A new method to study the number of colors in the final-state interactions of hadrons

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.

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 [1][2][3][4]. The phenomenology of these is complicated due the contribution from important final-state interactions (FSI) [5]. Dispersion relations are the natural way to include FSI, see e.g. [6,7]. For some of the light mesons, like the σ, κ, their existence has been confirmed [8][9][10][11] and accurate pole locations and ππ couplings, including also the ρ(770), have been given in Refs. [12][13][14]. Concerning the nature of the scalar mesons, there is a cornucopia of models [15][16][17][18][19][20][21][22][23][24][25][26][27][28]. 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 [29][30][31][32][33]. However, these analysis based on unitarized Chiral Perturbation theory (UχPT) 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 KK threshold, close to which the f 0 (980) is located. To solve this problem, we use an Omnès representation based on the phase of the relevant amplitudes, rather than the elastic phase shift [34,35]. There has been renewed interest in the study of the large N C limit [36,37] of the properties of resonances [38][39][40]. Weinberg [41] pointed out that resonant tetraquark states could exist due to the contribution of the leading order (LO) 'connected' diagrams to the Green functions. Their widths are O(N −1 C ), as narrow as ordinary mesons. They could be even narrower, with width of O(N −2 C ), when the flavor of the quarks is combined in different ways [42]. There are many other interesting discussions such as [43,44] and references therein. In this paper we focus on establishing a '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 ππ 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 (χPT) up to O(p 4 ). By expanding the amplitudes in the low-energy region, we match the dispersive and the χPT 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 ρ(770), f 2 (1270), σ(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 andqq components (for a recent review on hadronic molecules, see Ref. [45].
We first present our IJ = 00, 02, 11 partial waves of ππ → ππ calculated in a model-independent way. We start from: where Ω I J (s) is the Omnès function [46]: with ϕ I Jλ (s) the phase of the partial wave amplitude T I J (s), which has been given in previous amplitude analysis [34,35]. 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 where z I J is the Adler zero for the S-waves and 4M 2 π for P-and D-waves. The parameter n J is 1 for S-and Pwaves and 2 for D-waves. The parameters α i are given in Table I  shown in Fig. 1. We fit the amplitudes in the region of s ∈ [0, 4GeV 2 ], where the 'data' is as follows: χPT amplitudes at [0, 4M 2 π ], amplitudes of K-matrix and Roylike equation at [4M 2 π , 2GeV 2 ], and experiment data up to 4GeV 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 'prediction' region where s ∈ [−4M 2 π , 0]. From these amplitudes, we can extract the poles and residues on the second sheet.
The residue g f ππ and pole s R on the second Riemann sheet are defined as: The pole locations and their residues are listed in Table II. These are very similar from those of previous analyses [12,14,35,55,56]. For the f 0 (1370) and f 2 (1270), to find the pole closest to the physical sheet one needs to include the ππ, KK, 4π as coupled channels. However, notice that a Breit-Wigner resonance will always have shadow poles on other sheets. Although ππ 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) χPT, 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 'hadron loop' corrections, C . Third, in the χPT amplitudes the r.h.c is given entirely from one-loop integrals (the B functions) in the O(p 4 ) amplitudes. This is why we do not consider a matching with only LO χPT amplitudes. The imaginary part of the χPT amplitudes is N −2 C , while the real part of the amplitudes is of N −1 C . From this one sees that the phase of the amplitudes that enter the Omnès functions should be N −1 C . Indeed, any higher order N Cdependence of the phases such as N −2 C can be ignored, which will contribute finally at O(N −3 C ) to the amplitudes T I J (s) in Eq (1). Fourth, we choose the matching points to be the Adler zero for the S-waves and 2M 2 π 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 χPT amplitudes we expand each one. For the dispersive amplitude of Eq (1) we first expand the function P I J (s) as: where z I J is given in Eq. (3), while s m I J is the matching point. One readily finds where C k−1 l−1 are the binomial coefficients. With this transformation one can translate the parameters α I J l in Table I with , We note that ϕ I J (s) is proportional to N −1 C , and find Ω I J (s) = 1 + higher order loop corrections, these are at least N −2 C or even 'weaker'. As a result we simply have Finally we obtain the N C -dependence of the coefficients in Eq. (5). For the S-wave, we have For the P-and D-waves, we find The LECs and their N C -dependence in χPT are given by [49,50]. We also use the relation L 2 ∼ 2L 1 which is derived from the matching of χP T with RχT [57]. With these LECs and the input M π = 0.13957 GeV, M K = 0.496 GeV, M η = 0.54785 GeV, F π = 0.924 GeV and its N C dependence up to N −1 C [? ] we finally get: 0S (N C ) = −5.8358 1P (N C ) = 0.4062 1P (N C ) = 0.6126 0D (N C ) = −0.0864 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. 1 This is because in our model a N −1 C factor appears in the Omnès 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 UχPT. We only match to O(p 4 ) and lack more accurate N C -dependence, with which the peak of |T | will behave as 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 2 R ), where the N C dependence is canceled leading to T ≃ i/ρ. However, our pole trajectories suggest it has no obvious influence on the amplitudes locate elsewhere. The zero is not there and |T | is still O(N −1 C ). The trajectories of the pole locations and their residues g f ππ on the second Riemann sheet are obtained from Eq. (4), and plotted in Fig. 2 Adler zero for the S-waves and 2M 2 π for P-and D-waves. The step of ∆NC is 0.01 (just for illustration, of course, NC 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 2 π , 2M 2 π , 3M 2 π , 5M 2 π , 6M 2 π , 7M 2 π for the S-wave, and sa 0 S , M 2 π , 3M 2 π , 5M 2 π , 6M 2 π , 7M 2 π for the P-and D-waves, respectively. Note that for the matching points above threshold we have added i0.001 to avoid the singularity.
determine the uncertainties of our trajectories, we choose different matching points in (0, 8M 2 π ) and present the results in Fig. 2. 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.

ρ(770)
The pole trajectory of ρ(770) moves towards the real axis, similarly to what was found using UχPT [30,32,33]. The modulus of the residue decreases when N C increases. Such behavior confirms the widely acceptedqq structure. f 2 (1270) For f 2 (1270) our trajectory is quite similar to that of ρ(770). Again this confirms aqq structure. O(p 6 ) χPT amplitudes would be required to get a more precise N Cdependence.

σ(600)
For σ(600), its mass is O(1) and its width is O(N C ). To get such a wide width it could have a molecular component [42]. Notice in [33,35] the shadow pole in the third sheet suggests aqq component. So the σ is likely to be a mixed state including molecule,qq, 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 O(1), but also O(N −1/2 C ) or even O(N −1 C ). Such curved behavior of the trajectory is consistent with the mixing structure of molecule andqq. The relative strengths of these components can not be inferred from the analysis presented here.
f 0 (980) For f 0 (980), the pole moves rapidly to the real axis, slightly belowKK threshold. It is similar to that of ρ(770) and f 2 (1270), implying anss component. In contrast, in Refs [33,59] the pole moves to the real axis abovē KK 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 σ(600), it increases at first and then decreases, as a 'curve', implying that it must be a combination of N −1/2 C and N −1 C , etc. Note it is most likely to be KK molecular in other analysis [2,33,35]. Our findings support the idea that the f 0 (980) is a mixture of KK molecular andss 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 ππ scattering amplitude with FSI is constructed in a model-independent way. We match it with the amplitude of χPT in the lowenergy 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 finalstate interactions. We obtain the trajectory of poles and residues as N c changes. Those for both the ρ(770) and f 2 (1270) support, as expected, the standardqq structure. In contrast, the N C trajectory of the light scalar mesons, σ and f 0 (980), are consistent with each being a combination ofqq and multi-hadron (molecular) components. We stress that some of these conclusions might be modified when higher order corrections in the χPT amplitudes are accounted for.
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)