11institutetext:  School of Physics, Southeast University, Nanjing 211189,
China 22institutetext: Department of Physics and Hebei Advanced Thin Films
Laboratory, Hebei Normal University, Shijiazhuang 050024, China;

#  $N_{C}$ evolution of the light meson resonances

Zhi-Hui Guo 1 1 22<EMAIL_ADDRESS>
###### Abstract

The variation of the resonance poles, i.e., the mass and width, originating
from changing the values of $N_{C}$, the number of the QCD colors, provide an
intuitive theoretical method to discriminate the interior constituents of
hadrons. Many different approaches, including unitarized chiral amplitudes,
dispersive methods, quark models, linear-sigma-model-like framework, etc.,
have been widely used to determine the $N_{C}$ trajectories of the various
resonance pole positions. We focus on the discussions of the light-flavor
meson resonances in this work.

##  1 Introduction

Discerning the internal structures of the hadron resonances is a highly
nontrivial task, due to the quark confinement feature and the nonperturbative
nature of the QCD in the resonance energy region. Sophisticated theoretical
methods, based on the general principles of the S matrix theory (e.g.,
unitarity and analyticity) and chiral perturbation theory ($\chi$PT), have
been impressively developed over the last decades to improve the descriptions
of the various data from the experiments and lattice QCD Pelaez:2015qba ;
Yao:2020bxx ; Oller:2019opk ; Oller:2020guq . The hadronic states are
typically included as point-like particles 111The reference to point-like
particles here means that they are not treated as compounds with explicit
smaller constituent parts as compared to e.g., a quark-model description., in
the effective field theory approach, from which alone it is difficult to get
information about the interior structures of the hadrons. Consequently,
additional theoretical ingredients are usually needed to further discriminate
the possible constituents inside the various hadrons. One of such methods is
the compositeness relation developed by Weinberg in the study of the hadronic
bound-state deuteron Weinberg:1962hj , where the probability to find the two-
nucleon state in the deuteron, i.e., the compositeness coefficient, is
calculated. There exist several recipes to generalize the compositeness
relation to the resonance case from several groups Baru:2003qq ;
Hanhart:2011jz ; Hyodo:2011qc ; Aceti:2012dd ; Sekihara:2014kya ; Guo:2015daa
; Oller:2017alp ; Matuschek:2020gqe . The merit of the compositeness relation
approach is that one can give quantitative estimation of the probabilities to
find specific constituting bodies in the resonances. However, up to date, a
general agreement for the generalization of the compositeness to the resonance
is not reached yet Gao:2018jhk  and different recipes could give quite
different results for the compositeness coefficients. Other qualitative
methods to discriminate different constituent configurations of the various
hadrons include the changing of the QCD parameters, such as the number of
colors and the quark masses. Hadrons with different constituent configurations
are expected to exhibit different responses when changing these parameters.
The $\chi$PT naturally provides a reliable theoretical framework to
investigate the quark-mass dependences of the hadron properties Hanhart:2008mx
; Hanhart:2014ssa ; Liu:2012zya ; Torres:2014vna ; Guo:2015dha ;
RuizdeElvira:2017aet ; Ren:2012aj , which can be further examined by the
lattice calculations Aoki:2019cca .

Another powerful tool to study the possible interior structures of the hadron
resonances is the large $N_{C}$ QCD largenc , being $N_{C}$ the number of the
colors. By taking the scaling of the strong coupling of QCD as
$g_{s}^{2}/N_{C}$ when $N_{C}\to\infty$ and assuming the confinement, the
leading $N_{C}$ scaling of the masses and widths for the conventional meson
composed of a quark and an anti-quark ($\bar{q}q$) and the glueball composed
of pure gluons can be unambiguously determined largenc . E.g., the large
$N_{C}$ QCD predicts that the masses of both types of hadrons approach to
constants, and the widths of the $\bar{q}q$ meson and the glueball scale as
$1/N_{C}$ and $1/N_{C}^{2}$, respectively. Despite these leading $N_{C}$
scaling laws are obtained in the large $N_{C}$ QCD, they should hold with
corrections at finite values of $N_{C}$.

For the exotic mesons beyond the conventional $\bar{q}q$ picture, specially
those composed of two quarks and two antiquarks, the situation is more subtle.
It has long been thought that the tetraquark meson state is not allowed by the
large $N_{C}$ QCD Coleman:1985rnk . The main reason was that the leading-order
(LO) contributions of the correlation functions of the two tetraquark
operators are the disconnected parts, containing two closed quark loops, which
correspond to the two color-singlet mesonic states at the hadronic level.
Therefore it was concluded in Ref. Coleman:1985rnk  that the tetraquark meson
does not exist in the large $N_{C}$ QCD. However, in 2013 the long standing
view of the absence of tetraquark states was questioned in Ref.
Weinberg:2013cfa  and many other following works Knecht:2013yqa ;
Lebed:2013aka ; Cohen:2014tga ; Cohen:2014vta ; Maiani:2016hxw ; Lucha:2017gqq
. The key argument is that the subleading order connected parts of the two-
tetraquark-operator correlators, although $1/N_{C}$ suppressed comparing with
the leading disconnected contributions, could contain the possible tetraquark
meson poles, whose decay widths in the large $N_{C}$ limit can be proportional
to $1/N_{C}$ or even narrower ones for specific flavor contents
Weinberg:2013cfa ; Knecht:2013yqa ; Lebed:2013aka ; Cohen:2014tga ;
Cohen:2014vta ; Maiani:2016hxw ; Lucha:2017gqq . In this way, the tetraquark
mesons can have narrow widths and may be relevant for various physical
processes.

A much related type of hadron to the tetraquark state is the two-meson
molecule. A well accepted rule to distinguish the genuine tetraquark and two-
meson molecule is far to be reached. From the large $N_{C}$ point of view, the
hadron molecule made of two color-singlet objects may be likely to have a
large width when increasing $N_{C}$. Indeed in the large $N_{C}$ the
extraordinary scaling laws for the widths of the molecule-like resonances,
such as those with $O(1)$, $O(N_{C})$ or even higher powers of $N_{C}$, are
not found to be incompatible with the known properties of the dispersion
relations and effective theories Cohen:2014vta ; Jaffe:2007id ; Jaffe:2008zz .
In the $N_{C}>3$ case, there could exist another type of interesting hadronic
states, composed of $(N_{C}-1)$ quarks and $(N_{C}-1)$ antiquarks, whose mass
can behave as $O(N_{C})$ and width as $O(1)$ Jaffe:1981 ; Jaffe:2007id ;
Cohen:2014vta ; Cohen:2014tga . Similar large $N_{C}$ discussions can be also
extended to the baryons, whose masses and widths will scale as $N_{C}$ and
constants, respectively Dashen:1993jt ; Manohar:1998xv ; Goity:1996hk ;
Lutz:2001yb . In this short note, the topics on the baryons at large $N_{C}$
will not be covered and we focus on the $N_{C}$ properties of the meson
resonances, specially concentrating on the light-flavor meson resonances, and
the related physical quantities.

The classification of the various hadrons, according to the scaling laws of
their masses and widths discussed above, is inferred from the large $N_{C}$
QCD. In many cases, we are more interested in the physical hadrons living in
the real world with $N_{C}=3$. As a result, the $N_{C}$ trajectories of the
hadron properties obtained from $N_{C}=3$ to larger values of $N_{C}$, or even
to $N_{C}\to\infty$, can provide important information to discern the internal
structures of the physical resonances observed in experiments at $N_{C}=3$.
Nevertheless to extrapolate the hadron properties to $N_{C}>3$, especially to
rather large values of $N_{C}$, is not an easy task and in many cases it bears
big uncertainties. Furthermore, it is quite possible that the physical hadrons
are complicated mixtures of different types of constituents with different
$N_{C}$ behaviors. In the ideal case when the physical resonance has little
mixing of constituents with different $N_{C}$ behaviors, the $N_{C}$
trajectory of this resonance pole is expected to exhibit simple scaling laws
of the dominant constituent. E.g., the vector resonances, $\rho$ and
$K^{*}(892)$, belong to such category of resonances. The $N_{C}$ trajectories
of their resonance poles from various studies, as shown in later discussions,
follow almost exactly as that of the $q\bar{q}$ state, with their masses
behaving as constants and widths as $1/N_{C}$ in a wide range from $N_{C}=3$
to rather large values up to one hundred or even bigger ones. While in the
complex situation when the physical hadron resonance is a mixture of several
types of constituents with different $N_{C}$ behaviors, the $N_{C}$ trajectory
becomes complicated and it may show different trends in different $N_{C}$
regions. An even more subtle situation is that if at $N_{C}=3$ the hadron is
dominated by a constituent which is however suppressed at large $N_{C}$, its
trajectories in the region of large values of $N_{C}$ likely contain big
uncertainties. Since the hadron properties at $N_{C}=3$ are constrained by the
relevant experimental data, the theoretical uncertainties in the extrapolation
to the nearby values around $N_{C}\geq 3$ are believed to be under better
control than the situation with rather large values of $N_{C}$ Pelaez:2015qba
; Pelaez:2010er . Therefore it is advisory to separately address the results
in the regions near $N_{C}=3$ and those in large values of $N_{C}$. With this
in mind we address in this review the evolution in $N_{C}$ of the masses and
widths of various light meson resonances and related physical quantities.

##  2 State-of-art study of the $N_{C}$ trajectories for light meson
resonances

To keep track of the $N_{C}$ movement of the hadron resonance is by no means a
trivial matter, since one has to first determine the resonance properties from
the experimental data, corresponding to the initial point of the $N_{C}$
trajectory at $N_{C}=3$, and then to properly perform the extrapolations to
larger values of $N_{C}$. One of the practicable and reliable ways to
investigate the $N_{C}$ evolution of the hadron resonance is using unitarized
$\chi$PT in different forms, with Ref. Oller:1998zr  being the pioneer one to
signal out the remarkable unusual scaling of the poles of the lightest scalar
resonances in powers of 1/$N_{C}$ within that framework. A heavily used
approach along these lines is the inverse amplitude method (IAM) Pelaez:2003dy
, which relies on the combination of the $\chi$PT and the requirements of the
general properties of the scattering amplitudes, such as the unitarity and
analyticity. To set up the notations, we briefly recapitulate the methodology
how the $N_{C}$ trajectories of the resonance poles are obtained in the IAM
approach.

Chiral perturbation theory, the low energy effective realization of QCD, takes
the pseudo Nambu-Goldstone bosons (pNGBs), e.g., $\pi$ in the $SU(2)$ case and
$\pi,K,\eta$ in the $SU(3)$ version, as the dynamical fields. By construction
$\chi$PT respects the important spontaneous and explicit chiral symmetry
breaking patterns of QCD, and it has a rigorously organized power counting
rule to calculate the physical quantities order by order in perturbation
theory. The chiral power counting rule is based on the perturbative expansion
of the external momentum $p$ and the light quark masses $m_{q}$, counted as
$O(p^{2})$ Weinberg:1978kz ; Gasser:1983yg ; Gasser:1984gg .

At leading order in the $SU(3)$ $\chi$PT, all the free parameters in the
Lagrangian can be fixed by the pion weak decay constant in the chiral limit,
and the pion and kaon masses. In fact all the decay constants of the pNGBs,
denoted as $F$, share the same value at leading order. The mass of the $\eta$
at this order can be completely predicted, and the result coincides with the
celebrated Gell-Mann-Okubo relation $m_{\eta}^{2}=(4m_{K}^{2}-m_{\pi}^{2})/3$.
At next-to-leading order (NLO) of the $SU(3)$ $\chi$PT, twelve additional
$O(p^{4})$ local operators, each accompanied by one low energy constant (LEC),
will appear and eight of them are relevant to the meson-meson scattering. The
relevant LECs are labeled as $L_{i=1,2,\cdots,8}$ Gasser:1984gg . For the
perturbative NLO meson-meson scattering amplitudes, they include the tree-
level and one-loop diagrams. All the divergences of the one-loop diagrams can
be completely canceled by the infinite parts of the $O(p^{4})$ LECs. The
remaining unknown finite parts of the LECs, which are usually needed to be
fitted to the experimental or lattice data, together with the finite parts of
the one-loop diagrams, give the final expression for a physical quantity. To
confront the $\chi$PT with the large $N_{C}$ QCD, e.g. comparing the various
Green functions calculated in both schemes, enables one to obtain the leading
$N_{C}$ scaling of the parameters from $\chi$PT Gasser:1984gg . The results
are summarized as

$\displaystyle F^{2}\sim O(N_{C})\,,\quad L_{1}\sim O(N_{C})\,,\quad L_{2}\sim
O(N_{C})\,,\quad 2L_{1}-L_{2}\sim O(1)\,,\quad L_{3}\sim O(N_{C})\,,$
$\displaystyle L_{4}\sim O(1),\quad L_{5}\sim O(N_{C}),\quad L_{6}\sim
O(1),\quad L_{7}\sim O(1),\quad L_{8}\sim O(N_{C})\,.$ (1)

The $N_{C}$ scaling of $L_{7}$ written above is given by counting the number
of traces. In fact the $N_{C}$ counting of $L_{7}$ in the $SU(3)$ $\chi$PT is
more subtle, due to the contribution from the heavy singlet $\eta_{0}$
Gasser:1984gg ; Peris:1994dh , whose mass squared scales as $1/N_{C}$. One of
the proper ways to address this problem is the $U(3)$ $\chi$PT, which will be
discussed in detail later. Notice that it is possible that some of the LECs
may contain sizable subleading $N_{C}$ correction terms Ledwig:2014cla . And
the masses of the pNGBs behave as constants at large $N_{C}$, i.e.
$m_{\pi,K,\eta}^{2}\sim O(1)$.

The perturbative amplitudes from the $\chi$PT calculation alone can not
generate any resonance state. The requirement of the unitarity for the
partial-wave scattering amplitude provides a crucial guideline to extend the
perturbative $\chi$PT to the resonance energy region. The IAM is one of such
attempts to build unitarized partial-wave amplitudes and its explicit
expression for the two-meson scattering processes with definite isospin ($I$)
and angular momentum ($J$) takes the form

$\displaystyle\mathcal{T}_{IJ}(s)=T_{2,IJ}(s)\cdot\big{[}T_{2,IJ}(s)-T_{4,IJ}(s)\big{]}^{-1}\cdot
T_{2,IJ}(s)\,,$ (2)

where the subscripts $2$ and $4$ denote the chiral orders of the perturbative
amplitudes. In the couple-channel scattering, $T_{2,IJ}(s)$ and $T_{4,IJ}(s)$
in the above equations should be understood as matrices spanned in the
scattering-channel space. For simplicity, we will often omit the subscripts
$IJ$ when it is not necessary to clarify them. The partial-wave amplitude
$T_{2}(s)$ corresponds to the tree-level results calculated from the LO
Lagrangian, which are completely fixed by the weak decay constants of the
pNGBs and their masses. For the $O(p^{4})$ amplitude $T_{4}(s)$, it receives
contributions both from the one-loop diagrams calculated with the LO
Lagrangian and the tree-level diagrams from the NLO Lagrangian. The simple IAM
amplitude has been demonstrated to be quite powerful to describe the various
experimental scattering data Oller:1997ng ; Oller:1998hw ; GomezNicola:2001as
. After the unknown LECs are determined from the fits to data, one can
extrapolate the scattering amplitudes into the complex energy plane to look
for the resonance poles, the real and imaginary parts of which are usually
identified as the masses and half widths of the resonances, respectively. For
the IAM in the $SU(3)$ case, the scalar resonances $f_{0}(500)/\sigma$,
$K_{0}^{*}(700)$, $f_{0}(980)$ and $a_{0}(980)$ and the vector resonances
$\rho(770)$ and $K^{*}(892)$ can be naturally obtained Oller:1997ng ;
Oller:1998hw ; GomezNicola:2001as . From the $N_{C}$ point of view, the LECs
determined from the fits, correspond to their values at $N_{C}=3$. Though the
resulting values of the LECs from the unitarized $\chi$PT should be
qualitatively comparable to those from the standard $\chi$PT Gasser:1983yg ;
Bijnens:2014lea , moderate variations between the two approaches are also
within expectations, because the LECs might receive noticeable contributions
from higher orders obtained by iterating procedure. Nevertheless, the values
of the LECs from the standard $\chi$PT provide a useful guide for the
unitarized-$\chi$PT fits. It is advisory to rely on the reasonable unitarized
fits, that lead to comparable values of LECs to the standard $\chi$PT, to
perform the $N_{C}$ studies by imposing the leading $N_{C}$ scaling shown in
Eq. (2).

The resonance poles from the IAM amplitudes are determined by the LECs, such
as the $F$ and $L_{i=1,2,\cdots,8}$. Since the $N_{C}$ scalings of the various
LECs are known (2), one can implement them to get the LECs at different values
of $N_{C}$ through

$\displaystyle L_{i}(N_{C})=L_{i}^{\rm Exp}\frac{N_{C}}{3}\,,$ (3)

for $L_{i}\sim O(N_{C})$ and

$\displaystyle L_{i}(N_{C})=L_{i}^{\rm Exp}\,,$ (4)

for $L_{i}\sim O(1)$, where $L_{i}^{\rm Exp}$ denote the values obtained from
the phenomenological fits to the experimental data. The decay constant $F$
will be scaled as

$\displaystyle F(N_{C})=F^{\rm Exp}\sqrt{\frac{N_{C}}{3}}\,.$ (5)

By taking into account the results of the LECs obtained at different values of
$N_{C}$, one can then update the calculations of the resonance poles
correspondingly. In such a way, the paths of the resonance poles at different
values of $N_{C}$ can be tracked. Slight different strategies are adopted by
several research groups to address the $N_{C}$ trajectories of the various
light-flavor meson resonances. Below we highlight the main findings from
different groups.

###  2.1 Resonance trajectories from the IAM with $SU(3)$ $\chi PT$ at one
loop

The first attempt to the investigate the $N_{C}$ evolution of the resonance
poles in the IAM approach is carried out in Refs. Pelaez:2003dy ;
Pelaez:2004xp , which is based on the one-loop two-meson scattering amplitudes
of the $SU(3)$ $\chi$PT GomezNicola:2001as . The coupled-channel IAM formalism
is used to fit many experimental scattering data, including the phase shifts
and inelasticities from the $\pi\pi\to\pi\pi$, $\pi\pi\to K\bar{K}$, $\pi
K\to\pi K$ in different partial waves and isospin channels, and also the
$\pi\eta$ event distributions GomezNicola:2001as . For the isoscalar-scalar
($IJ=00$) case, three couple channels, i.e. $\pi\pi,K\bar{K},\eta\eta$, are
included in the IAM amplitudes, and the two scalar resonances $\sigma$ and
$f_{0}(980)$ are found to be compatible with nowadays determinations
Tanabashi:2018oca . For the $(I,J)=(1,1)$ case, $\pi\pi$ and $K\bar{K}$
channels are incorporated, and the vector resonance $\rho(770)$ found in the
complex energy plane agrees well with the PDG Tanabashi:2018oca . For the
$(I,J)=(1/2,1)$ and $(1/2,0)$ cases, the $K\pi$ and $K\eta$ channels are
included. The vector $K^{*}(892)$ resonance is nicely reproduced and a clear
resonance pole for the scalar $K^{*}_{0}(700)/\kappa$ is confirmed. For the
$(I,J)=(1,0)$ case, two couple channels $\pi\eta$ and $K\bar{K}$ are taken
into account, and a resonant pole is found for the $a_{0}(980)$. Other
experimental scattering data in the elastic-channel cases, including the
$\pi\pi$ with $(I,J)=(2,0)$ and $K\pi$ with $(I,J)=(3/2,0)$, are also used to
constrain the $O(p^{4})$ LECs, although resonances do not appear in these two
cases.

To impose the leading $N_{C}$ scalings for various LECs in Eq. (2), the
movements of the resonance pole positions by varying the values of $N_{C}$ are
illustrated in Refs. Pelaez:2003dy ; Pelaez:2004xp . The masses of the vector
resonances $\rho(770)$ and $K^{*}(892)$ are barely changed when increasing the
values of $N_{C}$ up to 30, while their widths clearly decrease and nicely
exhibit the $1/N_{C}$ scaling law. Those $N_{C}$ behaviors of the vector
resonances are well compatible with the expectations for the $\bar{q}q$ meson
resonances. The results in the vector channels also provide solid evidences
that the resulting $N_{C}$ trajectories of the resonances from the IAM study
do make sense, since there is little doubt that the $\rho(770)$ and
$K^{*}(892)$ are the well established hadrons overwhelmingly dominated by the
$\bar{q}q$ components Tanabashi:2018oca . In contrast, the $N_{C}$ evolutions
for the scalar resonances $\sigma$, $f_{0}(980)$, $\kappa$ and $a_{0}(980)$
are rather different from the vectors. When increasing the values of $N_{C}$
up to 30, it is found that the masses of the $\sigma$, together with the
uncertainties, tend to increase. With moderate uncertainties, the masses of
the $\kappa$ are also found to increase. The widths of the $\sigma$ and
$\kappa$ both steadily keep increasing at the rates between $\sqrt{N_{C}}$ and
$N_{C}$. Based on these observations, it was concluded that the $\sigma$ and
$\kappa$ resonances are likely to be dominated by the tetraquark or two-meson
components, instead of the $\bar{q}q$ constituents Pelaez:2003dy ;
Pelaez:2004xp .

In a later study Sun:2005uk , similar $\sigma$ trajectories are obtained as
those in Refs. Pelaez:2003dy ; Pelaez:2004xp  in the region of $N_{C}\lesssim
30$. By further taking very large values of $N_{C}$ in Ref. Sun:2005uk , say
several hundreds or more, the $\sigma$ poles are found to fall down to the
positive or negative real axis in the complex $s$ plane, depending on the
values of the LECs. This shows that for some of the LECs the decay width of
the $\sigma$ could vanish for such huge $N_{C}$, indicating that a $q\bar{q}$
seed could exist in the formation of the $\sigma$. Therefore it is concluded
in Ref. Sun:2005uk  that the $\sigma$ seems a $\bar{q}q$ resonance surrounded
with important $\pi\pi$ clouds. The rather different $\sigma$ pole
trajectories in the region around $N_{C}=3$, as compared to the $\rho$, imply
that the $\pi\pi$ or tetraquark components seem playing much more important
roles than the case of $\rho$ for $N_{C}$ not far away from 3. As mentioned in
the Introduction and discussed in detail in Refs. Pelaez:2015qba ;
Pelaez:2010er , it could bring in big theoretical uncertainties when taking
rather large values of $N_{C}$ to determine the resonance properties. E.g., it
is estimated that the 10-20% error bars at $N_{C}=3$ in the unitarized
$\chi$PT Pelaez:2010er ; Salas-Bernardez:2020hua  can be amplified to 100%
errors when taking $N_{C}=30$ or 15. Also a slight change of the values of the
LECs could result in rather different behaviors of the pole trajectories at
large $N_{C}$, as discussed in Ref. Sun:2005uk . Although the resonance
trajectories at large $N_{C}$ may reveal some components that are not the
dominant ones at $N_{C}=3$, one has to first overcome the somewhat big
uncontrollable theoretical uncertainties to reach more definite conclusions in
the large $N_{C}$ region.

Due to the closeness of the $K\bar{K}$ threshold, the $N_{C}$ behaviors of the
$f_{0}(980)$ and $a_{0}(980)$ are more complicated than those of the $\sigma$
and $\kappa$. Their poles in the complex energy plane were found to be
difficult to track for larger values of $N_{C}$ Pelaez:2004xp . According to
the $N_{C}$ evolution of the scattering amplitudes, the $f_{0}(980)$ signal
can be peaks, dips, or even disappear for different values of $N_{C}$. The
scattering amplitudes relevant for the $a_{0}(980)$ resonances turn out to be
sensitive to the regularization scale $\mu$. When taking $N_{C}=25$, there can
be a peak around the $a_{0}(980)$ energy region for $\mu=550$ MeV, or a rather
smooth amplitude for $\mu=770$ MeV. The $N_{C}$ behaviors of the amplitudes
relevant for the $f_{0}(980)$ and $a_{0}(980)$ show that most likely the
$\bar{q}q$ components play small roles in these two scalar resonances
Pelaez:2003dy ; Pelaez:2004xp . Nevertheless, it is still difficult to reach
definite conclusions about what the dominant constituents inside the
$f_{0}(980)$ and $a_{0}(980)$ are.

###  2.2  Shadow poles in the [1,1] Padé of the $SU(3)$ $\chi PT$

In Refs. Dai:2011bs ; Dai:2012kf , the authors propose to use the pole
counting rule Morgan:1992ge  to analyze the numbers of resonance poles at
different values of $N_{C}$. The one-loop IAM or the [1,1] Padé amplitudes
based on the one-loop $SU(3)$ $\chi$PT are used to revise the analyses of the
resonance poles, especially emphasizing the relevant shadow poles in different
Riemann sheets (RSs) for the resonances. According to the pole counting rule,
an elementary resonance with dominant $\bar{q}q$ component is characterized by
the presence of all the relevant shadow poles in different RSs. In contrast,
for the molecular type of hadron resonance, usually there is only one resonant
pole. The shadow poles in the all the relevant RSs for the vector resonances
of $\rho(770)$ and $K^{*}(892)$ can be easily tracked for different values of
$N_{C}$, which reconfirm their $\bar{q}q$ nature. Definite conclusion is not
obtained for the $a_{0}(980)$ due to the large uncertainties of the data
Dai:2012kf . No relevant shadow poles are found for $f_{0}(980)$ when varying
the values of $N_{C}$, and it supports the $K\bar{K}$-molecule explanation of
the $f_{0}(980)$. For the $\sigma$ and $\kappa$, although the shadow poles in
different RSs are found at $N_{C}=3$ and also for larger values of $N_{C}$,
the poles in different RSs turn out to be located in somewhat different
positions Dai:2011bs ; Dai:2012kf . It can be concluded that the $\sigma$ and
$\kappa$ only roughly meet the “Breit-Wigner criteria” Dai:2011bs ; Dai:2012kf
222I would like to thank Ling-Yun Dai for useful discussions on these two
references.. The poles of $\sigma$ and $\kappa$ tend to drive away from the
real $s$ axis when increasing the values of $N_{C}$ up to moderate values not
far away from three, which are clearly different from the $N_{C}$ trajectories
of the well established $q\bar{q}$ resonances $\rho$ and $K^{*}(892)$. This
indicates that the $q\bar{q}$ seeds are unlikely the dominant components in
the $\sigma$ and $\kappa$. Furthermore, the poles of the $\sigma$ and $\kappa$
are found to fall down to the negative real axis in the $s$ plane when taking
huge values of $N_{C}$, indicating that they become irrelevant at large
$N_{C}$.

###  2.3 Fates of the resonance poles from the $SU(2)$ two-loop IAM

The sensible results obtained in the one-loop IAM study are further verified
in the two-loop IAM formalism Sun:2005uk ; Pelaez:2006nj . Up to now the IAM
study at two-loop order has been only extended to the $SU(2)$ $\chi$PT, in
which the additional $O(p^{6})$ LECs relevant to the two-meson scattering are
still manageable Bijnens:1995yn . The $SU(2)$ two-loop IAM quantitatively
confirms the $N_{C}$ behaviors for the $\rho(770)$ from the $SU(3)$ one-loop
study Pelaez:2006nj . While for the $\sigma$, it is found that a $\bar{q}q$
component with the mass around $1$ GeV could play relevant roles in its
formation, although the seed of scalar state around 1 GeV is unlikely to be
important for the $\sigma$ at $N_{C}=3$. When increasing the values of
$N_{C}$, the two-meson loops are further suppressed and the $\bar{q}q$ seed
becomes more important, which drives in the large values of $N_{C}$ the
$\sigma$ pole down to the real axis around 1 GeV Pelaez:2006nj ; Sun:2005uk ,
rather different from its values at $N_{C}=3$. Nevertheless it is found that
the $\bar{q}q$ source at $1$ GeV in the $\sigma$ resonance plays quite
important roles to fulfill the semilocal duality RuizdeElvira:2010cs . It is
also further verified in Ref. Pelaez:2006nj  that with reasonable NNLO
parameters it is impossible to make the $\sigma$ scale as a conventional
$q\bar{q}$ meson with an $N_{C}$ dependent mass close to the its physical mass
at $N_{C}=3$, without destroying the well established $N_{C}$ behaviors of the
$\rho$ resonance.

While the $N_{C}$ trajectories of the $\rho(770)$ are quite robust, the
$N_{C}$ paths of the $\sigma$ pole are subject to many ingredients, such as
the subleading correction of the $N_{C}$ scaling of the LECs, higher order
amplitudes, slight different fit strategies and so on. The large uncertainties
of the $\sigma$ trajectories with varying $N_{C}$ can be easily seen in Fig. 2
of Ref. RuizdeElvira:2010cs . By tuning the regularization scale $\mu$ from
500 MeV to 1000 MeV, which can be considered as one part of the subleading
$N_{C}$ scaling corrections, rather different fates for the $\sigma$ poles
show up. For $\mu=500$ MeV and $1000$ MeV, the $\sigma$ poles tend to move to
the negative and positive real axis in the complex energy squared $s$ plane,
respectively. For $\mu=770$, the $\sigma$ poles seem running deeply in the
complex plane for rather large values of $N_{C}$. A deep explanation for the
large uncertainties of the $\sigma$ trajectories is obtained by studying the
analytical pole positions in the chiral and large $N_{C}$ limit Sun:2005uk ;
Nieves:2009ez ; Cohen:2014vta , which reveals a crucial combination of the
chiral LECs, i.e. $25L_{2}+11L_{3}$, that will determine the fates of the
$\sigma$ in the large $N_{C}$ limit. Intriguingly the phenomenologically
preferred values of the chiral LECs almost exactly lead to the critical value
for the combination Sun:2005uk ; Nieves:2009ez ; Cohen:2014vta . Unless the
chiral LECs and their subleading $N_{C}$ scalings can be precisely determined,
more definite conclusions on the $\sigma$ trajectories for large values of
$N_{C}$ are impossible, although the behaviors of the $\sigma$ poles for the
mild values around $N_{C}=3$ are found to be robust from different analyses.

###  2.4  The $N_{C}$ movements of light resonances from the dispersive Omnès
approach

The dispersive Omnès function approach provides a stringent way to include the
final-state strong interactions of the elastic two-body scattering
Omnes:1958hv . In Refs. Dai:2017uao ; Dai:2018fmx , the $N_{C}$ evolutions of
the light-flavor meson resonances, including the $\sigma$, $f_{0}(980)$,
$\rho(770)$ and $f_{2}(1270)$, and the semilocal duality in the finite energy
sum rules are studied within the Omnès approach. To use the phases of the
amplitudes, instead of the phase shifts, and to neglect the left-hand cuts,
the single-channel $\pi\pi$ amplitudes up to 2 GeV are parameterized by the
products of the polynomial terms and the Omnès functions, which will be simply
denoted as Omnès amplitudes in later discussion. Through the matching between
the Omnès amplitudes and the $\chi$PT ones in the low energy region, the
parameters in the Omnès amplitudes can be given in terms of the $\chi$PT LECs.
Since the $N_{C}$ scalings of the $\chi$PT LECs are known (2), it is possible
to indirectly determine the $N_{C}$ scaling laws of the Omnès parameters. The
$N_{C}$ movements of the various resonance poles can be then tracked
Dai:2017uao . For the vector $\rho(770)$ and tensor $f_{2}(1270)$, they are
confirmed to behave as the standard $\bar{q}q$ meson, with their masses
approaching to constants and their widths decreasing as $1/N_{C}$. The
$\sigma$-pole trajectory is found to run deeply in the complex energy plane,
with the mass roughly behaving as $O(1)$ and the width as $O(N_{C})$ when
increasing the values of $N_{C}$. It is argued that $\sigma$ could be a
mixture between different sates, including molecule, tetraquark and
$\bar{q}q$. The $f_{0}(980)$ pole, though running away from the real axis for
$N_{C}<6$, finally tends to fall down to the real axis below the $K\bar{K}$
threshold for large values of $N_{C}$. It is advocated that the $f_{0}(980)$
can be a mixture of the $K\bar{K}$ molecule and the $s\bar{s}$ bare state
Dai:2017uao . Following the method proposed in Ref. RuizdeElvira:2010cs , the
semilocal duality is studied by varying the $N_{C}$ and is found to be
satisfied up to large values of $N_{C}$. The effects from the $\rho(770)$ are
found to be balanced by the $f_{0}(980)$, $f_{0}(1370)$ and $f_{2}(1270)$,
while the $\sigma$ resonance plays little role. This result agree with the
findings in Refs. Guo:2012ym ; Guo:2012yt , where it is found that the
$f_{0}(980)$, by turning into a bound state of 1 GeV at large $N_{C}$, is the
one that compensates the vector contributions. While in Ref.
RuizdeElvira:2010cs , it is claimed that the $\sigma$, which eventually
becomes a bound state around 1 GeV in the two-loop IAM, cancels the effect
from the $\rho(770)$.

###  2.5 Linear-sigma-model like study of the strange scalar resonances

The two light-flavor scalar resonances with a strange quark/antiquark,
$\kappa$ and $K^{*}_{0}(1430)$, including their $N_{C}$ trajectories, are
studied within the linear-sigma-model like framework in Ref.
Wolkanowski:2015jtc . Both the derivative and non-derivative types of
interaction operators are taken into account to calculate the decay widths and
spectral functions, which in turn are used to obtain the phase shifts. The two
resonances are generated by one bare state with mass around 1.1-1.3 GeV. It
concludes that the $\kappa$ resonance could be a companion state of the
$K^{*}_{0}(1430)$. To rescale the coupling constants in terms of $N_{C}$, the
resonances poles of the $\kappa$ and $K^{*}_{0}(1430)$ with varying $N_{C}$
are then studied. The $K^{*}_{0}(1430)$ pole moves to the real axis for large
$N_{C}$, behaving like the standard $\bar{q}q$ meson. In contrast, the pole
corresponding to the $\kappa$ resonance moves deeply in the complex plane when
increasing the $N_{C}$; hence, it provides another hint that $\kappa$ is
unlikely dominated by the $\bar{q}q$ seed Wolkanowski:2015jtc ; Pelaez:2004xp
.

###  2.6 Light-flavor scalars from the unitarized-quark-model study

Unitarized quark-model-like approach provides another type of theoretical
frameworks to address the scalar resonances Tornqvist:1982yv ; Achasov:1994iu
; vanBeveren:2006ua ; Giacosa:2006tf ; Zhou:2010ra ; Lukashov:2019dir , which
usually needs to introduce a bare $q\bar{q}$ seed, in addition to the two-pNGB
continuum contributions. The mass of the bare $q\bar{q}$ seed from these
studies is found to be typically close to or even larger than 1 GeV.

In Ref Zhou:2010ra , the unitarized quark model simultaneously includes the
$\bar{q}q$ bare states and the two-pNGB continuum contributions to study the
light-flavor scalar meson spectra. The Adler zeros from the $\chi$PT are also
taken into account as additional constraints. Large amount of experimental
scattering data up to 2 GeV are fitted to fix the unknown parameters. A large
number of scalar resonance poles are then obtained, including the isoscalars
$\sigma$, $f_{0}(980)$, $f_{0}(1370)$, $f_{0}(1500)$, $f_{0}(1710)$,
$f_{0}(2020)$, the isovectors $a_{0}(980)$, $a_{0}(1450)$, $a_{0}(2020)$ and
the strange scalars $\kappa$, $K^{*}_{0}(1430)$, $K^{*}_{0}(1950)$. It is
found that when increasing the values of $N_{C}$ the resonance poles of
$\sigma$, $f_{0}(980)$, $\kappa$ and $a_{0}(980)$ move away from the real
axis, indicating that the bare $\bar{q}q$ seeds play minor roles in these
scalars. For all other heavier scalar resonances, they fall down exactly to
the mass positions of the $q\bar{q}$ seeds in the real axis at large values of
$N_{C}$, implying that the $\bar{q}q$ seeds are crucial in their formations.

###  2.7  The $N_{C}$ evolutions of the axial-vector resonances

The $N_{C}$-trajectory study is also extended to the axial-vector resonances
in Ref. Geng:2008ag , where the axial-vector resonances $a_{1}(1260)$,
$h_{1}(1170)$, $h_{1}(1380)$, $b_{1}(1235)$, $b_{1}(1285)$ and $K_{1}(1270)$
are simultaneously generated via the unitarization of the vector and pNGB
scattering amplitudes. Only the contact interactions between the vectors and
pNGBs are included in the study of Ref. Geng:2008ag , without considering any
bare state exchange contribution. It turns out that all the poles of the
dynamically generated axial-vector resonances move away from the real axis and
run deeply in the complex energy plane, when increasing the values of $N_{C}$.
Therefore it is concluded that all the axial-vector resonances, $a_{1}(1260)$,
$h_{1}(1170)$, $h_{1}(1380)$, $b_{1}(1235)$, $b_{1}(1285)$ and $K_{1}(1270)$,
studied in Ref. Geng:2008ag , are very unlikely to be the conventional
$\bar{q}q$ mesons.

###  2.8 $N_{C}$ dependences of the light resonances in elastic $\pi\pi$
scattering from the resonance chiral theory

The elastic $\pi\pi$ scattering from the resonance chiral theory, which
explicitly includes the bare resonance states and the pNGBs as the dynamical
degrees of freedom, is explored to unravel the $N_{C}$ paths of the $\sigma$
and $\rho(770)$ resonances in Refs. Sannino:1995ik ; Nieves:2011gb . The
pioneer work of this type of studies to address the broad $\sigma$ resonance
was carried out in Ref. Sannino:1995ik , which advocates a tetraquark
interpretation of the bare $\sigma$ state, belonging to a subleading $1/N_{C}$
effect, with a bare mass around 500 MeV. In Ref. Nieves:2011gb , the main
motivation to include the full propagators of the bare resonance exchanges is
to keep all the leading $N_{C}$ terms without any truncation up to a specific
chiral order. To impose the leading large $N_{C}$ high energy constraints in
the chiral limit, the elastic $S$\- and $P$-wave $\pi\pi$ scattering
amplitudes are extrapolated to larger values of $N_{C}$, and the $N_{C}$
variations of the $\sigma$ and $\rho(770)$ poles are tracked in the complex
energy plane. The $\rho(770)$ is robustly confirmed to become a $\bar{q}q$ at
large $N_{C}$. Less clear conclusions are made for the $N_{C}$ movement of the
$\sigma$ pole, since it is sensitive to the model parameters from different
fit procedures Nieves:2011gb . This fact is also noticed in the chiral-limit
IAM study in Ref. Nieves:2009kh , where the large cancellation of the LECs are
found in the scalar isoscalar channel and a rather precise determination of
the LECs would be needed to give definite conclusion for the $N_{C}$ fate of
the $\sigma$. It is pointed out that the two-pion constituent dominates the
$\sigma$ at $N_{C}=3$, but the $\bar{q}q$-like bare scalar state becomes
important when $N_{C}\to\infty$ Nieves:2011gb .

###  2.9 $N_{C}$ spectra of light meson resonances in lattice QCD

In the last decade there has been an important progress of lattice QCD
calculations on the $N_{C}$ dependence of the meson spectra Lucini:2012gg ;
Lucini:2013qja . However, most of the current lattice simulations on the
$N_{C}$ study on the resonance spectra are still quite coarse. E.g. many of
the lattice results on the spectra of the meson resonances with varying
$N_{C}$ are obtained with the quenched approximations, unphysically large
quark masses, or very small volumes DelDebbio:2007wk ; Hietanen:2009tu ;
Bali:2013kia ; DeGrand:2016pur ; Perez:2020fqn .

The quenched lattice QCD, which simplifies the numerical simulations by
excluding the effects of the dynamical sea quarks, recovers the full QCD in
the limit of $N_{C}\to\infty$, since the quark-loop contributions are
$1/N_{C}$ suppressed. In the quenched case by neglecting the quark loops, the
large $N_{C}$ QCD predicts that the subleading corrections for the hadronic
observables behave as $1/N_{C}^{2}$, in contrast to the correction by
including the dynamical fermions starting already at the order of $1/N_{C}$.
By taking several finite values of $N_{C}$ in the simulations ranging from 3
to 17, the subleading corrections of the hadron masses from the quenched QCD
studies nicely exhibit the quadratic scaling in the $1/N_{C}$ expansion
DelDebbio:2007wk ; Bali:2013kia . Currently the $N_{C}$ study of the hadron
spectra from the unquenched lattice QCD with dynamical fermions is still
sparse, and the exploratory unquenched simulation in Ref. DeGrand:2016pur
seems confirming the expected $1/N_{C}$ scaling in several hadronic
observables, including the masses and decay constants. However, larger finite
volumes are clearly needed to reach definite conclusions DeGrand:2016pur .
Moreover, up to date, most of lattice calculations focus on the the masses
when varying $N_{C}$, and the discussions on the $N_{C}$ scalings of the decay
widths are still absent. Recently, relatively finer unquenched lattice
simulations have been done for the mass and decay constant of the pion, and
the non-leptonic kaon decay process Hernandez:2019qed ; Donini:2020qfu .
Future unquenched lattice simulations with relatively large volumes by varying
the numbers of $N_{C}$ could provide important constraints for the $\chi$PT
community to further pin down the $N_{C}$ fates of the various resonances,
specially the scalar ones.

###  2.10 $N_{C}$ trajectories of the scalar charmed meson states

The $N_{C}$ trajectories of the resonance poles are now widely focused on the
light-flavor hadrons, and the discussions on the heavy-flavor resonances are
still rare. In Ref. Guo:2015dha , the scalar charmed mesons are studied in the
pNGB-$D$ scattering processes with different isospin and strangeness quantum
numbers. Although the main topics of this note focus on the light-flavor
hadrons, we briefly discuss the findings of the $N_{C}$ behaviors of the
charmed scalar meson resonances in the former reference. The scalar charmed
meson states, including $D^{*}_{s0}(2317)$ and $D^{*}_{0}(2400)$, are
important objects in the hadron phenomenological study. The $N_{C}$
trajectories of their poles can provide valuable information about the
internal structures for these two mesons. The algebraic approximation of the
$N/D$ method is used to unitarize the NLO chiral amplitudes. The LECs and the
subtraction constants are fitted to the lattice scattering lengths Liu:2012zya
obtained at different quark masses. The physical $D^{*}_{s0}(2317)$
corresponds to a bound state pole below the $DK$ threshold. Two poles, with
$I=1/2$ and strangeness zero, corresponding to the $D^{*}_{0}(2400)$, are
found in the complex energy plane. The lighter pole is located around 2.1 GeV
and the heavier one is around 2.4 GeV. Interesting $N_{C}$ trajectory for the
$D^{*}_{s0}(2317)$ is unveiled in Ref. Guo:2015dha . At $N_{C}=3$, the
$D^{*}_{s0}(2317)$ is just a bound state pole at 2.3 GeV. When increasing the
values of $N_{C}$ up to 6, this bound-state pole approaches to the $DK$
threshold, which becomes a virtual pole in the second RS for $N_{C}=7$. In the
meantime, another virtual pole also appears near the threshold energy region,
and the two virtual poles collide each other to become a pair of resonance
poles when keeping increasing the values of $N_{C}$. The resonance poles seem
going deep in the complex energy plane for large $N_{C}$. For the two poles
corresponding to the $D^{*}_{0}(2400)$, both of them move away from the real
axis in the complex energy plane when increasing the values of $N_{C}$
Guo:2015dha . The $N_{C}$ behaviors of the $D^{*}_{s0}(2317)$ and
$D^{*}_{0}(2400)$ poles imply that the $\bar{q}q$ constituents are unlikely to
play dominant roles in their formation.

###  2.11 Resonance observables with enhanced $1/N_{C}$ suppression

In Refs. Nieves:2009kh ; Nebreda:2011cp , the authors propose to directly
study the resonance observables with enhanced $1/N_{C}$ suppression, namely
the subleading $1/N_{C}$ correction includes at most the $1/N_{C}^{2}$ term
compared to the leading one. It is nicely demonstrated in Ref. Nieves:2009ez
that by performing the expansions of the real and imaginary parts of the
inverse scattering $T$ matrix separately, it leads to the $1/N_{C}^{2n+1}$
types of power corrections in each expansion series. Equivalently the phase
shift $\delta(M_{R}^{2})$ and its derivative $\delta^{\prime}(M_{R}^{2})$,
with the resonance pole $s_{R}=M_{R}^{2}-iM_{R}\Gamma_{R}$, are also found to
fall in the aforementioned category of observables Nieves:2009ez . Other
observables with even further enhanced $1/N_{C}$ suppression are studied and
scrutinized discussions using precise $\pi\pi$ and $K\pi$ phase shifts from
the Roy dispersive analyses are carried out for the $\sigma$ and $\kappa$
resonances Nebreda:2011cp . The key finding is that the coefficients of the
enhanced $1/N_{C}$ correction terms in the expansion series of the $\rho$ and
$K^{*}(892)$ channels turn out to be natural values around one or less, which
are the expected results for the $q\bar{q}$ mesons. However the corresponding
parameters in the $\sigma$ and $\kappa$ cases are found to be huge, around two
orders larger than the ones from the vector channels, indicating that it is
very unnatural to interpret the $\sigma$ and $\kappa$ as $q\bar{q}$ states
Nebreda:2011cp . It should be stressed that in this kind of studies one does
not really need to tune $N_{C}$. The $1/N_{C}$ simply offers an expansion
parameter for the physical observables. This kind of approach does not rely on
the unitarized chiral amplitudes and hence provides a different theoretical
framework to gain insights into internal structures of the resonances based on
the $1/N_{C}$ expansion.

##  3 $N_{C}$ evolvements of the light meson dynamics in the $U(3)$ chiral
theory

If the number of colors is taken large, the quark loop graph which gives rise
to the $U_{A}(1)$ anomaly is suppressed ua1anomaly ; Kaiser:2000gs . Local
non-invariance of the quark condensates under the $U(1)$ axial transformations
leads to a ninth Goldstone boson in the large $N_{C}$ limit ua1nc . 333Witten
argued that the instanton contribution tHooft:1976rip  to the $\eta^{\prime}$
mass scales as $e^{-N_{C}}$ Witten:1978bc  while the confinement contributions
would give $m_{\eta^{\prime}}^{2}\sim 1/N_{C}$. However, it is argued in Ref.
Schafer:2002af  that the contribution from larger instantons to the
$\eta^{\prime}$ can scale as $\sim 1/N_{C}$ as well. The large $N_{C}$
treatment of the QCD $U_{A}(1)$ anomaly is specially attractive to the
$\chi$PT practitioners, since the massive singlet $\eta_{0}$ state, whose
large mass is believed to originate from the $U_{A}(1)$ anomaly, can be then
systematically incorporated in the chiral effective theory. Although the mass
squared $M_{0}^{2}$ of the singlet $\eta_{0}$ is large around 1 GeV2 at
$N_{C}=3$, it can be treated as a small expansion parameter for large $N_{C}$,
since $M_{0}^{2}$ scales as $1/N_{C}$ ua1nc . As a result the singlet
$\eta_{0}$ would become massless and turn to the ninth Goldstone boson in the
chiral and large $N_{C}$ limits, implying that the complete dynamical meson
fields of the low energy QCD for $N_{C}\to\infty$ should include
simultaneously the nine pNGBs $\pi,K,\eta$ and $\eta^{\prime}$, not just the
octet states. It should be pointed out that all the previous studies on the
$N_{C}$ movements of the light meson resonances in the former section have
neglected the singlet $\eta_{0}$ as an explicit degree of freedom. The $U(3)$
chiral theory explicitly includes the nine pNGBs $\pi,K,\eta$ and
$\eta^{\prime}$ ua1nc , dynamically taking the QCD $U_{A}(1)$ anomaly into
account, therefore it provides a more appropriate theoretical framework to
investigate the $N_{C}$ evolution of the various resonances. In Refs.
Guo:2011pa ; Guo:2012ym ; Guo:2012yt , systematical calculations up to the
one-loop level have been carried out in the framework of the $U(3)$ chiral
theory with explicit resonance exchanges, which are further used in many other
phenomenological discussions after including the finite-volume and the finite-
temperature effects Guo:2016zep ; Gao:2019idb . The discussions of the $N_{C}$
trajectories of the light-flavor meson resonances and related physical
quantities are thoroughly addressed in detail. We briefly recapitulate the key
results here.

To set up the consistent power counting rules to simultaneously include the
nine pNGBs $\pi,K,\eta$ and $\eta^{\prime}$, the $U(3)$ $\chi$PT introduces
the triple-expansion formalism, also simply denoted as $\delta$ expansion,
namely $\delta\sim p^{2}\sim m_{q}\sim 1/N_{C}$ HerreraSiklody:1996pm ;
Kaiser:2000gs . The LO $U(3)$ chiral Lagrangian consists of three independent
$O(\delta^{0})$ operators

$\displaystyle\mathcal{L}_{0}=\frac{F^{2}}{4}\langle
u_{\mu}u^{\mu}\rangle+\frac{F^{2}}{4}\langle\chi_{+}\rangle+\frac{F^{2}}{3}M_{0}^{2}\ln^{2}{\det
u}\,,$ (6)

where the chiral tensors are given by

$\displaystyle
U=u^{2}=e^{i\frac{\sqrt{2}\Phi}{F}}\,,\qquad\chi=2B(s+ip)\,,\qquad\chi_{\pm}=u^{\dagger}\chi
u^{\dagger}\pm u\chi^{\dagger}u\,,$ $\displaystyle
u_{\mu}=iu^{\dagger}D_{\mu}Uu^{\dagger}\,,\qquad
D_{\mu}U\,=\,\partial_{\mu}U-i(v_{\mu}+a_{\mu})U\,+iU(v_{\mu}-a_{\mu})\,,$ (7)

and the matrix of the nonet pNGB fields reads

$\Phi\,=\,\left(\begin{array}[]{ccc}\frac{1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{6}}\eta_{8}+\frac{1}{\sqrt{3}}\eta_{0}&\pi^{+}&K^{+}\\\
\pi^{-}&\frac{-1}{\sqrt{2}}\pi^{0}+\frac{1}{\sqrt{6}}\eta_{8}+\frac{1}{\sqrt{3}}\eta_{0}&K^{0}\\\
K^{-}&\overline{K}^{0}&\frac{-2}{\sqrt{6}}\eta_{8}+\frac{1}{\sqrt{3}}\eta_{0}\end{array}\right)\,.$
(8)

$F$ is the LO pNGB decay constant, with the normalization $F_{\pi}=92.1$ MeV.
The QCD $U_{A}(1)$ anomaly is given by the last term in Eq. (6), which
provides the LO mass $M_{0}$ to the singlet $\eta_{0}$ even at the chiral
limit. The leading $N_{C}$ scaling for $M_{0}^{2}$ is $O(1/N_{C})$ ua1nc .
Within the $\delta$-expansion framework, the order of $M_{0}^{2}$ is counted
the same as the $m_{\pi}^{2},m_{K}^{2}$ and $m_{\eta_{8}}^{2}$. At leading
order, the mixing between the physical $\eta$, $\eta^{\prime}$ and the
$\eta_{0}$, $\eta_{8}$ can be described by one-mixing-angle formula

$\displaystyle\left(\begin{array}[]{c}\eta\\\
\eta^{\prime}\end{array}\right)\quad=\quad\left(\begin{array}[]{cc}\cos\theta&-\sin\theta\\\
\sin\theta&\cos\theta\end{array}\right)\quad\left(\begin{array}[]{c}\eta_{8}\\\
\eta_{0}\end{array}\right)\,.$ (15)

Beyond the leading order, one usually needs to introduce the two-mixing-angle
scheme and interested readers are referred to Refs. Guo:2015daa ; Gu:2018swy
for a complete NNLO $U(3)$ $\chi$PT study of the $\eta$-$\eta^{\prime}$
mixing. By diagonalizing the $\eta_{0}-\eta_{8}$ mixing term at leading order
of Eq. (6), the LO masses for the physical $\eta$ and $\eta^{\prime}$ states
and their mixing angle $\theta$ take the form Guo:2011pa

$\displaystyle m_{\eta}^{2}$ $\displaystyle=$
$\displaystyle\frac{M_{0}^{2}}{2}+m_{K}^{2}-\frac{\sqrt{M_{0}^{4}-\frac{4M_{0}^{2}\Delta^{2}}{3}+4\Delta^{4}}}{2}\,,$
(16) $\displaystyle m_{\eta^{\prime}}^{2}$ $\displaystyle=$
$\displaystyle\frac{M_{0}^{2}}{2}+m_{K}^{2}+\frac{\sqrt{M_{0}^{4}-\frac{4M_{0}^{2}\Delta^{2}}{3}+4\Delta^{4}}}{2}\,,$
(17) $\displaystyle\sin{\theta}$ $\displaystyle=$
$\displaystyle-\left(\sqrt{1+\frac{\big{(}3M_{0}^{2}-2\Delta^{2}+\sqrt{9M_{0}^{4}-12M_{0}^{2}\Delta^{2}+36\Delta^{4}}\big{)}^{2}}{32\Delta^{4}}}~{}\right)^{-1}\,,$
(18)

where all the masses here correspond to their LO values and
$\Delta^{2}=m_{K}^{2}-m_{\pi}^{2}$.

Already at LO, it is interesting to illustrate the $N_{C}$ evolutions of the
masses of the pNGBs and the $\eta$-$\eta^{\prime}$ mixing angle. To take the
physical values for the masses of the pion and kaon and fix $M_{0}=820.0$ MeV
at $N_{C}=3$ from the recent determination Gu:2018swy , the $N_{C}$ variations
of the pNGBs’ masses and the mixing angle are shown in Fig. 1 for a wide range
of $N_{C}$ from 3 to 30. The most obvious changes happen for the $m_{\eta}$
and the LO mixing angle $\theta$, which significantly reduces to around 250
MeV and approaches to the ideal mixing value around $-50^{\circ}$
respectively, when increasing the values of $N_{C}$ up to 30. Due to the large
mass of the kaon, the mass of the $\eta^{\prime}$ still remains quite large
around 700 MeV for $N_{C}=30$. When including the higher order contributions
Guo:2011pa ; Guo:2012ym ; Guo:2012yt ; Guo:2015xva , the $N_{C}$ evolutions of
the masses and the LO mixing angle in Fig. 1 are found to be barely affected,
e.g. see Figs. 8 and 9 of Ref. Guo:2011pa . In contrast, the LO masses of the
$\pi,K$ and $\eta$ will be simply constants within the $SU(3)$ $\chi$PT. To
our knowledge, Refs. Guo:2011pa ; Guo:2012ym ; Guo:2012yt  are the first works
in literature to take into account the striking $N_{C}$ evolutions of the pNGB
masses and the LO mixing angle $\theta$ to determine the $N_{C}$ trajectories
of the resonance poles.

Figure 1:  The LO results for the $N_{C}$ evolutions of the pNGBs’ masses
(left panel) and the $\eta$-$\eta^{\prime}$ mixing angle (right panel).

The two-pNGB scattering amplitudes have been fully calculated up to the one-
loop level Guo:2011pa ; Guo:2012ym ; Guo:2012yt , whose partial-wave
amplitudes are then unitarized with an approximated $N/D$ approach. In the on-
shell approximation of the $N/D$ method Oller:1998zr ; Oller:2000fj ;
Oller:1999me ; Lacour:2009ej , the left-hand contributions from the crossed
channels are perturbatively included by matching the unitarized partial-wave
amplitudes with their perturbative expressions calculated at one loop in
$U(3)$ $\chi$PT Guo:2011pa ; Guo:2012ym ; Guo:2012yt . The $N/D$ approach
overcomes the IAM problem when the inverse of the matrix $T_{2}-T_{4}$ in Eq.
(2) is singular. The explicit bare resonance contributions within the
resonance chiral theory Ecker:1988te , including the scalar octet plus
singlet, the vector nonet and the pseudoscalar resonance nonet, are considered
both in the $s$\- and crossed channels in the calculation Guo:2011pa ;
Guo:2012ym ; Guo:2012yt . Vast amount of experimental scattering data,
including the phase shifts and inelasticities of the $\pi\pi$ and $K\pi$
scattering with various isospin and angular momenta numbers, and also the
$\pi\eta$ event distributions, are fitted to obtain the unknown parameters in
our theoretical models, such as the subtraction constants arising from the
unitarization procedure, the bare resonance masses and couplings, and two pure
$U(3)$ LECs that can not be contributed by the considered resonance
Lagrangians. By extrapolating the unitarized partial-wave amplitudes into the
complex energy plane, the various resonance poles are searched in different
Riemann sheets. We have found the poles for a large number of light-flavor
resonances below 1.5 GeV, including the isoscalar scalar resonances $\sigma$,
$f_{0}(980)$ and $f_{0}(1370)$, the isovector scalar resonances $a_{0}(980)$
and $a_{0}(1450)$, the strange scalar resonances $\kappa$ and
$K_{0}^{*}(1430)$, and the vector resonances $\rho(770)$, $K^{*}(892)$ and
$\phi(1020)$. Their masses and widths Guo:2012yt  are nicely compatible with
the world average values from PDG Tanabashi:2018oca .

Following the similar recipes in Eqs. (3), (4) or (5), the leading $N_{C}$
scaling of the bare resonance parameters including the masses and couplings is
implemented in Ref. Guo:2011pa , where we only consider the values of the bare
parameters determined from the fits during the $N_{C}$ scaling procedure.
Later on Ref. Nieves:2011gb  advocated imposing the leading $N_{C}$ scaling of
the bare resonance parameters by also additionally taking into account the
high energy constraints and the relations dictated by the large $N_{C}$ QCD.
In Refs. Guo:2012ym ; Guo:2012yt , we have explicitly taken into account such
constraints on some important couplings and masses when varying $N_{C}$.
Regarding the subtraction constants introduced in the unitarization procedure,
it is argued in Ref. Guo:2011pa  that they should behave as constants at the
leading $N_{C}$ order, because any variation in them due to a change of the
subtraction point in the unitarity loop functions is $O(N_{C}^{0})$. In
addition, the $N_{C}$ variations of the pNGBs’ masses and the mixing angle,
e.g. those in Fig. 1, which are ignored in the previous $N_{C}$ studies, are
carefully considered in the determination of the resonance pole trajectories
for a wide range of $N_{C}$ from 3 to 30.

To qualitatively clarify the effects of the characteristic $N_{C}$ variations
of the $U(3)$ chiral theory, as opposed to the $SU(3)$ case, we have proposed
a procedure to mimic the $SU(3)$ $\chi$PT to study the $N_{C}$ evolutions of
the various resonance pole positions by using the $U(3)$ amplitudes. The
procedure includes three parts: (a) to fix the LO mixing angle $\theta=0$
throughout; (b) to freeze the $m_{\pi},m_{K},m_{\eta}$ at their physical
values; (c) to fix the mass of the singlet $\eta_{0}$ by its LO result in Eq.
(17). The $N_{C}$ trajectories from this procedure will be denoted as $SU(3)$
in the following discussions.

Figure 2:  The movements of the $\sigma$ poles with varying $N_{C}$ from 3 to
30 Guo:2012yt . See the text for details about the differences of the $U(3)$
and $SU(3)$ results. The left panel shows the pole positions in the energy
plane and the right panel gives the pole movements in the energy squared $s$
plane.  Figure 3:  The movements of the $\kappa$ and $a_{0}(980)$ poles in the
complex energy plane with varying $N_{C}$ from 3 to 30 Guo:2012yt . See the
text for details about the differences of the $U(3)$ and $SU(3)$ results.

The $N_{C}$ paths of the $\sigma$ resonance are illustrated in Fig. 2
Guo:2012yt . First, the physical $\sigma$ pole at $N_{C}=3$ is close to those
from the use of Roy-like equations Tanabashi:2018oca . To identify the real
and imaginary parts of the pole position as the mass and half width,
respectively, the movements of the $\sigma$ pole with varying $N_{C}$ are
shown in the left panel of Fig. 2. The pole trajectories in the complex $s$
plane are shown in the right panel. When increasing the values of $N_{C}$ up
to around 15, the $\sigma$ pole position keeps running away from the real axis
in the energy squared $s$ plane. For $N_{C}>15$, its pole starts to move
towards the real axis, but in our present study the $\sigma$ pole tends to
move to the negative real $s$ axis, instead of the positive one, implying a
loss of relevance of the physical signal attached to the $\sigma$ resonance at
large $N_{C}$ within our approach. This also implies that the dominant
components in the $\sigma$ resonance are not the $\bar{q}q$. This result is
roughly similar with the one-loop $SU(3)$ IAM study Pelaez:2003dy ;
Pelaez:2004xp . The $\sigma$ poles from the $U(3)$ and $SU(3)$ cases are
quantitatively close for $N_{C}\leq 10$ and for large values of $N_{C}$ the
two schemes still give qualitatively similar curves.

Regarding the pole variations of the $\kappa$ and $a_{0}(980)$ Guo:2012yt , as
shown in Fig. 3, roughly speaking they share comparable trends as the
$\sigma$, meaning that their poles tend to move deeper into the complex energy
plane when increasing the values of $N_{C}$, instead of falling down to the
real axis. According the $N_{C}$ behaviors in Fig. 3, it is unlikely that
there are important $\bar{q}q$ components inside the $\kappa$ and $a_{0}(980)$
at large $N_{C}$. This seems also consistent with the initial setups of our
theoretical models, since the $\sigma$, $a_{0}(980)$ and $\kappa$ do not have
any corresponding bare state in the resonance chiral Lagrangians and they are
mainly generated from the strong interactions among the pNGBs. Both the $U(3)$
and $SU(3)$ treatments give qualitatively comparative $N_{C}$ trajectories to
the $\kappa$ and $a_{0}(980)$, though the deviations for the $\kappa$ poles
are relatively larger. It is pointed out that the $N_{C}$ curves for the
$\kappa$ and $a_{0}(980)$ are also somewhat sensitive to the different fit
strategies. For example, one can see some obvious differences of the $\kappa$
trajectories between the results in Refs. Guo:2012yt ; Guo:2011pa , although
in both cases the $\kappa$ pole dilutes deeply into the complex energy plane.

Figure 4:  The movements of the resonance poles for the $\rho(770)$,
$f_{0}(980)$, $K^{*}(892)$ and $K^{*}_{0}(1430)$ in the complex energy plane
with varying $N_{C}$ from 3 to 30 Guo:2012yt . See the text for details about
the differences of the $U(3)$ and $SU(3)$ results.

The $N_{C}$ trajectories of the resonances $\rho(770)$ and $f_{0}(980)$ in the
coupled $\pi\pi$ scattering, and the $K^{*}(892)$ and $K^{*}_{0}(1430)$ in the
coupled $K\pi$ scattering, are illustrated in Figs. 4 Guo:2012yt . A striking
difference of the $N_{C}$ pole movements between the
$\sigma,\kappa,a_{0}(980)$ in Figs. 2 3 and the
$\rho(770),f_{0}(980),K^{*}(892),K^{*}_{0}(1430)$ in Fig. 4 is that the latter
ones approach to the positive real axis in the complex energy plane for large
values of $N_{C}$, which is a characteristic feature of a standard $\bar{q}q$
meson. The vector resonances $\rho(770)$ and $K^{*}(892)$, exhibit perfect
behaviors for $\bar{q}q$-like mesons, with their masses approaching to a
constant and their widths vanishing precisely as $1/N_{C}$ when increasing the
values of $N_{C}$. The differences caused by the $U(3)$ and $SU(3)$ theories
are small for the vector resonances, in particular the deviations for the
$\rho(770)$ are almost invisible for all values of $N_{C}$. The $f_{0}(980)$
in our study receives important contributions both from the $\bar{K}K$ channel
and a bare singlet scalar state, whose mass turns out to be around 1 GeV from
the fits Guo:2011pa ; Guo:2012yt . Were the role played by the bare singlet
scalar dominant, one would expect that the $N_{C}$ paths of the $f_{0}(980)$
look similar to the vector ones. In the actual situation, the widths of the
$f_{0}(980)$ tend to increase for $N_{C}<8$, which hints that the $\bar{q}q$
component is subdominant and probably the molecular component of the
$\bar{K}K$ is the dominant one in this region. For the case of $N_{C}\geq 8$
the $f_{0}(980)$ pole moves towards the real axis with decreasing width and
more or less constant mass, indicating that the $\bar{q}q$ type bare state
starts to play the predominant role at large $N_{C}$. For the heavier excited
scalar resonances $K^{*}_{0}(1430)$, $f_{0}(1370)$ and $a_{0}(1450)$, they
receive important contributions from the bare scalar octet in the resonance
chiral Lagrangian, whose mass is determined to be around 1.4 GeV from the fits
Guo:2011pa ; Guo:2012yt . For large values of $N_{C}$, their pole positions
all run to the real axis with vanishing widths and we explicitly show the
results for the $K^{*}_{0}(1430)$ in Fig. 4. Other $N_{C}$ curves for the
$f_{0}(1370)$ and $a_{0}(1450)$ are quite similar and can be found in Refs.
Guo:2011pa ; Guo:2012yt . A general feature for the heavier excited scalar
resonances is that their $N_{C}$ trajectories are moderately sensitive to the
different treatments of the $U(3)$ and $SU(3)$ theories. Although different
prescriptions of the $U(3)$ and $SU(3)$ will not alter the final trends of the
pole movements, they do lead to obviously different $N_{C}$ paths for the
heavier excited scalar resonances, as one can see the panels of the
$f_{0}(980)$ and $K^{*}_{0}(1430)$. Similar conclusions are also obtained for
the $f_{0}(1370)$ and $a_{0}(1450)$ Guo:2011pa ; Guo:2012yt .

The interesting $N_{C}$ evolutions of the various resonance poles will
definitely influence other related physical quantities. We have focused on the
semilocal duality in the $\pi\pi$ scattering and the Weinberg spectral sum
rules in the scalar and pseudoscalar sectors in Refs. Guo:2012yt . Their
variations with different values of $N_{C}$ are specially paid attention to.
For the fixed-$t$ meson-meson scattering amplitudes, the semilocal duality
refers to the duality relation between the Regge theory and the hadronic
system. In $\pi\pi$ scattering, the semilocal duality relation provides a
useful object to study the interplays between the vector and scalar
resonances, and it can be explicitly verified through the finite-energy sum
rules RuizdeElvira:2010cs . Due to the absence of the $s$-channel resonances
in the $\pi\pi$ amplitude with $I=2$, the Regge theory predicts vanishing
results for the isotensor amplitude. The duality relation then requires a
strong cancellation of the resonances from the crossed channels in the
hadronic side. In Refs. Guo:2012ym ; Guo:2012yt , we have used the semilocal
duality to study the possible relations between the scalar
($\sigma,f_{0}(980),f_{0}(1370)$) and the vector ($\rho(770)$) resonances. At
$N_{C}=3$, to fulfill the duality relation the $\sigma$ resonance generally
plays important roles in balancing the $\rho(770)$. For large values of
$N_{C}$, it is found that a scalar resonance at around 1 GeV is needed to
balance the $\rho(770)$ contribution to satisfy the semilocal duality relation
RuizdeElvira:2010cs . In our case, the contributions from the $\sigma$
gradually fade away, and it is the $f_{0}(980)$ that finally becomes the one
to cancel the effect from the $\rho(770)$ Guo:2012yt , through the
contribution to its composition of an elementary state introduced as a bared
field, as discussed above.

On the other hand, the Weinberg spectral sum rules of the $SS-PP$ types, with
$S$ the scalar currents and $P$ the pseudoscalar currents, offer another
theoretical tool to examine the relations between the scalar and pseudoscalar
resonances. The scalar spectral functions with different flavor contents are
calculated via the unitarized scalar form factors of the two-pNGB states
Guo:2012ym ; Guo:2012yt ; Meissner:2000bc ; Oller:2000ug . Similarly the
pseudoscalar spectral functions are constructed with the pseudoscalar form
factors contributed by the single-state exchanges of the pNGBs and the
pseudoscalar resonances Albaladejo:2010tj . According to the results of the
perturbative QCD and OPE, the $SS-PP$ spectral function integrals in the high
energy region tend to vanish in the chiral limit. This in turn implies the
cancellation between the scalar and pseudoscalar spectral integrals in the
nonperturbative region, if the $SS-PP$ Weinberg sum rules hold. By properly
performing the chiral extrapolation, the spectral sum rules are simultaneously
examined in the physical quark-mass situation and in the chiral-limit case.
Three different flavor types of the spectral functions are considered, namely
the $SU(3)$ singlet current $(\bar{u}\Gamma u+\bar{d}\Gamma d+\bar{s}\Gamma
s)/\sqrt{3}$, the $SU(3)$ octet $(\bar{u}\Gamma u+\bar{d}\Gamma
d-2\bar{s}\Gamma s)/\sqrt{6}$ and the isovector $(\bar{u}\Gamma
u-\bar{d}\Gamma d)/\sqrt{2}$, being $\Gamma=1$ for the scalar condensates and
$\Gamma=\gamma_{5}$ for the pseudoscalar condensates. The updated fits in Ref.
Guo:2012yt  indicate that the $SS-PP$ Weinberg spectral sum rules are well
satisfied for a wide range values of $N_{C}$, with the violation at the level
less than 16% Guo:2012yt . It is interesting to point out that the mechanisms
for the fulfillment of the $SS-PP$ Weinberg sum rules at different stages in
the evolution with $N_{C}$ are different. For the $SU(3)$ singlet spectral sum
rule, both the $\sigma$ and $f_{0}(980)$ scalar resonances are found to be
important to compensate the contributions from the pseudoscalar mesons at
$N_{C}=3$. For the $SU(3)$ octet case, in addition to the $\sigma$ and
$f_{0}(980)$, the $f_{0}(1370)$ also gives important contributions. For the
isovector $(\bar{u}\Gamma u-\bar{d}\Gamma d)/\sqrt{2}$ case at $N_{C}=3$, it
turns out that the rather prominent $a_{0}(980)$ peak gives more important
contributions than the $a_{0}(1450)$ to the scalar spectral integral. However,
as shown in Figs. 2 3 4, different scalar resonances show clearly different
$N_{C}$ trajectories, implying that their contributions to the scalar spectral
functions will also vary with different values of $N_{C}$. More precisely
speaking, the effects of the $\sigma$ and $a_{0}(980)$, though important at
$N_{C}=3$, start to diminish for large values of $N_{C}$, while the
contributions from the $f_{0}(980)$, $f_{0}(1370)$ and $a_{0}(1450)$ become
crucial in the large $N_{C}$ limit. Once the high-energy constraints on the
resonance bare couplings are properly imposed, the $SS-PP$ Weinberg spectral
sum rules are found to be perfectly fulfilled in the chiral limit at large
$N_{C}$ Guo:2012ym ; Guo:2012yt .

##  4 Summary and conclusions

In this work, we give an extensive review on the variations of the pole
positions of various light-flavor meson resonances and related physical
quantities by varying the number of colors $N_{C}$ of QCD. Although it is
difficult to give quantitative conclusions on the internal structures, the
$N_{C}$ trajectories of the resonance pole positions, namely the movements of
their masses and widths with varying $N_{C}$, can provide intuitive and, some
times, even compelling results to discern the interior constituents of
resonances.

Many theoretical studies from different research groups, including the inverse
amplitude method, the $N/D$ approach, unitarized quark models, dispersive
Omnès function, linear-sigma-model like scheme, etc., have been proposed to
examine the $N_{C}$ trajectories of meson resonance poles. A general agreement
from different models on the $N_{C}$ behaviors of the vector resonances, such
as the $\rho(770)$ and $K^{*}(892)$, has been reached. As expected for a
conventional $\bar{q}q$ meson, the masses approach to constants and the widths
nicely decrease as $1/N_{C}$ when taking larger values of $N_{C}$. The
movement in $N_{C}$ of the vector-resonance poles can be considered as a
calibration for the model.

The situation for the scalar resonances is more subtle. It is enlightening to
elaborate on their behaviors in different $N_{C}$ regions, namely the ones
near $N_{C}=3$ and those in the large $N_{C}$ case. A more or less robust
behavior revealed by different groups for the broad $\sigma$ or $\kappa$ is
that their poles seem not falling down to the real axis for $N_{C}$ not far
away from its physical value. However, the trajectories of the $\sigma$ and
$\kappa$ resonances at large values of $N_{C}$ turn out to be sensitive to
many factors, such as different fit strategies, higher order low energy
constants, different unitarization models, etc. It is found in some works that
the $\sigma$ and $\kappa$ poles fade away at large $N_{C}$. In other
situations, such as the two-loop inverse-amplitude-method case, the $\sigma$
pole finally turns back to the positive real axis at the large $N_{C}$,
indicating that there is a $q\bar{q}$ seed in the $\sigma$ with a mass around
1 GeV, which is rather different from the physical mass of the $\sigma$.
Compared to the sharp $N_{C}$ trajectories for the vector resonances, all of
these features found for the broad $\sigma$ and $\kappa$ imply that they do
not seem to correspond to the standard $q\bar{q}$ states, so that the
$q\bar{q}$ seeds are unlikely the dominant constituents.

The presence of the $\bar{K}K$ threshold makes the studies of the $f_{0}(980)$
and $a_{0}(980)$ more complicated, and the $N_{C}$ movements of the two
resonance poles from different studies also show different trends. The recent
$U(3)$ study on the heavier excited scalar resonances $f_{0}(1370)$,
$K^{*}_{0}(1430)$ and $a_{0}(1450)$ show that their pole positions move
towards the real axis with vanishing widths for large $N_{C}$. The finite-
volume energy levels related to the scalar resonances $\sigma$ and
$f_{0}(980)$ in the unquenched lattice QCD nowadays have been obtained with
great precision with $N_{C}=3$ Briceno:2016mjc ; Briceno:2017qmb . Future
similar unquenched lattice simulations but with different values of $N_{C}$
could provide crucial hints of the internal structures of the scalar
resonances.

## Acknowledgments

I would like to thank J. A. Oller and J. Ruiz de Elvira for the careful
reading of the manuscript and useful suggestions. This work is partially
funded by the Natural Science Foundation of China under Grant Nos. 11975090
and 11575052, the Natural Science Foundation of Hebei Province under Contract
No. A2015205205, and the Fundamental Research Funds for the Central
Universities.

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