sigma_{s}$, $\sigma_{c}$, $m_{u}$,
$m_{d}$, $m_{s}$, $m_{c}$, $f_{\pi}$, and $f_{K}$, can be fixed from recent
compilation of particle data group [26].

|  Meson | Mass [MeV] |  $h_{1}$ |  $\lambda_{1}$ |  $\lambda_{2}$ |  $g_{1}$ |  $\%$ Error
---|---|---|---|---|---|---|---
| This Work | PDG
Pseudo-Scalar | $m_{\pi}$ | $1300.58\pm 15$ | $1300.00\pm 100$ | $5.65$ | $13.49$ | $68.30$ | $5.50$ | $0.047$
$m_{K}$ | $496.90\pm 4$ | $493.68\pm 0.016$ | $5.65$ | $0.0$ | $42.0$ | $2.3$ | $0.648$
$m_{\eta_{N}}$ | $1458.91\pm 21$ | $1475\pm 4$ | $5.65$ | $13.49$ | $68.30$ | $7.60$ | $1.103$
$m_{\eta_{S}}$ | $1477.94\pm 21$ | $1475\pm 4$ | $5.65$ | $13.49$ | $68.30$ | $1.25$ | $0.199$
Scalar | $m_{a_{0}}$ | $1482.64\pm 15$ | $1474\pm 19$ | $5.65$ | $4.0$ | $68.30$ | $1.0$ | $0.583$
$m_{K_{0}^{*}}$ | $893.40\pm 4$ | $891.66\pm 0.26$ | $5.65$ | $1.0$ | $25.0$ | $1.8$ | $0.195$
$m_{\sigma_{N}}$ | $1206.77\pm 2$ | $1200-1500$ | $5.65$ | $1.0$ | $55.0$ | $7.60$ | $0.561$
$m_{\sigma_{S}}$ | $1720.46\pm 3$ | $1720\pm 20$ | $5.65$ | $1.0$ | $46.0$ | $7.60$ | $0.027$
Vector | $m_{\omega_{N}}$ | $784.42\pm 6$ | $782.65\pm 0.12$ | $-21.5$ | $13.49$ | $68.30$ | $7.60$ | $0.225$
$m_{\omega_{S}}$ | $1020.26\pm 5$ | $1019.46\pm 0.11$ | $-16.21$ | $13.49$ | $68.30$ | $7.60$ | $0.078$
$m_{K^{*}}$ | $1390.64\pm 20$ | $1418\pm 15$ | $5.65$ | $13.49$ | $68.30$ | $7.60$ | $1.967$
$m_{\rho}$ | $784.42\pm 14$ | $775.5\pm 38.8$ | $-21.5$ | $13.49$ | $68.49$ | $7.60$ | $1.137$
Axial-Vector | $m_{a_{1}}$ | $1289.17\pm 21$ | $1230\pm 30$ | $1.0$ | $1.8$ | $25$ | $1.8$ | $4.589$
$m_{1f}$ | $1427.26\pm 2$ | $1426.4\pm 0.9$ | $5.65$ | $1.8$ | $25$ | $2.3$ | $0.061$
$m_{K_{1}}$ | $1400.61\pm 5$ | $1403\pm 7$ | $5.65$ | $1.8$ | $25$ | $2.67$ | $0.171$
$m_{1N}$ | $1450.66\pm 5$ | $1453.4\pm 27$ | $5.65$ | $1.8$ | $25$ | $3.5$ | $0.189$
Tab. 1: Masses of sixteen non-charmed mesons compared with the recent compilation of the particle data group (PDG) [26].  |  Meson | Mass [MeV] |  $h_{1}$ |  $\lambda_{1}$ |  $\lambda_{2}$ |  $g_{1}$ |  $\%$ Error
---|---|---|---|---|---|---|---
| This Work | PDG
PS | $m_{D}$ | $1868.92\pm 2$ | $1869.6\pm 0.15$ | $5.65$ | $4.2$ | $68.30$ | $1.0$ | $0.036$
$m_{\eta_{s}}$ | $2977.46\pm 3$ | $2981\pm 1.1$ | $5.65$ | $13.49$ | $68.30$ | $2.9$ | $0.119$
$m_{D_{s}}$ | $1965.99\pm 21$ | $1968.49\pm 0.32$ | $5.65$ | $3.6$ | $68.30$ | $1.0$ | $0.127$
Scalar | $m_{\chi_{c0}}$ | $3448.23\pm 5$ | $3414.75\pm 0.3$ | $5.65$ | $13.49$ | $68.30$ | $7.60$ | $0.971$
$m_{D_{0}^{*}}$ | $2421.82\pm 7$ | $2403.38\pm 14$ | $5.65$ | $1.0$ | $68.30$ | $2.0$ | $0.761$
$m_{D_{0}^{*0}}$ | $2332.02\pm 8$ | $2317.8\pm 0.06$ | $5.65$ | $1.0$ | $62.0$ | $1.0$ | $0.610$
$m_{D_{s0}^{*}}$ | $2121.6\pm 6$ | $2112.3\pm 0.3$ | $5.65$ | $1.0$ | $52.0$ | $1.0$ | $0.438$
Vector | $m_{D^{0}}$ | $2016.07\pm 3$ | $2010.28\pm 0.13$ | $5.65$ | $13.49$ | $68.30$ | $26.5$ | $0.278$
$m_{J/\psi}$ | $3096.15\pm 12$ | $3096.92\pm 0.011$ | $183.0$ | $13.49$ | $68.30$ | $7.6$ | $0.025$
$m_{D_{s}^{*}}$ | $2109.98\pm 11$ | $2112.3\pm 0.5$ | $63.0$ | $13.49$ | $68.30$ | $7.6$ | $0.120$
AV | $m_{D_{s1}}$ | $3001.98\pm 6$ | $2997.4\pm 0.17$ | $98.0$ | $13.49$ | $68.30$ | $7.6$ | $0.152$
$m_{D_{1}}$ | $2846.77\pm 6$ | $2841.8\pm 0.48$ | $98.0$ | $13.49$ | $68.30$ | $7.6$ | $0.174$
$m_{\chi_{sc1}}$ | $3510.57\pm 5$ | $3510.60\pm 0.7$ | $170.0$ | $13.49$ | $68.30$ | $7.6$ | $0.001$
Tab. 2: Masses of thirteen charmed mesons compared with the recent compilation
of the particle data group (PDG) [26].

Our calculations for the masses of various meson states are summarized in
Tabs. 1 and 2. Our calculations, in which $h_{1}$, $\lambda_{1}$,
$\lambda_{2}$ and $g_{1}$ are the only free parameters, are in excellent
agreement with the recent compilation of the particle data group (PDG) [26].
The percent error is listed in the last column. The reason why some of these
free parameters play crucial roles in some meson states is to be found in the
corresponding analytical expressions of such meson states.

The section that follows is devoted to the final conclusions and outlook.

##  IV Conclusions and Outlook

We have constructed the meson states
$\langle\bar{q}q\rangle=\langle\bar{q}_{\ell}q_{r}-\bar{q}_{\ell}q_{r}\rangle\neq
0$ with and without charm quarks from the effective Lagrangian of the extended
linear-sigma model. With respect to their quantum numbers, orbital angular
momenta $J$, parity $P$, and charge conjugates $C$, the meson states could be
classified into pseudoscalar $J^{PC}=0^{-+}$, scalar $J^{PC}=0^{++}$, vector
$J^{PC}=1^{--}$, and axialvector $J^{PC}=1^{++}$. We have introduced
analytical expressions for the mass spectrum of sixteen non-charmed and
thirteen charmed meson states, section III.2. The Appendix gives further
details needed for the analytical analysis. The numerical analysis and the
corresponding free parameters are summarized in Tab. 1 and 2, in which the
calculations are confronted with the recent compilation of PDG. We conclude
that the masses of the sixteen non-charmed and thirteen charmed meson states
are in excellent agreement with PDG. Therefore, we also conclude that the eLSM
with its set of parameters represents a suitable theoretical approach for the
mass spectrum of non-charmed and charmed meson states, at vanishing
temperature. The temperature dependence of these meson states represent the
natural outlook to be accomplished elsewhere. Also, the in-medium
modifications of these meson masses either in dense or magnetic or electric
medium shall be derived elsewhere.

##  Appendix A Gell-Mann Matrices in SU(4)

In SU(4), the linearly independent traceless Hermitian Gell-Mass matrices are
defined as

$\displaystyle\lambda_{1}=\left(\begin{tabular}[]{c c c c}$0$&$1$&$0$&$0$\\\
$1$&$0$&$0$&$0$\\\ $0$&$0$&$0$&$0$\\\
$0$&$0$&$0$&$0$\end{tabular}\right),\qquad\lambda_{2}=\left(\begin{tabular}[]{c
c c c}$0$&$-i$&$0$&$0$\\\ $i$&$0$&$0$&$0$\\\ $0$&$0$&$0$&$0$\\\
$0$&$0$&$0$&$0$\end{tabular}\right),\quad$
$\displaystyle\lambda_{3}=\left(\begin{tabular}[]{c c c c}$1$&$0$&$0$&$0$\\\
$0$&$-1$&$0$&$0$\\\ $0$&$0$&$0$&$0$\\\
$0$&$0$&$0$&$0$\end{tabular}\right),\qquad\lambda_{4}=\left(\begin{tabular}[]{c
c c c}$0$&$0$&$1$&$0$\\\ $0$&$0$&$0$&$0$\\\ $1$&$0$&$0$&$0$\\\
$0$&$0$&$0$&$0$\end{tabular}\right),$ (92)
$\displaystyle\lambda_{5}=\left(\begin{tabular}[]{c c c c}$0$&$0$&$-i$&$0$\\\
$0$&$0$&$0$&$0$\\\ $i$&$0$&$0$&$0$\\\
$0$&$0$&$0$&$0$\end{tabular}\right),\qquad\lambda_{6}=\left(\begin{tabular}[]{c
c c c}$0$&$0$&$0$&$0$\\\ $0$&$0$&$1$&$0$\\\ $0$&$1$&$0$&$0$\\\
$0$&$0$&$0$&$0$\end{tabular}\right),\quad$
$\displaystyle\lambda_{7}=\left(\begin{tabular}[]{c c c c}$0$&$0$&$0$&$0$\\\
$0$&$0$&$-i$&$0$\\\ $0$&$i$&$0$&$0$\\\
$0$&$0$&$0$&$0$\end{tabular}\right),\qquad\lambda_{8}=-\frac{1}{\sqrt{3}}\left(\begin{tabular}[]{c
c c c}$1$&$0$&$0$&$0$\\\ $0$&$1$&$0$&$0$\\\ $0$&$0$&$-2$&$0$\\\
$0$&$0$&$0$&$0$\end{tabular}\right),$ (109)

and $\lambda_{0}=\bf{\mathtt{I}}/\sqrt{2}$ with $\bf{\mathtt{I}}$ is $3\times
3$ identity matrix [21].

##  Appendix B $4\times 4$ $\Phi$ fields

$\displaystyle\Phi$ $\displaystyle=$ $\displaystyle
T_{0}\sigma_{0}+T_{8}\sigma_{8}+T_{15}\sigma_{15}=\frac{1}{2}\lambda_{0}\sigma_{0}+\frac{1}{2}\lambda_{8}\sigma_{8}+\frac{1}{2}\lambda_{15}\sigma_{15}$
(122) $\displaystyle=$
$\displaystyle\frac{1}{2}\sqrt{\frac{1}{2}}\left(\begin{tabular}[]{c c c
c}$\sigma_{0}$&$0$&$0$&$0$\\\ $0$&$\sigma_{0}$&$0$&$0$\\\
$0$&$0$&$\sigma_{0}$&$0$\\\
$0$&$0$&$0$&$\sigma_{0}$\end{tabular}\right)+\frac{1}{2}\sqrt{\frac{1}{3}}\left(\begin{tabular}[]{c
c c c}$\sigma_{8}$&$0$&$0$&$0$\\\ $0$&$\sigma_{8}$&$0$&$0$\\\
$0$&$0$&$-2\sigma_{8}$&$0$\\\
$0$&$0$&$0$&$0$\end{tabular}\right)+\frac{1}{2}\sqrt{\frac{1}{6}}\left(\begin{tabular}[]{c
c c c}$\sigma_{15}$&$0$&$0$&$0$\\\ $0$&$\sigma_{15}$&$0$&$0$\\\
$0$&$0$&$-2\sigma_{15}$&$0$\\\
$0$&$0$&$0$&$-3\sigma_{15}$\end{tabular}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\begin{tabular}[]{c c c
c}$\frac{\sigma_{0}}{\sqrt{2}}+\frac{\sigma_{8}}{\sqrt{3}}+\frac{\sigma_{15}}{\sqrt{6}}$&$0$&$0$&$0$\\\
$0$&$\frac{\sigma_{0}}{\sqrt{2}}+\frac{\sigma_{8}}{\sqrt{3}}+\frac{\sigma_{15}}{\sqrt{6}}$&$0$&$0$\\\
$0$&$0$&$\frac{\sigma_{0}}{\sqrt{2}}-\frac{2\sigma_{8}}{\sqrt{3}}+\frac{\sigma_{15}}{\sqrt{6}}$&$0$\\\
$0$&$0$&$0$&$\frac{\sigma_{0}}{\sqrt{2}}-\sqrt{\frac{3}{2}}\sigma_{15}$\end{tabular}\right).$
(127)

Similarly, we get

$\displaystyle\Phi^{\dagger}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\begin{tabular}[]{c c c
c}$\frac{\sigma_{0}}{\sqrt{2}}+\frac{\sigma_{8}}{\sqrt{3}}+\frac{\sigma_{15}}{\sqrt{6}}$&$0$&$0$&$0$\\\
$0$&$\frac{\sigma_{0}}{\sqrt{2}}+\frac{\sigma_{8}}{\sqrt{3}}+\frac{\sigma_{15}}{\sqrt{6}}$&$0$&$0$\\\
$0$&$0$&$\sqrt{2}\left(\frac{\sigma_{0}}{2}-\sqrt{\frac{2}{3}}\sigma_{8}+\frac{\sigma_{15}}{2\sqrt{3}}\right)$&$0$\\\
$0$&$0$&$0$&$\sqrt{2}\left(\frac{\sigma_{0}}{2}-\frac{\sqrt{3}}{2}\sigma_{15}\right)$\end{tabular}\right)$
(132)

so that

$\displaystyle\Phi\Phi^{\dagger}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\left(\begin{tabular}[]{c c c
c}$\left(\frac{\sigma_{0}}{\sqrt{2}}+\frac{\sigma_{8}}{\sqrt{3}}+\frac{\sigma_{15}}{\sqrt{6}}\right)^{2}$&$0$&$0$&$0$\\\
$0$&$\left(\frac{\sigma_{0}}{\sqrt{2}}+\frac{\sigma_{8}}{\sqrt{3}}+\frac{\sigma_{15}}{\sqrt{6}}\right)^{2}$&$0$&$0$\\\
$0$&$0$&$2\left(\frac{\sigma_{0}}{2}-\sqrt{\frac{2}{3}}\sigma_{8}+\frac{\sigma_{15}}{2\sqrt{3}}\right)^{2}$&$0$\\\
$0$&$0$&$0$&$2\left(\frac{\sigma_{0}}{2}-\frac{\sqrt{3}}{2}\sigma_{15}\right)^{2}$\end{tabular}\right).\hskip
31.29802pt$ (137)

##  Appendix C Explicit Chiral Symmetry Breaking

The chiral symmetry in explicitly broken

$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{a=0}^{15}h_{a}T_{a}=h_{0}T_{0}+h_{8}T_{8}+h_{15}T_{15}$
(142) $\displaystyle=$ $\displaystyle\frac{1}{2}\left(\begin{tabular}[]{c c c
c}$\frac{h_{0}}{\sqrt{2}}+h_{3}+\frac{h_{8}}{\sqrt{3}}+\frac{h_{15}}{\sqrt{6}}$&$h_{1}-ih_{2}$&$h_{4}-ih_{5}$&$h_{9}-ih_{10}$\\\
$h_{1}+ih_{2}$&$\frac{h_{0}}{\sqrt{2}}-h_{3}+\frac{h_{8}}{\sqrt{3}}+\frac{h_{15}}{\sqrt{6}}$&$h_{6}-ih_{7}$&$h_{11}-ih_{12}$\\\
$h_{4}+ih_{5}$&$h_{6}+ih_{7}$&$\frac{h_{0}}{\sqrt{2}}+2\frac{h_{8}}{\sqrt{3}}+\frac{h_{15}}{\sqrt{6}}$&$h_{13}-ih_{14}$\\\
$h_{9}-ih_{10}$&$h_{11}+ih_{12}$&$h_{13}+ih_{14}$&$\frac{h_{0}}{\sqrt{2}}+\sqrt{\frac{2}{3}}h_{15}$\end{tabular}\right).$

##  Appendix D Wavefunction renormalization factors

Here, we summarize the wavefunction renormalization factors needed for
different meson masses,

$\displaystyle
Z_{\pi}=Z_{\eta}=\frac{m_{a_{1}}}{\sqrt{m_{a_{1}}^{2}-g_{1}^{2}\sigma_{x}^{2}}},\quad$
$\displaystyle\quad
Z_{k}=\frac{2m_{k_{1}}}{\sqrt{4m_{k_{1}}^{2}-g_{1}^{2}\left[\sigma_{x}+\sqrt{2}\sigma_{y}\right]^{2}}},$
(143) $\displaystyle
Z_{k_{s}}=\frac{2m_{K}}{\sqrt{4m_{K}^{2}-g_{1}^{2}\left[\sigma_{x}+\sqrt{2}\sigma_{y}\right]^{2}}},\quad$
$\displaystyle\quad
Z_{\eta_{s}}=\frac{m_{f_{1s}}}{\sqrt{m_{f_{1s}}^{2}-2g_{1}^{2}\sigma_{y}^{2}}},$
(144) $\displaystyle
Z_{\eta_{c}}=\frac{m_{\chi_{c1}}}{\sqrt{m_{\chi_{c1}}^{2}-2g1^{2}\sigma_{c}^{2}}},\quad$
$\displaystyle\quad
Z_{D}=\frac{2m_{D_{1}}}{\sqrt{4m_{D_{1}}-g_{1}^{2}\left(\sigma_{c}+\sqrt{2}\sigma_{c}\right)^{2}}},$
(145) $\displaystyle
Z_{\eta_{s}}=\frac{m_{f_{1s}}}{\sqrt{m_{f_{1s}}^{2}-2g_{1}^{2}\sigma_{y}^{2}}},\quad$
$\displaystyle\quad
Z_{D_{s}}=\frac{\sqrt{2}m_{D_{s_{1}}}}{\sqrt{2m_{D_{s_{1}}}^{2}-g_{1}^{2}(\sigma_{y}^{2}+\sigma_{c})^{2}}},$
(146) $\displaystyle
Z_{D_{0}^{*}}=\frac{2m_{D^{*}}}{\sqrt{4m_{D^{*}}-g_{1}^{2}(\sigma_{x}^{2}-\sqrt{2}\sigma_{c})^{2}}},\quad$
$\displaystyle\quad
Z_{D_{0}^{*0}}=\frac{2m_{D^{*0}}}{\sqrt{4m_{D^{*0}}^{2}-g_{1}^{2}(\sigma_{x}-\sqrt{2}\sigma_{c})^{2}}},$
(147) $\displaystyle
Z_{D_{s0}^{*}}=\frac{\sqrt{2}m_{D^{*}_{s}}}{\sqrt{2m_{D^{*}_{s}}^{2}-g_{1}^{2}\left(\sigma_{y}-\sigma_{c}\right)^{2}}},\quad$
(148)

where

$\displaystyle g_{1}^{2}$ $\displaystyle=$
$\displaystyle\frac{m_{a_{1}}^{2}}{f_{\pi}^{2}Z_{\pi}^{2}}\left(1-\frac{1}{Z_{\pi}^{2}}\right).$
(149)

###  D.1 Mesonic Potential of SU(4) Linear-Sigma Model

We start with the mesonic potential of the SU(3) linear-sigma model,

$\displaystyle U_{m}^{SU(3)}(\Phi)$ $\displaystyle=$ $\displaystyle
m^{2}\mathtt{Tr}\left(\Phi^{\dagger}\Phi\right)+\lambda_{1}\left[\mathtt{Tr}\left(\Phi^{\dagger}\Phi\right)\right]^{2}+\lambda_{2}\left[\mathtt{Tr}\left(\Phi^{\dagger}\Phi\right)\right]^{2}$
(150) $\displaystyle-$ $\displaystyle
c\left[\det\left(\Phi\right)+\det\left(\Phi^{\dagger}\right)\right]-\mathtt{Tr}\left[H\left(\Phi+\Phi^{\dagger}\right)\right].$

The mesonic potential of SU(4) has an analogous form as that of SU(3). For a
precise estimation of the mass spectrum, a new term must be added, namely
$-2\mathtt{Tr}\left[\epsilon\Phi^{\dagger}\Phi\right]$ so that

$\displaystyle U_{m}^{SU(4)}(\Phi)$ $\displaystyle=$
$\displaystyle\frac{m^{2}}{2}\left(\sigma_{x}^{2}+\sigma_{y}^{2}+\sigma_{15}^{2}\right)^{2}+\lambda_{1}\left[4\left(\sigma_{x}+\frac{\sigma_{15}}{\sqrt{6}}\right)^{4}+\left(\sqrt{2}\sigma_{y}+\frac{\sigma_{15}}{\sqrt{6}}\right)^{4}+\left(\sqrt{\frac{2}{3}}\sigma_{0}-\sqrt{\frac{3}{2}}\sigma_{15}\right)^{4}\right.$
(151)
$\displaystyle\left.+4\left(\sigma_{x}+\frac{\sigma_{15}}{\sqrt{6}}\right)^{2}\left(\sqrt{2}\sigma_{y}+\frac{\sigma_{15}}{\sqrt{6}}\right)^{2}+4\left(\sigma_{x}+\frac{\sigma_{15}}{\sqrt{6}}\right)^{2}\left(\sqrt{\frac{3}{2}}\sigma_{0}-\sqrt{\frac{3}{2}}\sigma_{15}\right)^{2}\right.$
$\displaystyle\left.+2\left(\sqrt{2}\sigma_{y}+\frac{\sigma_{15}}{\sqrt{6}}\right)^{2}\left(\sqrt{\frac{2}{3}}\sigma_{0}-\sqrt{\frac{3}{2}}\sigma_{15}\right)^{2}\right]$
$\displaystyle+$
$\displaystyle\lambda_{2}\left[2\left(\sigma_{x}+\frac{\sigma_{15}}{\sqrt{6}}\right)^{4}+\left(\sqrt{2}\sigma_{y}+\frac{\sigma_{15}}{\sqrt{6}}\right)^{4}+\left(\sqrt{\frac{2}{3}}\sigma_{0}-\sqrt{\frac{3}{2}}\sigma_{15}\right)^{4}\right]$
$\displaystyle-$
$\displaystyle\frac{c}{8}\left[\frac{2}{3}\sigma_{x}^{2}\sigma_{y}\sigma_{0}+\frac{\sigma_{y}\sigma_{15}^{2}\sigma_{0}}{3\sqrt{3}}+\frac{2\sqrt{2}}{3}\sigma_{x}\sigma_{y}\sigma_{15}\sigma_{0}+\frac{1}{3}\sigma_{x}^{2}\sigma_{15}\sigma_{0}+\frac{\sigma_{15}^{3}\sigma_{0}}{8}\right.$
$\displaystyle\left.+\frac{\sqrt{2}\sigma_{x}\sigma_{15}\sigma_{0}^{2}}{3\sqrt{3}}-\sqrt{3}\sigma_{x}^{2}\sigma_{y}\sigma_{15}-\frac{\sigma_{15}^{3}\sigma_{y}}{2\sqrt{3}}-\sqrt{2}\sigma_{x}\sigma_{y}\sigma_{15}^{2}-\frac{\sigma_{x}^{2}\sigma_{15}^{2}}{2}-\frac{\sigma_{15}^{4}}{12}-\frac{\sigma_{x}\sigma_{15}^{3}}{\sqrt{6}}\right].$

##  Appendix E Parameters of the SU(4) Linear-Sigma Model

We start with the potential of the SU(4) linear-sigma model, Eq. (151). The
global minimum, i.e., vanishing partial derivatives with respect to
$\sigma_{x}$, $\sigma_{y}$, and $\sigma_{c}$ lead to

$\displaystyle h_{x}$ $\displaystyle=$ $\displaystyle
m^{2}\sigma_{x}-\frac{c}{2}\sigma_{x}\sigma_{y}\sigma_{c}+\lambda_{1}\sigma_{x}\sigma_{y}^{2}+\lambda_{2}\sigma_{x}\sigma_{c}^{2}+\frac{1}{2}\left(2\lambda_{1}+\lambda_{2}\right)\sigma_{x}^{2},$
(152) $\displaystyle h_{y}$ $\displaystyle=$ $\displaystyle
m^{2}\sigma_{y}-\frac{c}{2}\sigma_{x}^{2}\sigma_{c}+\lambda_{1}\sigma_{x}^{2}\sigma_{y}+\lambda_{2}\sigma_{y}\sigma_{c}^{2}+\left(\lambda_{1}+\lambda_{2}\right)\sigma_{y}^{2},$
(153) $\displaystyle h_{c}$ $\displaystyle=$ $\displaystyle
m^{2}\sigma_{c}-\frac{c}{2}\sigma_{x}^{2}\sigma_{y}+\lambda_{1}\sigma_{x}^{2}\sigma_{c}+\lambda_{2}\sigma_{y}^{2}\sigma_{c}+\left(\lambda_{1}+\lambda_{2}\right)\sigma_{c}^{2},$
(154)

where $\lambda_{1}$ and $\lambda_{2}$ are respectively defined as

$\displaystyle\lambda_{1}$ $\displaystyle=$
$\displaystyle\frac{m_{\sigma}^{2}-m_{\pi}^{2}-m_{a_{0}}^{2}+m^{2}_{\eta}}{3f_{\pi}^{2}},$
(155) $\displaystyle\lambda_{2}$ $\displaystyle=$
$\displaystyle\frac{3\left(2f_{K}-f_{\pi}\right)m_{K}^{2}-3\left(2f_{K}-f_{\pi}\right)m_{\pi}^{2}-2\left(f_{K}-f_{\pi}\right)\left(m_{\eta^{\prime}}^{2}+m_{\eta}^{2}\right)}{\left(f_{K}-f_{\pi}\right)\left(3f_{\pi}^{2}+8f_{K}\left(f_{K}-f_{\pi}\right)\right)},$
(156)

where $f_{K}$ and $f_{\pi}$ are the decay constants of $K$ and $\pi$ mesons
which can be taken from the recent compilation of the particle data group
[27].

The U(1)A anomaly breaking term $C$ is fixed by $\lambda_{2}$ and the
difference of pion and Kaon masses,

$\displaystyle C$ $\displaystyle=$
$\displaystyle\frac{m_{K}^{2}-m_{\pi}^{2}}{f_{K}-f_{\pi}}-\lambda_{2}\left(2f_{K}-f_{\pi}\right).$
(157)

The external field $\Delta$ could be expressed from the Lagrangian term
$\mathtt{Tr}[\Delta(L^{\mu\nu}+L^{\mu\nu})]$

$\displaystyle\Delta$ $\displaystyle=$ $\displaystyle\left(\begin{tabular}[]{c
c c c}$\delta_{u}$&$0$&$0$&$0$\\\ $0$&$\delta_{d}$&$0$&$0$\\\
$0$&$0$&$\delta_{s}$&$0$\\\ $0$&$0$&$0$&$\delta_{c}$\end{tabular}\right),$
(162)

from which we deduce that

$\displaystyle\left(\begin{tabular}[]{c}$\delta_{u}$\\\ $\delta_{d}$\\\
$\delta_{s}$\\\ $\delta_{c}$\end{tabular}\right)$ $\displaystyle=$
$\displaystyle\left(\begin{tabular}[]{c}$m_{u}^{2}$\\\ $m_{d}^{2}$\\\
$m_{s}^{2}$\\\ $m_{c}^{2}$\end{tabular}\right).$ (171)

In the isospin-symmetric limit, it turns possible to set
$\delta_{u}=\delta_{d}=0$. In this case, for $\delta_{x}$, $\delta_{y}$, and
$\delta_{c}$, we might use the mass equations of vector mesons, for example,
$m^{2}_{\omega_{N}}$, $m^{2}_{\omega_{S}}$, and $m^{2}_{\chi_{c1}}$. Then, we
get

$\displaystyle\delta_{x}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[m^{2}_{\omega_{N}}-m_{1}^{2}+m_{0}^{2}-\frac{\sigma_{x}^{2}}{2}\left(h_{1}+h_{2}+h_{3}\right)-\frac{h_{1}}{2}\left(\sigma_{y}^{2}+\sigma_{c}^{2}\right)\right],$
(172) $\displaystyle\delta_{y}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[m^{2}_{\omega_{S}}-m_{1}^{2}+m_{0}^{2}-\frac{\sigma_{y}^{2}}{2}\left(\frac{h_{1}}{2}+h_{2}+h_{3}\right)-\frac{h_{1}}{2}\left(\sigma_{x}^{2}+\sigma_{c}^{2}\right)\right],$
(173) $\displaystyle\delta_{c}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[m^{2}_{\chi_{c1}}-m_{1}^{2}+m_{0}^{2}-2g_{1}^{2}\sigma_{c}^{2}-\sigma_{c}^{2}\left(\frac{h_{1}}{2}+h_{2}-h_{3}\right)-\frac{h_{1}}{2}\left(\sigma_{y}^{2}+\sigma_{y}^{2}\right)\right].$
(174)

For the mass parameters,

$\displaystyle m^{2}$ $\displaystyle=$ $\displaystyle
m_{\pi}^{2}-\frac{f_{\pi}^{2}}{2}\lambda_{2}+\frac{c}{2}\left(2f_{K}-f_{\phi}\right)-\lambda_{1}\left(\frac{1}{2}\left(2f_{K}-f_{\pi}\right)^{2}\right),$
(176) $\displaystyle m^{2}_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[m_{a_{0}}^{2}+m_{\sigma_{s}}^{2}-\lambda_{2}\left(\frac{3}{2}\sigma_{x}^{2}+3\sigma_{y}^{2}\right)\right],$
(177) $\displaystyle m^{2}_{1}$ $\displaystyle=$ $\displaystyle
m_{\omega_{s}}^{2}-\left(\frac{h_{1}}{2}+h_{2}+h_{3}\right)\sigma_{s}^{2}-\frac{h_{1}}{2}\sigma_{x}^{2}-2\delta_{s}.$
(178)

Finally, for the parameters $h_{1}$, $h_{2}$. and $h_{3}$, we recall the
potential of the SU(3) linear-sigma model, Eq. (150), where $\sigma_{u}$,
$\sigma_{d}$, and $\sigma_{s}$ can be related to $\sigma_{0}$, $\sigma_{3}$,
and $\sigma_{8}$,

$\displaystyle\sigma_{u}$ $\displaystyle=$
$\displaystyle\sqrt{2}\sigma_{0}+\sigma_{3}+\sigma_{8},$ (179)
$\displaystyle\sigma_{d}$ $\displaystyle=$
$\displaystyle\sqrt{2}\sigma_{0}-\sigma_{3}+\sigma_{8},$ (180)
$\displaystyle\sigma_{s}$ $\displaystyle=$
$\displaystyle\sigma_{0}-\sqrt{2}\sigma_{8}.$ (181)

Then, $h_{0}$, $h_{3}$, and $h_{8}$ become

$\displaystyle h_{0}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{6}}\left[f_{\pi}m_{\pi}^{2}+2f_{K}m_{k}^{2}\right],$
(182) $\displaystyle h_{3}$ $\displaystyle=$
$\displaystyle\left[m^{2}+\frac{c}{\sqrt{6}}\sigma_{0}-\frac{c}{\sqrt{6}}\sigma_{8}+\lambda_{1}\left(\sigma_{0}^{2}+\sigma_{3}^{2}+\sigma_{8}^{2}\right)+\lambda_{2}\left(\sigma_{0}^{2}+\frac{\sigma_{3}^{2}}{2}+\frac{\sigma_{8}^{2}}{2}+\sqrt{2}\sigma_{0}\sigma_{8}\right)\right]\sigma_{3},$
(183) $\displaystyle h_{8}$ $\displaystyle=$
$\displaystyle\frac{2}{3}\left[f_{\pi}m_{\pi}^{2}-2f_{K}m_{k}^{2}\right].$
(184)

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# Dual systolic graphs

Daniel Carmon and Amir Yehudayoff

###### Abstract.

We define a family of graphs we call dual systolic graphs. This definition
comes from graphs that are duals of systolic simplicial complexes. Our main
result is a sharp (up to constants) isoperimetric inequality for dual systolic
graphs. The first step in the proof is an extension of the classical
isoperimetric inequality of the boolean cube. The isoperimetric inequality for
dual systolic graphs, however, is exponentially stronger than the one for the
boolean cube.

Interestingly, we know that dual systolic graphs exist, but we do not yet know
how to efficiently construct them. We, therefore, define a weaker notion of
dual systolicity. We prove the same isoperimetric inequality for weakly dual
systolic graphs, and at the same time provide an efficient construction of a
family of graphs that are weakly dual systolic. We call this family of graphs
clique products. We show that there is a non-trivial connection between the
small set expansion capabilities and the threshold rank of clique products,
and believe they can find further applications.

##  1\. Introduction

The systole of a space is the smallest length of a cycle that can not be
contracted. Systoles were originally studied for manifolds, but our focus is
combinatorial. For graphs, the systole is the girth. A graph is systolic if it
locally looks like a tree. In a broader topological setting, a simplicial
complex111A collection of finite sets that is closed downwards. is systolic if
every short cycle in its one-skeleton is contractible (roughly speaking, there
are no short induced cycles). The systolic condition for simplicial complexes
can be thought of as a combinatorial definition of non-positive curvature.
Systolic inequalities play an important role in many areas of mathematics (see
[13, 7] and references within).

A particular systolic condition for simplicial complexes, identified by Davis
and Moussong, is the no empty square condition (see [18, 4, 10] and references
within). This condition implies that the one-skeleton does not contain an
induced cycle of length at most four. It was open whether simplicial complexes
satisfying this condition exist or not. Januszkiewicz and Świątkowski were the
first to construct such complexes [10, 11]. Their construction is deep and
sophisticated. Its importance is also evident from its application in machine
learning [3]. This application highlights the fundamental importance of
systoles. Roughly speaking, some systolic spaces lead to learning problems
that are difficult to solve due to a “global” reason but not a “local” reason.

The starting point of our work was to improve our understanding of the JS
constructions. A specific question that came up is what is special about duals
of systolic spaces? Every pure222All maximal simplexes have the same
dimension. simplicial complex defines a dual graph. The vertices of the graph
are the facets333Full-dimensional simplexes. in the complex. Two vertices are
connected by an edge if the two facets share a face of co-dimension one. The
fact that the graph is a dual of a complex automatically leads to non-trivial
behavior. We formalize such behavior using the notion of pseudo-cubes.

######  Definition.

We say that a graph $G=(V,E)$ is a pseudo-cube if the following conditions are
satisfied:

  1. (i)

It is $d$-regular.

  2. (ii)

It has an edge $d$-coloring; that is, there is $\chi:E\to[d]$ so that for all
$v\in V$, if $E_{v}$ is the set of $d$ edges touching the vertex $v$, then
$\chi(E_{v})=[d]$.

  3. (iii)

For every color $i\in[d]$, the graph obtained from $G$ by contracting all
edges of color $\neq i$ does not have self-loop.

The number $d$ is called the dimension of the pseudo-cube. Property (iii) is
called color independence. In other words, it means that for every color $i$
and every cycle in $G$, the number of edges of color $i$ along the cycle is
not one (either zero or at least two).

While this definition is simple to state, let us describe some examples, and
provide some intuition. A well-known example of a pseudo-cube is the
$d$-dimensional boolean cube $Q_{d}$. Its vertex set is $\\{0,1\\}^{d}$ and
two vertices are connected by an edge if they differ in a single entry. The
edge coloring comes from the $d$ dimensions or directions in the cube. The
color independence follows from the linear independence of the standard basis.

The definition of a pseudo-cube graph corresponds to the graph being dual to a
chromatic and non-branching pure complex. A simplicial complex is chromatic if
there is a coloring of its vertices such that each facet has a vertex of each
color. A pure simplicial is non-branching if every simplex of co-dimension one
is a face of exactly two facets. Such simplicial complexes allow us to encode
data in a useful manner. The general theme is to model “legal or possible
states” as topological spaces.

Consider, for example, the following scenario. There are three players
${\color[rgb]{0.05,0.05,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.05,0.05,1}A},{\color[rgb]{0.525,0.05,0.525}\definecolor[named]{pgfstrokecolor}{rgb}{0.525,0.05,0.525}B},{\color[rgb]{0.05,0.525,0.525}\definecolor[named]{pgfstrokecolor}{rgb}{0.05,0.525,0.525}C}$,
and a deck of four cards $1,2,3,4$. Each player gets a unique card. There are
$24$ possible assignments of cards to players. This scenario can be modeled as
a chromatic and non-branching simplicial complex (see Figure 1 and Example 3
in [22]). A vertex in the complex corresponds to the local view of a single
player (i.e., the vertices are pairs of the form
$({\color[rgb]{0.05,0.05,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.05,0.05,1}A},1)$).
The color of a vertex is the player’s name (the color of
$({\color[rgb]{0.05,0.05,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.05,0.05,1}A},1)$
is
${\color[rgb]{0.05,0.05,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.05,0.05,1}A}$).
The $24$ facets correspond to possible states of the world such as
$\\{({\color[rgb]{0.05,0.05,1}\definecolor[named]{pgfstrokecolor}{rgb}{0.05,0.05,1}A},2),({\color[rgb]{0.525,0.05,0.525}\definecolor[named]{pgfstrokecolor}{rgb}{0.525,0.05,0.525}B},1),({\color[rgb]{0.05,0.525,0.525}\definecolor[named]{pgfstrokecolor}{rgb}{0.05,0.525,0.525}C},4)\\}$.
Each facet has three colors. When a single player switches between her card
and the free card, we move to an adjacent facet. This simplicial complex turns
out to be a triangulation of the two-dimensional torus.

$1$$4$$1$$4$$1$$4$$1$$3$$2$$3$$2$$3$$2$$3$$1$$4$$1$$4$$1$$4$$1$ Figure 1. Four
cards to three players as a simplicial complex. The 24 triangles are facets,
and vertices of the same color and number are identified.

The translation to the topological language leads to important results in
computer science. A prominent example is the proof of the asynchronous
computability theorem in [8], where the different colors corresponded to
different processors in a distributed setting. In the recent work [3], such
simplicial complexes were used to model multiclass learning problems.

The following example provides some additional intuition. Imagine the unit
interval $I=[0,1]$ placed on the line $\mathbb{R}$. Let $x$ be the reflection
around $1\in\mathbb{R}$ and let $y$ be the reflection around $0\in\mathbb{R}$.
Both $x$ and $y$ are involutions. If we apply $x$ on $I$ we get $xI=[1,2]$,
and if we apply $y$ we get $yI=[-1,0]$. The group $D$ generated by $x$ and $y$
is the infinite dihedral group. If we apply all of $D$ on $[0,1]$, we get a
tiling of the line by unit intervals. This tiling defines a simplicial
complex. The dual graph is the Cayley graph of $D$ with $x,y$ as generators,
and is a pseudo-cube. The intermediate value theorem says that if we move
continuously in the line starting at some unit interval and coming back to the
same interval, then we must cross the same boundary of the interval more than
once. This topological property is abstracted by color independence.

The boolean cube is the dual graph of a non-branching and chromatic pure
complex. The vertices of this complex are the $2d$ elements in
$[d]\times\\{0,1\\}$. Each $x\in\\{0,1\\}^{d}$ defines a full dimensional
simplex $\sigma_{x}=\\{(i,x_{i}):i\in[d]\\}$. Two vertices
$x,y\in\\{0,1\\}^{d}$ are connected by an edge iff the common face
$\sigma_{x}\cap\sigma_{y}$ has co-dimension one. This simplicial complex is,
however, not systolic. For example, for $d=2$, this complex is an empty
square.

The JS complexes are systolic. Their dual graphs are therefore dual systolic.
The following definition provides a combinatorial abstraction.

######  Definition.

A graph $G$ is dual systolic of dimension $d$ if it is a pseudo-cube of
dimension $d$ so that for every color $i\in[d]$, the graph obtained from $G$
by contracting all edges of color $\neq i$ is simple (that is, there are no
double edges between vertices).

Stated differently, dual systolicity means that for every color $i$, the
following holds. Let $V_{1},V_{2}\subseteq V$ be two distinct connected
components with respect to edges of color $\neq i$. There is at most a single
edge of color $i$ between $V_{1}$ and $V_{2}$ in $G$.

How does this definition relate to systoles? The observation is that if a
simplicial complex has no empty squares then its dual graph is dual systolic.
Let $\mathcal{C}$ be a chromatic, non-branching and pure simplicial complex,
and let $G$ be its dual graph. The star of a vertex $v$ in $\mathcal{C}$ is
the set of all simplexes $\sigma\in\mathcal{C}$ so that
$\\{v\\}\cup\sigma\in\mathcal{C}$. The vertex $v$ is called the center of the
star. There is a one-to-one correspondence between stars in $\mathcal{C}$
whose center has color $i$ and connected components in the graph obtained by
deleting edges with color $i$ from $G$. Let $v_{1},v_{2}$ be two distinct
vertices of color $i$ in $\mathcal{C}$. Assume that the two corresponding
connected components have two or more $i$-edges between them. It follows that
the corresponding stars share two different simplexes $\sigma_{1},\sigma_{2}$
of co-dimension one. There are two distinct vertices $u_{1}\in\sigma_{1}$ and
$u_{2}\in\sigma_{2}$ with the same color. The square $v_{1},u_{1},v_{2},u_{2}$
has alternating colors so it is empty (because the complex is chromatic).

Figure 2. A systolic complex (on the left) and its dual graph (on the right).

The $d$-dimensional boolean cube $Q_{d}$ is not dual systolic for $d>1$; for
every $i\in[d]$, the the graph obtained from $Q_{d}$ by contracting all edges
of color $\neq i$ consists of $2^{d-1}$ parallel edges. In a sense, it is the
“least dual systolic” pseudo-cube. The dual graphs of the JS complexes are
dual systolic.

The definition of dual systolic graphs is simple and natural. The
constructions of Januszkiewicz and Świątkowski imply that dual systolic graphs
exist [10, 11]. It seems, however, quite complicated to efficiently build dual
systolic graphs. For example, we do not yet know of strongly polynomial time
algorithms for building them.

Figure 3. The dual systolic graph from Figure 1 with blue edges deleted.

###  1.1. Isoperimetry

The first property we study is the isoperimetric profile of dual systolic
graphs. Let $G=(V,E)$ be a graph. For $U\subseteq V$, denote by
$\partial(U)=\partial_{G}(U)$ the number of edges connecting $U$ and
$V\setminus U$ in $G$. The edge expansion of $U\neq\emptyset$ is

$\phi(U)=\phi_{G}(U)=\frac{\partial(U)}{|U|}.$

The isoperimetric profile of $G$ is the function defined by

$P(s)=P_{G}(s)=\min\\{\phi(U):|U|=s\\}.$

It provides a lot of information on the structure of $G$.

The first result we state bounds the isoperimetric profile of pseudo-cubes.
Samorodnitsky proved the same result for the boolean cube [20]. We observe
that his proof can be extended to all pseudo-cubes.444Here and below,
logarithms are in base two.

######  Theorem 1.

For every $d$-dimensional pseudo-cube $G$ and $s>0$,

$P_{G}(s)\geq d-\log s.$

The theorem is sharp for the boolean cube for all integers $s$ that are powers
of two. It can be thought of as saying that pseudo-cubes are small set
expanders. The study of small set expansion is motivated by its intimate
connection with the unique games conjecture, its potential to provide insights
into hardness of optimization problems, and its connection to metric
embeddings (see [19, 15, 1] and references within).

The boolean cube is the canonical example of a small set expander. The small
set expansion of various other graphs has been studied in several works. Noisy
versions of the boolean cube were studied in [12, 17]. Small set expansion for
noisy cubes is related to the "majority is stablest" theorem. The authors of
[2] considered a derandomized version of the noisy cube and achieved a better
trade-off between expansion and threshold rank (more on this later on). The
small set expansion of the multi-slice graph was studied in [6], and of the
Johnson graph was studied in [14]. Another line of research includes
characterizing the non-expanding subsets and showing that they are, in some
sense, negligible [5].

Dual systolicity leads to a much stronger bound on the isoperimetric profile.

######  Theorem 2.

For every $d$-dimensional dual systolic graph $G$ and $s>1$,

$P_{G}(s)\geq d-8(1+\log\log s).$

This is an exponential improvement over the previous theorem. Pseudo-cubes
have strong expansion, say of at least $\tfrac{d}{2}$, for sets of size at
most exponential in $d$. The same expansion for dual systolic graphs holds for
sets of size doubly exponential in $d$. This bound is sharp (up to the
constant before the $\log\log$) because the JS construction yields dual
systolic graphs of size doubly exponential in $d$.

Most proofs of small set expansion are analytic and go through
hypercontractivity. Our proof follows a different path. The bound for pseudo-
cubes uses information theory (following Samorodnitsky’s footsteps). The bound
for dual systolic graphs has three parts. The first part is the bound for
pseudo-cubes. The second part is identifying a dynamics that the isoperimetric
profile satisfies. On a high-level, there is a functional $\mathcal{F}$ so
that if $P$ is the isoperimetric profile of dual systolic graphs, then
${\mathcal{F}}(P)$ is also such a profile. The third part is finding the fixed
point of this dynamics. This dynamics can be thought of as a bootstrapping
mechanism for proving isoperimetric inequalities that relies on dual
systolicity. For more details, see Section 3.

###  1.2. Explicit constructions

The JS complexes are defined via a collection of groups. This collection of
groups is inductively constructed, but we do not know of an efficient
algorithm that implements this construction. In this section, we weaken the
dual systolic condition. This leads to explicit constructions and at the same
time we get the same isoperimetric behavior as for dual systolic graphs.

The first notion we define is that of a weak pseudo-cube. We replace color
independence with a weaker requirement.

######  Definition.

We say that a graph $G=(V,E)$ is a weak pseudo-cube if (i) it is $d$-regular,
(ii) it has an edge $d$-coloring, and (iii) the following two conditions hold.
First, the graph obtained from $G$ by contracting all edges of color $\neq d$
does not have self-loop. Second, let $G_{-d}$ denote the graph obtained from
$G$ by deleting all edges of color $d$. Every connected component in $G_{-d}$
is a $(d-1)$-dimensional pseudo-cube. This definition is inductive; for $d=1$,
a weak pseudo-cube is a perfect matching.

Property (iii) is called weak color independence. Weak pseudo-cubes satisfy
the same isoperimetric inequality as the one stated for pseudo-cubes above.

######  Theorem 3.

For every $d$-dimensional weak pseudo-cube $G$ and $s>0$,

$P_{G}(s)\geq d-\log s.$

We now define the weak notion of dual systolicity.

######  Definition.

A graph $G$ is weakly dual systolic if it is a weak pseudo-cube of dimension
$d$ so that (a) the graph obtained from $G$ by contracting all edges of color
$\neq d$ is simple, and (b) every connected component in $G_{-d}$ is weakly
systolic of dimension $d-1$. This definition is inductive; for $d=1$, a weakly
systolic graph is a perfect matching.

Weakly dual systolic graphs also satisfy the same strong isoperimetric
inequality as dual systolic graphs.

######  Theorem 4.

For every $d$-dimensional weakly dual systolic graph $G$ and $s>1$,

$P_{G}(s)\geq d-8(1+\log\log s).$

Even though we weakened the requirements we obtained the same isoperimetric
behavior. But we also obtained one more important thing. There are explicit
and simple constructions of weakly dual systolic graphs.

Before describing the construction, we answer a basic question. What is the
least size of a weakly systolic graph of dimension $d$? The size of
$d$-dimensional pseudo-cubes is at least roughly $\exp(d)$. The size of
$d$-dimensional dual systolic graphs is much larger, at least roughly
$\exp(\exp(d))$. Indeed, if we denote by $F(d)$ the minimal number of vertices
in a $d$-dimensional weakly dual systolic graph, we have the recurrence
relation: $F(1)=2$ and for $d>1$,

(1) $F(d)\geq F(d-1)\big{(}F(d-1)+1\big{)}.$

The inequality holds because if we look at a single connected component of
$G_{-d}$ of size $S$ then it has $S$ neighboring components, and the size of
each of these $S+1$ components is at least $F(d-1)$.

######  Claim 5.

For every integer $d\geq 1$, there is an explicit weakly dual systolic graph
$CP^{(d)}$ of dimension $d$ with $n^{(d)}$ vertices. In addition $n^{(1)}=2$
and $n^{(d)}=n^{(d-1)}(n^{(d-1)}+1)$ for $d>1$.

(a)

(b)

(c)

(d)

Figure 4. The graph $CP^{(3)}$ is displayed in the top-left corner. The other
three images show this graph when we delete edges of a specific color. Only
the green color on the bottom-right “partitions the graph” in the desired way.

The graph $CP^{(d)}$ is called the $d$-dimensional clique product. The
construction of clique products is by repeated applications of replacement
products of cliques [21, 9]. See Figure 4 for an example.

The clique products show that inequality (1) is in fact an equality. The
$d$-dimensional clique product is the smallest possible $d$-dimensional weakly
systolic graph. In some sense, clique products are the dual systolic versions
of the boolean cubes (which are the smallest possible pseudo-cubes). Below we
describe one application of clique products. It seems reasonable that they can
be used in future applications as well.

###### Proof of Claim 5.

Take $CP^{(1)}$ to be a single edge. Construct $CP^{(d)}$ from $CP^{(d-1)}$ by
taking its replacement product with the clique of size $n^{(d-1)}+1$.
Specifically, take $n^{(d-1)}+1$ disjoint copies of $CP^{(d-1)}$, and connect
node $i\in\\{1,2,\ldots,n^{(d-1)}\\}$ in copy $j\in\\{0,1,\ldots,n^{(d-1)}\\}$
to node $-i$ in copy $i+j$, where the operations are modulo $n^{(d-1)}+1$.
Color all these new edges with color $d$. ∎

###  1.3. An application

The following question came up in the study of the unique games conjecture:
what is the relationship between threshold rank and small set expansion?

Let $G$ be a regular graph with $n$ vertices, and let $M$ be its normalized
adjacency matrix. The matrix $M$ is symmetric and has $n$ eigenvalues (the
maximum eigenvalue is one). The threshold rank of $G$ with parameter
$\epsilon>0$, denoted by $TR_{1-\epsilon}(G)$, is defined to be the number of
eigenvalues of $M$ that are at least $1-\epsilon$.

Threshold rank is important in algorithms for the unique games conjecture [16,
1]. The authors of [1] bounded from above the threshold rank of small set
expanders. The authors of [2] constructed small set expanders with relatively
large threshold rank. A full understanding of the relationship between
threshold rank and small set expansion remains an open problem. The clique
products are (somewhat) small set expanders with high threshold rank.

######  Theorem 6.

Let $d>2$, let $CP=CP^{(d)}$ and let $n=n^{(d)}$. Let $0<k\leq\tfrac{d}{2}$ be
an integer and $\epsilon\geq\frac{2k}{d}$. Then,

$TR_{1-\epsilon}(CP)\geq\frac{n^{1-2^{-k}}}{2}.$

For $k=1$, we get a lower bound of $\Omega\left(\sqrt{n}\right)$ on the
threshold rank with $\epsilon=\tfrac{2}{d}$. In other words, we have a graph
of size $n$ so that the threshold rank $TR_{1-\epsilon}$ is polynomially
large, for $\epsilon\leq\frac{2}{1+\log\log n}$ that tends to zero as $n$
tends to infinity. When we think of $\epsilon$ as a constant, the integer $k$
becomes $\tfrac{\epsilon d}{2}$, and the threshold rank is almost as high as
it can be ($TR_{1-\epsilon}$ is at least $n^{1-o(1)}$).

A rough comparison can be made for example with Theorem 4.14 in [2]. Their
construction depends on two parameters. If we plug-in $\epsilon$ as one of
their parameters, and $\tfrac{1}{2}$ as the second parameter, we see that the
threshold rank of $CP^{(d)}$ is larger than in their construction, but the
small set expansion properties in their construction is better. The two
constructions are incomparable.

The ideas developed in our paper can perhaps lead to better constructions in
the small set expansion versus threshold rank question. A specific approach
that seems reasonable is weakening the dual systolicity condition; instead of
requiring at most one edge between two connected components, we can upper
bound the number of edges between components by some other number. This
weakening still leads to small set expansion behavior, and can potentially
lead to a wider range of parameters.

###  1.4. A discussion

The dual systolicity condition translates the systolic notion from [10, 11] to
the language of graphs. There are stronger systolic criteria for simplicial
complexes than the one considered here (i.e., the $k$-large condition). We
believe that further exploring these stronger criteria could lead to stronger
guarantees and to powerful properties. Our work is only a first step in this
direction.

##  2\. Isoperimetry of weak pseudo-cubes

In this section we prove bounds on the isoperimetric profile of weak pseudo-
cubes. The proof we present here uses information theory, and is inspired by
Samorodnitsky’s proof for the hypercube [20]. The main difficulty we face is
identifying the correct “coordinate system” to use.

###### Proof of Theorem 3.

Let $G=(V,E)$ be a $d$-dimensional weak pseudo-cube, and let $U$ be a non-
empty set of vertices. For $i\in[d]$, denote all inner $i$-edges of $U$ by

$e_{i}(U):=\\{e\in E:e\subseteq U,\chi(e)=i\\}.$

Because

$d|U|=\partial(U)+2\sum_{i\in[d]}e_{i}(U),$

we see that

$\phi(U)=d-\frac{2\sum_{i\in[d]}e_{i}(U)}{|U|}.$

It remains to prove that

(2) $\frac{2\sum_{i=1}^{d}e_{i}(U)}{|U|}\leq\log|U|.$

Let $X$ be a uniformly random vertex in $U$ so that the Shannon entropy of $X$
is $H(X)=\log|U|$. For every $I\subseteq[d]$, denote by $X_{I}$ the set

$X_{I}=\\{v\in V:\text{ $v$ is reachable from $X$ using colors in $I$}\\}.$

The crucial observation is that for all $e,i$ so that $\Pr[X_{\\{i\\}}=e]>0$,
we have

$H(X|X_{\\{i\\}}=e)=1_{e\in e_{i}(U)}.$

In other words, when $e\in e_{i}(U)$, the entropy is one, and when $e\not\in
e_{i}(U)$, the entropy is zero. Taking expectation, we see that

$H(X|X_{\\{i\\}})=\frac{2e_{i}(U)}{|U|}.$

Inequality (2) can thus be written as

$\sum_{i\in[d]}H(X|X_{\\{i\\}})\leq H(X).$

To show this, we observe that for all $i>1$, weak color independence implies
that

$H(X|X_{\\{i\\}})\leq H(X_{\\{1,..,i-1\\}}|X_{\\{i\\}}).$

When $H(X|X_{\\{i\\}}=e)$ is one, the distribution of $X_{\\{1,..,i-1\\}}$
conditioned on $X_{\\{i\\}}=e$ is uniform between two options (so that the
r.h.s. is one as well). The set $X_{I}$ is determined by the set
$X_{I^{\prime}}$ whenever $I^{\prime}\subseteq I$. The data processing
inequality thus implies that for all $i>1$,

$H(X_{\\{1,\ldots,i-1\\}}|X_{\\{i\\}})\leq
H(X_{\\{1,\ldots,i-1\\}}|X_{\\{1,2,\ldots,i\\}}).$

By the chain rule,

$\displaystyle\sum_{i\in[d]}H(X|X_{\\{i\\}})$
$\displaystyle=H(X|X_{\\{1\\}})+\sum_{i>1}H(X|X_{\\{i\\}})$ $\displaystyle\leq
H(X|X_{\\{1\\}})+\sum_{i>1}H(X_{\\{1,\ldots,i-1\\}}|X_{\\{1,2,\ldots,i\\}})$
$\displaystyle=H(X,X_{\\{1\\}},X_{\\{1,2\\}},\ldots,X_{\\{1,\ldots,d-1\\}}|X_{\\{1,\ldots,d\\}})$
$\displaystyle\leq H(X).\qed$

##  3\. Isoperimetry for weak dual systolic graphs

This section is devoted for analyzing the isoperimetric profile of weakly dual
systolic graphs.

###  3.1. A dynamic

The key step in the proof is identifying a dynamic that the isoperimetric
profile satisfies. This leads to the following definition.

######  Definition.

A function $g:[1,\infty)\rightarrow[0,\infty)$ is a dimension-independent
bounding function if for every weakly dual systolic graph $G=(V,E)$ of
dimension $d$, and every $s>0$,

$P_{G}(s)\geq d-g(s).$

The following lemma describes the underlying dynamics. For $\epsilon>0$,
define a functional $\mathcal{F}_{\epsilon}$ by

$\big{[}\mathcal{F}_{\epsilon}(g)\big{]}(s)=g(2^{\frac{4}{\epsilon}})+\epsilon\log
s.$

######  Lemma 7.

Let $g$ be a dimension-independent bounding function that is monotonically
non-decreasing so that $g(2)\geq 1$. Then, for every $\epsilon>0$, the
function $\mathcal{F}_{\epsilon}(g)$ is also a dimension-independent bounding
function.

###### Proof.

Let $G$ be a $d$-dimensional weakly dual systolic graph, and let $U$ be a set
of vertices of size $s>0$. We need to prove that

(3) $\phi(U)=\phi_{G}(U)\geq d-g(2^{\frac{4}{\epsilon}})-\epsilon\log s.$

The proof is by induction on $d$ and $s$. There are two induction bases. If
$s=1$, then $\phi(U)=d$ and the inequality holds. If $d=1$ and $1<s$, then the
r.h.s. of (3) is non-positive. It remains to perform the inductive step. Let
$V_{1},\ldots,V_{L}$ be the connected components of the graph $G_{-d}$
obtained from $G$ by deleting all edges of color $d$. Decompose $U$ to
$U_{1},\ldots,U_{L}$ defined by $U_{\ell}=U\cap V_{\ell}$. Assume that only
the first $U_{1},\ldots,U_{k}$ are non-empty (and the rest are empty). If
$k=1$, then induction on $d$ completes the proof, because if $G^{\prime}$ is
the graph $G$ induces on $V_{1}$ then $\phi(U)\geq\phi_{G^{\prime}}(U)+1$ and
$G^{\prime}$ has dimension $d-1$. So, we can assume that $k>1$. For each
$i\in[k]$, we can consider $U_{i}$ and $U_{\neg i}=\bigcup_{j\neq i}U_{j}$.
For convenience, let $p_{i}=\tfrac{|U_{i}|}{s}$ and let $q_{i}=1-p_{i}$. Both
$U_{i}$ and $U_{\neg i}$ are smaller than $U$, so that induction implies that

$\displaystyle\frac{\partial(U_{i})}{p_{i}s}$ $\displaystyle\geq
d-g(2^{\frac{4}{\epsilon}})-\epsilon\log(p_{i}s),$
$\displaystyle\frac{\partial(U_{\neg i})}{q_{i}s}$ $\displaystyle\geq
d-g(2^{\frac{4}{\epsilon}})-\epsilon\log(q_{i}s).$

How does $\partial(U)$ relate to these quantities? The edges between $U_{i}$
and $U_{\neg i}$ are counted both in $\partial(U_{i})$ and in
$\partial(U_{\neg i})$. All these edge have color $d$. The number of edges
between $U_{i}$ and $U_{\neg i}$ is denoted by $e(U_{i},U_{\neg i})$.
Therefore,

$\partial(U)=\partial(U_{i})+\partial(U_{\neg i})-2e(U_{i},U_{\neg i}).$

Combining the three former equations gives us:

$\displaystyle\frac{\partial(U)}{s}$ $\displaystyle\geq
d-g(2^{\frac{4}{\epsilon}})-\epsilon p_{i}\log(p_{i}s)-\epsilon
q_{i}\log(q_{i}s)-\frac{2e(U_{i},U_{\neg i})}{s}$
$\displaystyle=d-g(2^{\frac{4}{\epsilon}})-\epsilon\log(s)+\epsilon
h(p_{i})-\frac{2e(U_{i},U_{\neg i})}{s},$

where $h$ is the binary entropy function. The missing part for us can thus be
summarized by the inequality:

(4) $\frac{2e(U_{i},U_{\neg i})}{s}\leq\epsilon h(p_{i}).$

Notice that we did not assume anything about the index $i$, and so it is
enough to find any index for which (4) is satisfied. The analysis is
partitioned between three cases.

Case I: There exists $i\in[k]$ so that $p_{i}\leq 2^{-\frac{2}{\epsilon}}$.

In this case, we do not even need to use dual systolicity. Instead, we can use
the bound $e(U_{i},U_{\neg i})\leq p_{i}s$ that holds due to proper coloring.
Plugging this in (4), we get that it is enough to show $2p_{i}\leq\epsilon
h(p_{i})$. So, it is enough to show that $2p_{i}\leq\epsilon
p_{i}\log\frac{1}{p_{i}}$. This is immediately satisfied by the assumption of
the case.

Case II: $2^{\frac{4}{\epsilon}}\leq s$ and for all $i\in[k]$, we have
$2^{-\frac{2}{\epsilon}}<p_{i}$.

Because $\sum_{i\in[k]}p_{i}=1$, we know that $k\leq 2^{\frac{2}{\epsilon}}$.
Systolicity implies that $e(U_{i},U_{\neg i})\leq k$ for all $i$. This means
that it suffices to prove for some $i$ that

$\frac{2\cdot 2^{\frac{2}{\epsilon}}}{s}\leq\epsilon h(p_{i}).$

Because $k>1$, we can choose $i$ so that $p_{i}\leq\frac{1}{2}$. This means we
can treat the binary entropy as an increasing function, and lower bound it by
$h(2^{-\frac{2}{\epsilon}})$. It suffices to show that

$\frac{2\cdot 2^{\frac{2}{\epsilon}}}{\epsilon s}\leq\frac{2}{\epsilon}\cdot
2^{-\frac{2}{\epsilon}},$

which indeed holds.

Case III: $s<2^{\frac{4}{\epsilon}}$.

Because $g$ is monotonic non-decreasing, we have $g(s)\leq
g(2^{\frac{4}{\epsilon}})$. So,

$\phi(U)\geq d-g(s)\geq d-g(2^{\frac{4}{\epsilon}})-\epsilon\log s.\qed$

###  3.2. The fixed point

Lemma 7 provides a bootstrapping mechanism. We can start with an initial
bounding function $g_{0}$. The pseudo-cube bound allows use to choose
$g_{0}(x)=\log x$. From $g_{0}$, we can get a family of new bounds
$g_{1,\epsilon}=\mathcal{F}_{\epsilon}(g_{0})$ parametrized by $\epsilon$. For
every set-size $s$, we can choose an $\epsilon$ that gives the best possible
bound for that $s$. This leads to a new bounding function $g_{1}$. We can plug
$g_{1}$ into the same mechanism and get a stronger bound $g_{2}$, and so
forth. The fixed-point of this dynamics is the best possible bound this proof
gives (which turns out to be optimal up to constant factors). The following
lemma describes this mechanism.

######  Lemma 8.

Let $\ell>0$ be an integer. Suppose $g(s)=c\log^{\frac{1}{\ell}}s$ is a
dimension-independent bounding function where $c\geq 1$. Then, the following
function is also a dimension-independent bounding function:

$g^{\prime}(m)=c^{\prime}\log^{\frac{1}{\ell+1}}s,$

where

(5)
$c^{\prime}=\big{(}4c^{\ell}\ell\big{)}^{\frac{1}{\ell+1}}+\Big{(}4\Big{(}\frac{c}{\ell}\Big{)}^{\ell}\Big{)}^{\frac{1}{\ell+1}}\geq
1.$

###### Proof.

Plugging $g$ into $\mathcal{F}_{\epsilon}$ gives for every $\epsilon>0$, a new
function

$\displaystyle f_{\epsilon}(s)$
$\displaystyle=c\log^{\frac{1}{\ell}}(2^{\frac{4}{\epsilon}})+\epsilon\log s$
$\displaystyle=c\Big{(}\frac{4}{\epsilon}\Big{)}^{\frac{1}{\ell}}+\epsilon\log
s.$

For every $s$, we wish to find the best possible $\epsilon=\epsilon(s)$. An
elementary calculation leads to the following choice

$\displaystyle\epsilon(s)=\Big{(}\frac{c4^{\frac{1}{\ell}}}{\ell\log
s}\Big{)}^{\frac{\ell}{\ell+1}}.$

Plugging this $\epsilon$ gives

$\displaystyle f^{\prime}(s)$
$\displaystyle=c4^{\frac{1}{\ell}}\Big{(}\frac{\ell\log
s}{c4^{\frac{1}{\ell}}}\Big{)}^{\frac{1}{\ell+1}}+\Big{(}\frac{c4^{\frac{1}{\ell}}}{\ell\log
s}\Big{)}^{\frac{\ell}{\ell+1}}\log s$
$\displaystyle=\Big{(}c4^{\frac{1}{\ell}}\Big{(}\frac{\ell}{c4^{\frac{1}{\ell}}}\Big{)}^{\frac{1}{\ell+1}}+\Big{(}\frac{c4^{\frac{1}{\ell}}}{\ell}\Big{)}^{\frac{\ell}{\ell+1}}\Big{)}\log^{\frac{1}{\ell+1}}s.$

It remains to verify that $c^{\prime}\geq 1$. This follows by a direct
substitution of $c=1$ that gives
$(4\ell)^{\frac{1}{\ell+1}}+(\frac{4}{\ell^{\ell}})^{\frac{1}{\ell+1}}>1$. ∎

###### Proof of Theorem 4.

We build a sequence of dimension-independent bounding functions
$g_{0},g_{1},\ldots$ using Lemma 8. Theorem 3 allows to choose $g_{0}(s)=\log
s$, which we can plug into Lemma 8. The lemma leads to a sequence of constants
$c_{0}=1$, and $c_{\ell+1}$ is defined from $c_{\ell}$ via (5). A simple
induction on $\ell$ shows that $c_{\ell}\leq 4\ell$. The induction base is
true, and the induction step is

$\displaystyle\big{(}4(4\ell)^{\ell}\ell\big{)}^{\frac{1}{\ell+1}}+\Big{(}4\Big{(}\frac{4\ell}{\ell}\Big{)}^{\ell}\Big{)}^{\frac{1}{\ell+1}}\leq
4\ell+4.$

We thus have the following family of dimension-independent bounding functions:

(6) $g_{\ell}(s)=4\ell\log^{\frac{1}{\ell}}s.$

For each $s>1$, we can use $\ell$ that suits us best. Plugging
$\ell=\lceil\log\log s\rceil$ gives

$\displaystyle g_{\ell}(s)$ $\displaystyle=4\lceil\log\log
s\rceil\log^{\frac{1}{\lceil\log\log s\rceil}}s$ $\displaystyle\leq
4(1+\log\log s)\log^{\frac{1}{\log\log s}}s$ $\displaystyle=8(1+\log\log
s).\qed$

##  4\. Threshold rank of clique products

###### Proof of Theorem 6.

Let $\epsilon=\frac{2k}{d}$ with a positive integer $k\leq\tfrac{d}{2}$. Let
$M$ denote the normazlied adjacency matrix of $CP^{(d)}$. Let
$\lambda_{1}\geq\lambda_{2}\geq\ldots\geq\lambda_{T}$ be the eigenvalues of
$M$ which are larger than $1-\epsilon$. Let $u_{1},\ldots,u_{T}$ be the
corresponding normalized eigenvectors (which are orthogonal). Denote by $U$
the span of $u_{1},\ldots,u_{T}$. The graph $CP^{(d)}$ is composed of multiple
copies of $CP^{(d-k)}$. The number of copies is $\tfrac{n^{(d)}}{n^{(d-k)}}$.
For each copy $j$, let $\tilde{v}_{j}$ be the indictor vector of the vertices
in copy $j$, and let $v_{j}=\frac{\tilde{v}_{j}}{\|\tilde{v}_{j}\|}$. We claim
two things:

  1. (1)

For every $t$,

$\sum_{j}(\langle u_{t},v_{j}\rangle)^{2}\leq 1.$

  2. (2)

For every $j$,

$\sum_{t}(\langle u_{t},v_{j}\rangle)^{2}\geq\tfrac{1}{2}.$

The first item holds because the vectors $v_{1},v_{2},\ldots$ are orthonormal.
To prove the second item, fix $j$, and denote by $w_{1}$ the projection of
$v_{j}$ to $U$ and by $w_{2}$ the projection of $v_{j}$ to $U^{\perp}$. So,
$\|w_{1}\|^{2}=\sum_{t}(\langle u_{t},v_{j}\rangle)^{2}$ and
$\|w_{1}\|^{2}+\|w_{2}\|^{2}=1$. The main point is that

$\displaystyle\langle v_{j},Mv_{j}\rangle=1-\tfrac{k}{d}.$

The last calculation is

$\displaystyle\langle v_{j},Mv_{j}\rangle$ $\displaystyle=\langle
w_{1},Mw_{1}\rangle+2\langle w_{1},Mw_{2}\rangle+\langle w_{2},Mw_{2}\rangle$
$\displaystyle=\langle w_{1},Mw_{1}\rangle+\langle w_{2},Mw_{2}\rangle$
$\displaystyle\leq\|w_{1}\|^{2}+\big{(}1-\tfrac{2k}{d}\big{)}(1-\|w_{1}\|^{2})$
$\displaystyle=1-\tfrac{2k}{d}+\tfrac{2k}{d}\|w_{1}\|^{2},$

so that $\|w_{1}\|^{2}\geq\frac{k}{d}\cdot\frac{d}{2k}=\frac{1}{2}$, as
needed.

Now, taking the sum

$\frac{n^{(d)}}{2n^{(d-k)}}\leq\sum_{j,t}(\langle u_{t},v_{j}\rangle)^{2}\leq
T.$

Recalling the recurrence relation in Equation 1, and applying it $k$ times, we
have

$\big{(}n^{(d-k)}\big{)}^{2^{k}}<n^{(d)}$

so that

$T\geq\frac{\big{(}n^{(d)}\big{)}^{1-2^{-k}}}{2}.\qed$

## Acknowledgement

We wish to thank Irit Dinur, Nir Lazarovich, and Roy Meshulam for helpful and
insightful discussions.

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