A new method to study the number of colors in the final-state interactions of hadrons
Ling-Yun Dai$^a$\footnote{l.dai@fz-juelich.de
Ulf-G. Mei{\ss
We match the $\pi\pi\to\pi\pi$ scattering amplitudes of Chiral Perturbation Theory with those from dispersion relations that respect analyticity and coupled channel unitarity, as well as accurately describing experiment. Their dependence on the number of colors ($N_C$) is obtained. By varying $N_C$ the trajectories of the poles and residues (the couplings to $\pi\pi$) of light mesons, the $\sigma$, $f_0(980)$, $\rho(770)$ and $f_2(1270)$ are investigated. Our results show that the method proposed is a reliable way to study the $N_C$ dependence in hadron-hadron scattering with final-state interactions.
largeNc_letter_7_1.tex
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\begin{document}

\title{ A new method to study the number of colors in the final-state interactions of hadrons }
\author{Ling-Yun Dai$^a$\footnote{l.dai@fz-juelich.de}}
\author{Ulf-G. Mei{\ss}ner$^{b,a}$\footnote{meissner@hiskp.uni-bonn.de} }
\affiliation{ \small $^a$Institute for Advanced Simulation, Institut f\"ur Kernphysik
   and J\"ulich Center for Hadron Physics, Forschungszentrum J\"ulich, D-52425 J\"ulich, Germany\\
$^b$ Helmholtz Institut f\"ur Strahlen- und Kernphysik and Bethe Center
 for Theoretical Physics, Universit\"at Bonn, D-53115 Bonn, Germany\\
}

\date{\today}


\begin{abstract}
We match the $\pi\pi\to\pi\pi$ scattering amplitudes of Chiral Perturbation Theory with those
from dispersion relations that respect analyticity and coupled channel unitarity,  as well as
accurately describing experiment. Their  dependence on the number of colors ($N_C$)
is obtained. By varying $N_C$ the trajectories of the poles and residues (the couplings to
$\pi\pi$) of light mesons, the $\sigma$, $f_0(980)$, $\rho(770)$ and $f_2(1270)$ are investigated. Our results show that the method proposed is a reliable way to study the $N_C$ dependence in
hadron-hadron scattering with final-state interactions.
\end{abstract}

\pacs{11.55.Fv, 11.80.Et, 12.39.Fe, 13.60Le}
\keywords{Dispersion relations, Partial-wave analysis, Chiral Lagrangian, meson production}


\maketitle



The lightest scalar mesons are rather interesting as they have the same quantum numbers as
the QCD vacuum. The nature of them is still a mystery~\cite{MRP07Rev,MRP10,jaffe4q,Pelaez15}.
The phenomenology of these is  complicated due the contribution from important final-state interactions
(FSI)~\cite{AMP-FSI}.
Dispersion relations are the natural way to include FSI, see e.g.~\cite{Kang:2013jaa,Chen2015}.
For some of the light mesons, like the $\sigma$, $\kappa$, their existence has been
confirmed~\cite{zheng00,colangelo01,zhou04,Moussallam06} and accurate pole locations
and $\pi\pi$ couplings, including also the $\rho(770)$, have been given in
Refs.~\cite{caprini06,Descotes04,PelaezPRL}.
%A Regge trajectory study reveals the non-ordinary quark-antiquark structure of sigma~\cite{Nebreda2014} makes it more interesting.
Concerning the nature of the scalar mesons, there is a cornucopia of models \cite{vanbeveren,schechter,Baru2003,maiani,Hooft08,Zhou2011,MNW2,Guo2011,Guo2012,Nebreda2014,Jacob2014a,Yao2015,Guo2015,Wilson2016}. Among them
the large $N_C$ trajectories of the poles are an effective diagnostics to distinguish ordinary from  non-ordinary
quark-antiquark structure as considered
in~\cite{Pelaez04,Pelaez06,MRP11,Sun,DLY11}. However, these analysis based on
unitarized Chiral Perturbation theory (U$\chi$PT) lack crossing symmetry. Unitarization itself
will also generate spurious poles and cuts.
In contrast, dispersion relations respect analyticity, but including coupled channel unitarity
and the $N_C$ dependence is difficult.
Clearly, both analyticity and coupled channel unitarity are critical in the region of the
$\overline{K}K$ threshold, close to which the $f_0(980)$ is located. To solve this problem, we
use an Omn\`es representation based on the phase of the relevant amplitudes,
rather than the elastic phase shift~\cite{DLY-MRP14,DLY2015}.
%
There has been renewed interest in the study of the  large $N_C$ limit~\cite{'tHooft:1973jz,'tHooft:1974hx}
of the properties of resonances~\cite{Witten:1979kh,Coleman1985,Cohen1998}.
Weinberg~\cite{Weinberg2013} pointed out that resonant tetraquark states could exist due to the
contribution of the leading order (LO) \lq connected' diagrams to the Green functions.
Their widths are $\mathcal{O}(N_C^{-1})$, as narrow as ordinary mesons.
They could be even narrower, with width of $\mathcal{O}(N_C^{-2})$, when the flavor of the quarks
is combined in different ways~\cite{Knecht2013}.
There are many other interesting discussions such as~\cite{Lebed2013,Cohen2014} and references therein.
%Following~\cite{Knecht2013} the width of isoscalar $0^{++}$ resonances, if they are tetraquark ($[\bar{q}q][\bar{q}q]$), will be $\mathcal{O}(N_C^{-2})$.
%The question is then how to \lq practical' these , is it possible to inject color into the analytical amplitudes, thus we can find the $N_C$ dependence of the resonances.
In this paper we focus on establishing a \lq practical' way to study the $N_C$ dependence of
the scattering amplitudes, built into dispersion relations.
Resonances appearing in the intermediate states are also studied.

In this letter we first use dispersive methods to obtain the $\pi\pi$ scattering amplitudes
up to 2~GeV. We construct the amplitudes in a model-independent way,   which is both analytic
and respects coupled channel unitarity.
We also recalculate the analytical expressions of $IJ=00,02,11$ waves in $SU(3)$ Chiral Perturbation
Theory ($\chi$PT) up to $\mathcal {O}(p^4)$. By expanding the amplitudes in the low-energy region,
we match the dispersive and the $\chi$PT amplitudes and also introduce the $N_C$-dependence
into the dispersive amplitudes. This $N_C$ dependence is automatically transferred to the
high-energy region, where the FSI are implemented by the dispersion relation.
We give the trajectories of the poles and residues by varying $N_C$.
The behavior of the $\rho(770)$, $f_2(1270)$, $\sigma(600)$ and $f_0(980)$ show that this
is a reliable way to study the number of colors in hadron-hadron scattering. The $N_C$ trajectory of
the light scalar mesons supports a mixed structure of hadronic molecule and
$\bar{q}q$ components (for a recent review on hadronic molecules, see Ref.~\cite{Guo:2017jvc}.


We first present our $IJ=00,02,11$ partial waves of $\pi\pi\to\pi\pi$ calculated
in a model-independent way. We start from:
\be\label{eq:TP}
T^I_J(s)=P^I_J(s)\Omega^I_J(s).
\ee
where $\Omega^I_{J}(s)$ is the Omn\`es function~\cite{Omnes1958}:
\be\label{eq:Omnes}
\Omega^I_{J}(s)=\exp\left(\frac{s}{\pi} \int^\infty_{s_{th}} ds' \frac{\varphi^I_{J}(s')}{s'(s'-s)}\right) \,\, .
\ee
with $\varphi^I_{J\lambda}(s)$ the phase of the partial wave amplitude $T^I_{J}(s)$, which has been given in previous amplitude analysis~\cite{DLY-MRP14,DLY2015}.
This phase is known from experiment up to roughly 2~GeV. The function $P^I_J(s)$ includes
the effect of the left-hand cut (l.h.c) and corrections that come from the distant right-hand
cut (r.h.c) above 2~GeV.
Other information is provided by chiral dynamics that fixes the Adler zeros in the S-waves,
and the approach to threshold of the S-, P-, and D-waves in terms of scattering lengths and
effective ranges. We therefore parameterize the $P^I_J(s)$ as
\be\label{eq:P}
P^I_J(s)=(s-z^I_J)^{n_J}\sum_{k=1}^n {\alpha^I_J}_k (s-4M_\pi^2)^{k-1}\,,
\ee
where $z^I_J$ is the Adler zero for the S-waves and $4M_\pi^2$ for P- and  D-waves.
The parameter $n_J$ is 1 for S- and P-waves and 2 for D-waves.
The parameters $\alpha_i$ are given in Table~\ref{tab:para;alpha}. The units of the
$\alpha_k$ are chosen to ensure that the amplitude $T^I_J(s)$ is dimensionless.
\begin{table}[htbp]
\hspace{-1.5cm}
\vspace{-0.0cm}
{\footnotesize
\begin{center}
\tabcolsep=0.11cm
\begin{tabular}  {|c|c|c|c|c|c|c|}
\hline
                                                                & ${T_{11}}^0_S$  &  ${T_{11}}^0_D$ & ${T_{11}}^1_P$ \rbox \\[0.5mm] \hline
\rule[0.32cm]{0.1cm}{0cm}$\alpha_1$\rule[0.32cm]{0.1cm}{0cm}    & 2.4051          & 0.2972          & 0.4283           \\ \hline
\rule[0.32cm]{0.1cm}{0cm}$\alpha_2$\rule[0.32cm]{0.1cm}{0cm}    & -1.9451         & -0.9354         &-0.2976           \\ \hline
\rule[0.32cm]{0.1cm}{0cm}$\alpha_3$\rule[0.32cm]{0.1cm}{0cm}    & 1.5068(39)      &  2.3171(2)      & 0.6278(17)       \\ \hline
\rule[0.32cm]{0.1cm}{0cm}$\alpha_4$\rule[0.32cm]{0.1cm}{0cm}    & -0.8590(20)     & -2.8838(2)      &-0.7225(11)       \\ \hline
\rule[0.32cm]{0.1cm}{0cm}$\alpha_5$\rule[0.32cm]{0.1cm}{0cm}    & 0.2942(6)       &  1.8719(1)      & 0.3820(4)        \\ \hline
\rule[0.32cm]{0.1cm}{0cm}$\alpha_6$\rule[0.32cm]{0.1cm}{0cm}    & -0.0531(2)      & -0.6520(1)      & -0.0912(1)       \\ \hline
\rule[0.32cm]{0.1cm}{0cm}$\alpha_7$\rule[0.32cm]{0.1cm}{0cm}    & 0.0039(1)       &  0.1156(1)      & 0.0081(1)        \\ \hline
\rule[0.32cm]{0.1cm}{0cm}$\alpha_8$\rule[0.32cm]{0.1cm}{0cm}    & -               & -0.0082(1)      & -             \\ \hline
\end{tabular}
\caption{\label{tab:para;alpha}The fit parameters of our Fit, as given in Eq.~(\ref{eq:P}).
The errors are given by MINUIT and notice that the $\alpha_{1,2}$ are
fixed by the scattering lengths and slope parameters~\cite{DLY2015,KPY,Ke4}.}
\end{center}
}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The fit results are shown in Fig.~\ref{Fig:T}. We fit the amplitudes in the region of
$s\in[0,4\rm{GeV}^2]$, where the \lq data' is as follows: $\chi$PT amplitudes at
$[0,4M_\pi^2]$, amplitudes of K-matrix and Roy-like equation at $[4M_\pi^2,2\rm{GeV}^2]$,
and experiment data up to $4\rm{GeV}^2$. Though we do not have the left-hand cuts directly
in our fit, their contribution are correctly implemented in the physical region, as we fit to
the results of Roy-like equation which keeps the crossing symmetry. The fits are
of high quality, even in our \lq prediction' region where $s\in [-4M_\pi^2,0]$.
From these amplitudes, we can extract  the poles and residues on the second sheet.
The residue $g_{f\pi\pi}$ and pole $s_R$ on the second Riemann sheet are defined as:
\bea
T^{II}(s)=\frac{g_{f\pi\pi}^{2}}{s_R-s}\,.\label{eq;g}
\eea
\begin{figure}[th]
\includegraphics[width=0.48\textwidth,height=0.45\textheight]{T_3.eps}
\caption{\label{Fig:T} The fit of $\pi\pi$ scattering amplitudes are in the left column. The violet and magnetic bands are from K-Matrix~\cite{DLY-MRP14}. The olive and light grey bands in the low energy region are from $\chi$PT~\cite{Gasser1984,Gasser1985,Bijnens1994,Pelaez02,DLY11}. The cyan and green bands are from CFDIV~\cite{KPY}.  The Cern-Munich data is from~\cite{CERN-Munich} and the OPE and OPE-DP data are from~\cite{OPE1973}. The absolute values of the amplitudes by varying $N_C$ are in the right column. The black solid, magenta dashed, and blue dotted lines are for $N_C=3,4,5$, respectively. }
\end{figure}
The pole locations and their residues are listed in Table~\ref{tab:poles}.
%The error is estimated in two parts, one is from the MINUIT and the other part is from systematic error from different solutions with the same quality.
\begin{table}[tbp]
%\vspace{-1mm}
{\footnotesize
%\begin{center}
%\hspace{-10mm}
\begin{tabular}{|c || c |c |@{}c @{}|@{}c@{} |}
%\cline{2-4}\multicolumn{1}{c|}{~~}
\hline
\rule[-0.3cm]{0cm}{0.8cm}\multirow{2}{*}{\rule[-1cm]{0cm}{2cm}State}  &  pole locations  &\multicolumn{2}{c|}{$g_{f\pi\pi}=|g_{f\pi\pi}|e^{i\varphi}$~}  \\
\cline{3-4}%\noalign{\smallskip}
\rule[-0.3cm]{0cm}{0.8cm}    & (MeV)   & $~|g_{f\pi\pi}|~(GeV)~$ & $~\varphi~(^\circ)~$   \\
\hline\hline
$\sigma/f_0(500)$ & $436.2(12.2)-i260.7(6.8)$
     & $0.45(0.02)$   & $74(2)$      \rbox \\[0.5mm] \hline
$f_0(980)$        & $997.7(1.1)-i21.7(1.9)$
     & $0.27(0.02)$   & $84(3)$    \rbox \\[0.5mm] \hline
$f_0(1370)$        & $1431.6(34.6) -i185.4(22.4)$
     & $0.78(0.21)$   & $-47(18)$    \rbox \\[0.5mm] \hline
%$f_0(1370)$/       & \multirow{2}{*}{$1431.6(34.6) -i185.4(22.4)$}
%    & \multirow{2}{*}{$0.78(0.21)$}   & \multirow{2}{*}{$-47(18)$}   \rbox \\[-5mm]
%$f_0(1500)$?       & \multirow{2}{*}{} & \multirow{2}{*}{}
%    & \multirow{2}{*}{}     \rbox \\[0.5mm] \hline
$f_2(1270)$       & $1278.3(7.0) -i79.3(17.8)$
        & $0.50(0.05)$   & $0.7(4.3)$    \rbox \\[0.5mm] \hline
$\rho(770)$       & $762.4(3.9) -i68.7(6.3) $
    & $0.34(0.01)$   & $12(3)$                     \rbox \\[0.5mm] \hline
\end{tabular}
\caption{\label{tab:poles}The pole locations and residues on the second Riemann sheet.  }
%\end{center}
}
\end{table}
These are very similar from those of previous
analyses~\cite{caprini06,DLY-MRP14,PelaezPRL,Rusetsky2011,PDG16}.
For the $f_0(1370)$ and $f_2(1270)$, to find the pole closest to the physical sheet one needs to
include the $\pi\pi$, $\overline{K}K$, 4$\pi$ as coupled channels. However, notice that a Breit-Wigner
resonance will always have shadow poles on other sheets.
Although $\pi\pi$ is the dominant decay channel of the $f_2(1270)$, our $f_2(1270)$ on
the second sheet is not far away from the physical one.

Having analytically calculated the partial wave amplitudes of $IJ=00,11,02$ within
one-loop SU(3) $\chi$PT,
we can match our dispersive results to these and so fix their $N_C$ dependence.
We note several points about this matching:
First, though the matching is done in the low-energy region,
this $N_C$ dependence is transferred to  high-energy region.
As the FSI of hadrons at higher energy region corresponds to the \lq hadron loop'
corrections, 
they could be translated into the higher order corrections (quark loops) in large $N_C$ QCD,
and suppressed by an extra factor of $N_C^{-1}$.
%
Second, all the $\chi$PT amplitudes are calculated up to $\mathcal{O}(p^4)$, $\mathcal{O}(p^6)$ or higher orders are ignored.
This contains the $N_C$ dependence of $N_C^{-1}$ and $N_C^{-2}$.
Consequently we only perform the matching up to $N_C^{-2}$.
%
Third, in the $\chi$PT amplitudes the r.h.c is given entirely from  one-loop
integrals (the $B$ functions) in the $\mathcal{O}(p^4)$ amplitudes.
This is why we do not consider a matching with only LO $\chi$PT amplitudes.
The imaginary part of the $\chi$PT amplitudes is $N_C^{-2}$, while the real part of the
amplitudes is of $N_C^{-1}$.
From this one sees that the phase of the amplitudes that enter the Omn\`es functions
should be $N_C^{-1}$.
Indeed, any higher order $N_C$-dependence of the phases such as $N_C^{-2}$ can be ignored,
which will contribute finally at $\mathcal{O}(N_C^{-3})$ to
the amplitudes $T^I_J(s)$ in Eq~(\ref{eq:TP}).
%
Fourth, we choose the matching points to be the Adler zero for the S-waves and $2M_\pi^2$ for
the P- and D-waves. This avoids  $s=0$ where the l.h.c starts
and the threshold where the r.h.c starts.


To match the dispersive and $\chi$PT amplitudes we expand each one. For the dispersive
amplitude of Eq~(\ref{eq:TP}) we first expand the function $P^I_J(s)$ as:
\be\label{eq:P;expand}
P^I_J(s)=(s-{z}^I_J)^{n_J}\sum_{k=1}^n {\beta^I_J}_k (s-{s_m}^I_J)^{k-1}\,,
\ee
where $z^I_J$ is given in Eq.~(\ref{eq:P}), while ${s_m}^I_J$ is the matching point. One readily finds
\bea
{\beta^I_J}_k=\sum_{l=k}^{n}C_{l-1}^{k-1}{\alpha^I_J}_l({s_m}^I_J-4M_\pi^2)^{l-k}\;,\label{eq:beta}
\eea
where $C_{l-1}^{k-1}$ are the binomial coefficients.
With this transformation one can translate the parameters ${\alpha^I_J}_l$ in
Table~\ref{tab:para;alpha} into ${\beta_k}^I_J$.
Then we expand the Omn\`es function in the low-energy region as:
\bea
\Omega^I_J(s)=\exp\left(\sum_{k=1}^{n} \omega_k({s_m}^I_J)(s-{s_m}^I_J)^{k-1}\right)\;,
\eea
with
\bea
\omega_1({s_m}^I_J)&=&\frac{1}{\pi}\int\frac{\varphi^I_J(s')~ds'}{s'(s'-{s_m}^I_J)}\;,\no\\
\omega_k({s_m}^I_J)&=&\frac{1}{\pi}\int\frac{\varphi^I_J(s')~ds'}{(s'-{s_m}^I_J)^{k}}\;, \;\;\text{for}~~ k>1\;.
\eea
We note that $\varphi^I_J(s)$ is proportional to $N_C^{-1}$, and find
\bea\label{eq:Omnes;expand}
\Omega^I_J(s)=1+\sum_{k=1}^{n} \frac{3}{N_C}\left(\omega_k({s_m}^I_J)(s-{s_m}^I_J)^{k-1}\right)+\cdots\;.
\eea
For the $\chi$PT amplitudes, they are only calculated up to $\mathcal{O}(p^4)$,
so we expand the amplitudes up to $(s-s_m)^2$.
%
For higher order  $\chi PT$ amplitudes, either from the higher order expansions of the $B$
functions in $\mathcal{O}(p^4)$ or from higher order loop corrections,  these are
at least $N_C^{-2}$ or even \lq weaker'. As a result we simply have
\be
{\beta^I_J}_k\sim \frac{1}{N_C^2}\;,\;\;\text{for}\;k\geq3\;.
\ee
Finally we obtain the $N_C$-dependence of the coefficients in Eq.~(\ref{eq:P;expand}).
For the S-wave, we have
\bea
P^I_S(s,N_C)&=&(s-s_0)\sum_{k=1}^n \frac{d^{(k)}_{IS}(N_C)}{d^{(k)}_{IS}(3)}{\beta^I_S}_k (s-s_0)^{k-1}  \no\\
            &+&d^{(0)}_{IS}(N_C)\;,
\eea
For the P- and D-waves, we find
\bea
P^{I}_{J}(s,N_C)=(s-4M_\pi^2)^J\sum_{k=1}^n \frac{d^{(k)}_{IJ}(N_C)}{d^{(k)}_{IJ}(3)}{\beta^I_J}_k (s-s_0)^{k-1},\no\\
\eea
%
The LECs and their $N_C$-dependence in $\chi$PT are given by~\cite{Gasser1984,Gasser1985}.
%%%%
We also use the relation $L_2\sim 2L_1$ which is derived from the matching of $\chi PT$
with $R\chi T$~\cite{Ecker1:1989}.
With these LECs and the input $M_\pi=0.13957$~GeV, $M_K=0.496$~GeV, $M_\eta=0.54785$~GeV,
$F_\pi=0.924$~GeV and its $N_C$ dependence up to $N_C^{-1}$ \footnote{We note that ~\cite{Jacob2015} points out the sub-leading $N_C$ dependence of LECs may be sizable. Here we ignore it as the matching is done at low energy region.}
we finally get:
\bea
d^{(0)}_{0S}(N_C)&=&-0.0077\frac{3}{N_C}+0.0077\frac{9}{N_C^2}\;,\no\\
d^{(1)}_{0S}(N_C)&=& 2.6205\frac{3}{N_C}+6.7444\frac{9}{N_C^2}\;,\no\\
d^{(2)}_{0S}(N_C)&=&-5.8358\frac{3}{N_C}+19.2587\frac{9}{N_C^2}\;,\no\\
d^{(1)}_{1P}(N_C)&=& 0.4062\frac{3}{N_C}-0.6776\frac{9}{N_C^2}\;,\no\\
d^{(2)}_{1P}(N_C)&=& 0.6126\frac{3}{N_C}-1.8309\frac{9}{N_C^2}\;,\no\\
d^{(1)}_{0D}(N_C)&=&-0.0864\frac{3}{N_C}+0.4595\frac{9}{N_C^2}\;,\label{eq:Nc}
\eea
We are aware that these coefficients can be modified by higher order corrections, but this
goes beyond the accuracy of the present calculation.



We test the stability of the amplitudes by varying $N_C$. The shape of the amplitudes are
rather stable, and only the magnitude varies dramatically, as shown in Fig.~\ref{Fig:T}
This is because in our model a $N_C^{-1}$ factor appears in the Omn\`es function.
An exponential function of course converges fast.
Consequently, poles move rapidly  either towards the real energy axis, or far away,
much faster than the behavior using U$\chi$PT.
We only match to $\mathcal{O}(p^4)$ and lack more accurate $N_C$-dependence, with which the peak of $|T|$ will behave as $\mathcal{O}(1)$ for $P$ and $D$ waves. This is because for Breit-Wigner particles, there is a zero of the real part amplitudes ($s=M_R^2$), where the $N_C$ dependence is canceled leading to
$T\simeq i/\rho$. However, our pole trajectories suggest it has no obvious influence on the
amplitudes locate elsewhere. The zero is not there and $|T|$ is still $\mathcal{O}(N_C^{-1})$.
%
The trajectories of the pole locations and their residues $g_{f\pi\pi}$ on the second Riemann
sheet are obtained from Eq.~(\ref{eq;g}), and plotted in Fig.~\ref{Fig:pole}.
\begin{figure}[ht]
\includegraphics[width=0.48\textwidth,height=0.6\textheight]{pole_5.eps}
\caption{\label{Fig:pole} The trajectories of pole locations and their residues by varying $N_C$. The black filled circle and lines represent poles and residues, respectively. The matching points are chosen as follows: Adler zero for the S-waves and $2M_\pi^2$ for P- and D-waves.
The step of $\Delta N_C$ is 0.01 (just for illustration, of course, $N_C$ should be an integer).
The magenta dashed, red dotted, orange dash-dotted, olive dashed, blue dotted, and dark cyan
lines represent the results for different matching points: $M_\pi^2$, $2M_\pi^2$, $3M_\pi^2$, $5M_\pi^2$, $6M_\pi^2$, $7M_\pi^2$ for the S-wave, and ${s_a}^0_S$, $M_\pi^2$, $3M_\pi^2$, $5M_\pi^2$, $6M_\pi^2$, $7M_\pi^2$ for the P- and D-waves, respectively. Note that for the
 matching points above threshold we have added $i0.001$ to avoid the singularity.}
\end{figure}
To determine the uncertainties of our trajectories, we choose different matching points in
$(0, 8M_\pi^2)$ and present the results in Fig.~\ref{Fig:pole}.
They are similar to each other.
This is consistent with the assumption that the matching does not depend on the details of
the matching points. Next, we discuss the various resonances within the accuracy of our approach.

\vspace{1mm}
\noindent $\mathbf{\rho(770)}$\\
The pole trajectory of $\rho(770)$ moves towards the real axis, similarly to what was
found using U$\chi$PT~\cite{Pelaez06,DLY11,MRP11}.
The modulus of the residue decreases when $N_C$ increases. Such behavior confirms the widely
accepted $\bar{q}q$ structure.

\vspace{1mm}
\noindent $\mathbf{f_2(1270)}$\\
For $f_2(1270)$ our trajectory is quite similar to that of $\rho(770)$.
Again this confirms a $\bar{q}q$ structure.   $\mathcal{O}(p^6)$ $\chi$PT amplitudes
would be required to get a more precise $N_C$-dependence.


\vspace{1mm}
\noindent $\mathbf{\sigma(600)}$\\
For $\sigma(600)$, its mass is $\mathcal{O}(1)$ and its width is $\mathcal{O}(N_C)$.
To get such a wide width it could have a molecular component~\cite{Knecht2013}.
Notice in~\cite{DLY11,DLY-MRP14} the shadow pole in the third sheet suggests a $\bar{q}q$ component.
So the $\sigma$ is likely to be a mixed state including molecule, $\bar{q}q$, etc.
The modulus of the residue increases, reaching the peak at roughly $N_C=3.5$ and then
decreases. It implies that the residue should not only contain $\mathcal{O}(1)$,
but also $\mathcal{O}(N_C^{-1/2})$ or even $\mathcal{O}(N_C^{-1})$.
Such curved behavior of the trajectory is consistent with the mixing
structure of molecule and $\bar{q}q$. The relative strengths of these components
can not be inferred from the analysis presented here.

\vspace{1mm}
\noindent $\mathbf{f_0(980)}$\\
For $f_0(980)$, the pole moves rapidly to the real axis, slightly below $\bar{K}K$ threshold.
It is similar to that of $\rho(770)$ and $f_2(1270)$, implying an $\bar{s}s$ component.
In contrast, in Refs~\cite{Uehara04,DLY11} the pole moves to the real axis
above $\bar{K}K$ threshold and goes onto the fourth Riemann sheet.
We may need higher order $1/N_C$ corrections, especially that caused by kaon loops,
to obtain a more accurate pole trajectory.
The residue behaves like that of the $\sigma(600)$, it increases at first and
then decreases, as a \lq curve',
implying that it must be a combination of $N_C^{-1/2}$ and $N_C^{-1}$, etc.
Note it is most likely to be $\overline{K}K$ molecular in other analysis~\cite{MRP10,DLY11,DLY-MRP14}.
Our findings support the idea that the $f_0(980)$ is a mixture of $\overline{K}K$ molecular and
$\bar{s}s$ components. At present, we can not quantify the relative strengths of these
components.



In this letter we present a new method to study the large $N_C$ behaviour of resonances.
The $\pi\pi$ scattering amplitude with FSI is constructed in a model-independent way.
We match it with the amplitude of $\chi$PT in the low-energy region, which gives the
$N_C$-dependence of the coefficients and phase in the dispersive approach. This is a
reliable way to study the $N_C$-dependence with final-state interactions.
We obtain the trajectory of poles and residues as $N_c$ changes.
Those for both the $\rho(770)$ and $f_2(1270)$ support, as expected, the standard
$\bar{q}q$ structure. In contrast, the $N_C$ trajectory of the light scalar mesons,
$\sigma$ and $f_0(980)$, are consistent with each being a combination of
$\bar{q}q$ and multi-hadron (molecular) components. We stress that some of these
conclusions might be modified when higher order corrections in the $\chi$PT amplitudes
are accounted for.






\begin{acknowledgments}
We are very grateful to M.R. Pennington for many valuable suggestions and discussions
to improve the paper. Special thanks to J. Ruiz de Elvira for his thoughtful and
critical reading of the manuscript. Helpful discussions with Z.H. Guo  are also acknowledged.
This work is supported by the DFG (SFB/TR 110, ``Symmetries and the Emergence of
Structure in QCD'') and by the Chinese Academy of Sciences (CAS) President's
International Fellowship Initiative (PIFI) (Grant No. 2017VMA0025)
\end{acknowledgments}









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