Test of semi-local duality in a large $N_C$ framework

In this paper we test the semi-local duality based on the method of Ref.[1] for calculating final-state interactions at varying number of colors ($N_C$). We compute the amplitudes by dispersion relations that respect analyticity and coupled channel unitarity, as well as accurately describing experiment. The $N_C$ dependence of the $\pi\pi\to\pi\pi$ scattering amplitudes is obtained by comparing these amplitudes to the one of chiral perturbation theory. The semi-local duality is investigated by varying $N_C$. Our results show that the semi-local duality is not violated when $N_C$ is large. At large $N_C$, the contributions of the $f_2(1270)$, the $f_0(980)$ and the $f_0(1370)$ cancel that of the $\rho(770)$ in the finite energy sum rules, while the $f_0(500)$ has almost no effect. This gives further credit to the method developed in Ref.[1] for investigating the $N_C$ dependence of hadron-hadron scattering with final-state interactions. This study is also helpful to understand the structure of the scalar mesons.


I. INTRODUCTION
The 1=N C expansion [1,2] provides an effective diagnostic to differentiate the ordinary from the nonordinary quark-antiquark structure of the mysterious scalars; see e.g., [3][4][5][6][7]. In the physical world, i.e., at N C ¼ 3, there should be local duality [8][9][10][11][12][13]. This means that Regge exchange in the crossed channel is dual to the contribution of resonances in the direct channel. One thus only needs to add either the Regge term or the direct channel resonances in a given calculation. An explicit model shows that there is no interference between these two contributions [8]. Indeed, in the high-energy region the overlap of the resonances is much stronger, leading to a smooth amplitude. Such a smooth amplitude is similar to the one generated by Regge poles in the t-channel. The cross section is therefore more readily described by the Regge exchange in the crossed channel rather than by lots of resonances in the direct (s-) channel. However, in the real world, things are more complicated as the widths of the resonances are finite and only semilocal duality is fulfilled [6]. Through finite-energy sum rules (FESR) the equivalence between the resonances in the direct s-channel and the Regge poles in the crossed t-channel holds on the average [11,12].
In the pioneering work of [6], the semilocal duality is tested in the large N C case and it is shown to be useful for investigating the structure of the light scalar mesons. The scattering amplitudes are obtained by unitarized chiral perturbation theory (UχPT) and the N C dependence of the pertinent low-energy constants is taken over to the amplitudes. The FESR are tested by tuning N C up to 30 or 100. They found that the f 0 ð500Þ (often also called the σ) should contain a subdominantqq component and this ensures that the semilocal duality is fulfilled up to N C ¼ 15-30. This was later used to constrain the meson-meson scattering amplitudes calculated within Uð3Þ unitary χPT [14]. The semilocal duality could be fulfilled very well up to N C ¼ 30. The relation between local duality and exotic states is also discussed in Ref. [15].
On the other hand, final-state interactions (FSI) play an important role in hadron phenomenology, especially when the energy is not very far away from the threshold of a pair or triplet of hadrons. For different models to describe the FSI, see e.g., [16][17][18][19][20][21][22][23][24][25][26][27][28]. In our earlier paper, a new method to study the large N C behavior of the FSI was proposed [29].
The N C dependence is generated based on the fact that the tangent of the phase is proportional to 1=N C , that is, tan φ ∼ Oð1=N C Þ, where ReT ∼ Oð1=N C Þ and ImT ∼ Oð1=N 2 C Þ are naturally given by chiral perturbation theory (χPT). The trajectories of the widths of the ρ and the f 2 quantitatively behave as 1=N C , which confirms the reliability of the method. Following that work, a natural extension is to check whether the semilocal duality is satisfied using this method. This paper is organized as follows: In Sec. II we use a dispersive method to obtain the I ¼ 2 ππ scattering partial waves up to s ∼ 4 GeV 2 , which were not considered in [29]. The amplitudes are constructed analytically and respect the coupled channel unitarity and give a good description of the experimental data. In Sec. III we introduce the N C dependence into the dispersive amplitudes following Ref. [29]. The semilocal duality is tested by tuning N C up to 180. We find that it works well when N C is large. The contributions of each resonance that appear in the amplitudes are also studied. Finally we give a brief summary in Sec. IV.

II. SCATTERING AMPLITUDES AND N C DEPENDENCE
In Ref. [29], the ππ scattering amplitudes with IJ ¼ 00, 11, 02 (with I=J the total isospin/angular momentum) have already been given. Here, we focus on the isospin-2 waves with IJ ¼ 20, 22 to complete the analysis. All these waves are certainly needed for testing the semilocal duality. We use (for more details on the method, see [29]) with z I J to be either the Adler zero for the S-wave or 4M 2 π for the D-wave. Similarly, n J is 1 for the S-wave and 2 for the D-wave. The fitted parameters α i are given in Table I. The units of the α k are chosen to guarantee the amplitude T I J ðsÞ to be dimensionless. The fit amplitudes are shown in Fig. 1 for the energy region of s ∈ ½0; 4 GeV 2 . What we fit to are the following contributions: χPT amplitudes for ½0; 4M 2 π [36][37][38][39], amplitudes of the Roy-type equation analysis at ½4M 2 π ; 2 GeV 2 [34], and experiment up to 4 GeV 2 [33]. We also plot the amplitudes in the region of s ∈ ½−4M 2 π ; 0. Here the real part of our amplitudes is in good agreement with that of χPT [Oðp 4 Þ], and the imaginary part vanishes, which is consistent with the imaginary part of the χPT amplitudes as the latter is rather small. These indicate the high quality of the fit.
As is well known, the Roy-type equation analysis embodies crossing symmetry, 1 which is lacking in Eq. (1). Therefore, following [41], we fit our amplitudes to the "data" on the real axis as well as the amplitudes given by the Roy-like equation [34] in the complex s-plane. As shown in Fig. 2, the two T 2 0 amplitudes are compatible with each other except for the region where s is too large (either Re½s > 1.0 GeV 2 or Im½s < −0.3 GeV 2 ). We note that the amplitudes on the upper half of the s-plane are readily obtainable from the ones on the lower side according to the Schwarz reflection principle. The distribution of contours is in good agreement and moreover, their gradient variations are compatible with each other, as shown by the shading of the color from blue to red. Nevertheless, above 1.0 GeVour amplitudes are a bit different from that of the dispersive analysis, while both of them are compatible with the data; see Fig. 1. Also our amplitudes in the bottom-right direction, where either Re½s or Im½s is large, are becoming less consistent with differences ≤0.1.
Now that these amplitudes are obtained for the physical world, that is for N C ¼ 3, we can introduce the N C Notice that the D-wave is absent in the Roy-like equation analysis [34] and the I ¼ 2 D-wave is very small; we thus do not discuss it here. For higher partial waves we refer to Ref. [40]. dependence. Apparently, the real part of the χPT amplitude is OðN −1 C Þ and the imaginary part is OðN −2 C Þ up to any order. Therefore, we generate the N C dependence as [29] φðs; N C Þ ¼ arctan 3 and P I J ðs; N C Þ ¼ It is not difficult to check that at large N C the phase, which would return back to the phase shift in a single channel, will jump by π around s ¼ M 2 R where φðM 2 R Þ ¼ 90°. This is consistent with the large N C property of a simple Breit-Wigner formalism or the "narrow resonance pole" on the second Riemann sheet [42]. All the complicated higherorder N C dependence is ignored for simplicity. By increasing N C , the magnitude of the I ¼ 2 S-wave and D-wave become smaller and smaller, as shown in Fig. 1. This is consistent with the fact that there is no resonance in the I ¼ 2 channel.

III. SEMILOCAL DUALITY
We follow Ref. [6] to calculate the variation of the FESR by tuning N C . It is well known that the t-channel ππ scattering amplitudes could be written in terms of the s-channel ones according to the crossing relations where I denotes the total isospin (I ¼ 0, 1, 2) of the ππ-system and C st is the orthogonal crossing matrix The s-channel amplitude is composed of a complete set of partial waves with θ s the s-channel scattering angle in the center of mass frame. Of course, I s þ J should be even as required by Bose symmetry and isospin conservation. Higher partial waves are less known and we restrict our amplitudes up to the D-waves. We introduce the function  [36][37][38][39]. The cyan and green bands are from CFDIV [34]. The OPE and OPE-DP data are from [33]. Right column: The absolute values of the amplitudes by varying N C are shown. The black solid, orange dashed, blue dotted and magenta dashed-dotted lines are for N C ¼ 3, 6, 12, 120, respectively.
To test duality in the large N C limit, we first estimate it at The upper and lower limits of the integration are chosen as The R I n ðtÞ of our amplitudes and that of the Regge amplitudes are given in Table II. As can be seen, our calculation with the D-waves is compatible with that of Regge exchange [6] within the uncertainties. For more discussions about the Regge analysis, we refer to [6]. 2 The difference between the Regge and our amplitudes as well as the difference between our two results (with or without D-waves) are much more obvious at n ¼ 0 than those at n ¼ 3. This tells us that the D-and even higher partial waves cannot be ignored at small n. In contrast, for large n the low-energy amplitudes will dominate the integration and the contribution of resonances could be less important. We thus pay attention to n ¼ 1-3 only and include the D-waves in the next sections. As a support, the R I n ðtÞ of the Regge analysis and ours (with D-waves) are closest to each other for n ¼ 2, 3. Also we find that the F-waves have tiny contributions only. It is instructive to plot each F I n ðν; tÞ amplitude for different values of N C ; see Fig. 3. Notice that the peaks around ffiffi ffi ν p ¼ 0.85, 1.25 GeV at N C ¼ 120 are caused by the IJ ¼ 00 wave; cf. Fig. 1 of Ref. [29]. They are related to the f 0 ð980Þ and the f 0 ð1370Þ in the large N C limit, respectively. From Fig. 3 one notices that when n is large and N C is not too large, the low energy amplitudes (including the σ), and the ρ have much larger contributions, while the f 2 ð1270Þ contributes most at n ¼ 0. Only in the I t ¼ 2 amplitude is the contribution of the ρ negative, which will cancel other contributions such as those from the σ in the low-energy region and from the f 2 ð1270Þ in the high-energy region. This makes sure that F I t ¼2 n ðν; tÞ is superconvergent, being much smaller than the corresponding function for I t ¼ 0 or I t ¼ 1. Note that resonances/ Regge exchanges are built from qq and multiquark contributions. When N C is large, the pole of theqq state will fall down to the real axis on the s-plane (zero width), while the multiquark component will disappear. 3 Consequently, TABLE II. Comparison of the R I t n ðtÞ ratios between our amplitudes and that of Regge exchange. The latter is given by [6]. S,P,D represent our work up to the D-waves, and S,P represent our work with only S-and P-waves.
2 It is worth noting that in [6] the scattering lengths of IJ ¼ 11, 02 waves are calculated in the Regge parametrization with n ¼ 2, 3. They are in perfect agreement with that obtained by the dispersive analysis. This certainly confirms the semilocal duality at N C ¼ 3, especially when n ¼ 2, 3. In Ref. [43] the nonlinear Regge trajectory of the f 0 ð500Þ is obtained and this supports its nonordinary nature. 3 Nowadays it is believed that tetraquarks could be as narrow as the conventionalqq resonances [44] or even narrower [45]; however, these will not change our conclusion as we do not have any tetraquarks in the I ¼ 2 amplitude. the I s=t ¼ 2 amplitude is superconvergent at large N C as it does not contain any resonances or Regge poles. Also the ratio of the I t ¼ 1 FESR compared to that of I t ¼ 0 should be 2=3 in the large N C limit. These are analyzed in [6] within UχPT and we will check them with our method to generate the N C dependence in what follows.
Further, it is helpful to use the definition As discussed before, semilocal duality means that F 10 n ðtÞ should be 2=3, and F 21;20 n ðtÞ should be rather small. The values of these FESR for different values of N C are given in Table III. For convenience, we plot F 10 n ðtÞ and F 21 n ðtÞ by tuning N C up to N C ¼ 180; see Fig. 4.
Following [29], we simply assume that the whole contribution of the N −2 C corrections is roughly one third of that of N −1 C , while the correlation between each polynomial is not discussed, despite the fact that the first two terms of the polynomials are fixed by the scattering lengths and slope parameters. In principal, the complete separated N −2 C dependence of each polynomial in Eq. (5) could be obtained by matching with χPT, if χPT is calculated up to higher orders. However, this is not yet available. One certainly needs a more careful analysis of the N −2 C corrections. 4 These higher-order N C corrections contribute most to the uncertainties at large N C , estimated by randomly choosing 2=N C þ 3=N 2 C and/or 4=N C − 3=N 2 C to replace 3=N C in Eqs. (4) and (5) for each partial wave. The other contributions to the uncertainties are the higher partial TABLE III. Duality by tuning N C , given here for N C ¼ 3, 6, 12, 120.  4 We note that Ref. [46] points out that the subleading corrections of the low-energy constants (LECs) may be sizable as L i =N C , which is consistent with our assumptions. In Refs. [3,47], the uncertainty caused by the regularization scale μ is discussed, which is not required here. waves, for instance the IJ ¼ 13 wave, whose amplitude has been given in [34], and the systematic uncertainties from different solutions of the scattering amplitudes. The combination of all three parts is collected as the total uncertainty; see Table III for N C ¼ 3, 6, 12, 120. The uncertainties are also shown as error bars in Fig. 4. We note that the uncertainty of the FESR increases as N C increases. See e.g., F 10;21 2 ð4M 2 π Þ in Table III, with the upper limit ν max ¼ 2 GeV 2 . This is because the uncertainty given by higher-order N C corrections is important. Besides, the uncertainty of the F 21 n ðtÞ is larger than that of F 10 n ðtÞ. The reason is that the cancellation happens at I t ¼ 2 and they are more sensitive to the relative difference between partial waves. The uncertainty coming from the upper limit ν max ¼ 1 GeV 2 is smaller than that of ν max ¼ 2 GeV 2 ; this is caused by the important D-waves. We will discuss this later.
The results are quite different when the upper limit is chosen to be 1 or 2 GeV 2 , especially in terms of n ¼ 0, 1. As an example, for F 21 0;1 ðtÞ the signs of the results with these two upper limits are even opposite. For n ¼ 2, 3, the differences are still distinct but smaller. This is consistent with our analysis that the IJ ¼ 02 wave contributes a lot in the resonance region at small n (0 or 1). In Fig. 3, the peaks of the f 2 ð1270Þ in F I 0 ðν; tÞ are much larger than those in F I 2 ðν; tÞ. While at large n (n ¼ 2, 3), the contribution of the IJ ¼ 02 wave in the resonance region is still important but smaller. Therefore, we consider the upper limit of 2 GeV 2 as the optimal choice.
The change of the results with different n of F 10 n ðtÞ is smaller than that of F 21 n ðtÞ in the large N C case. For example, with upper limit ν max ¼ 2 GeV 2 and N C ¼ 120, the difference between F 10 0 ð4M 2 π Þ and F 10 3 ð4M 2 π Þ is 0.22, while the difference between F 21 0 ð4M 2 π Þ and F 21 3 ð4M 2 π Þ is 0.93. The reason is that the contribution from the ρ dominates both amplitudes in I t ¼ 0 and I t ¼ 1 below 1 GeV, and the relative signs between different resonances [ρ, f 2 ð1270Þ etc.] are positive, while in I t ¼ 2 the relative sign is negative; see also Eq. (6). This can be checked in Fig. 3, by comparing the lines of 2 ffiffi ffi ν At large N C , none of the absolute values of F 21 n ðtÞ is larger than 0.6, and all F 10 n ðtÞ are distributed in the region [0.6,0.8]. For n ¼ 2, with the upper limit 2 GeV 2 and N C ¼ 180, both F 10 2 ð4M 2 π Þ ¼ 0.64 AE 0.04 and F 21 2 ð4M 2 π Þ ¼ −0.14 AE 0.16 are very close to the expected value, 2=3 and 0, respectively. Similarly, we have F 10 2 ð0Þ ¼ 0.65 AE 0.03 and F 21 2 ð0Þ ¼ −0.08 AE 0.15, even a bit closer. We note that the two kinds of results, with t ¼ 0 or t ¼ 4M 2 π , are rather similar to each other. We thus only discuss the case with t ¼ 4M 2 π in the next sections. For n ¼ 1, 3 the situation is not so good, but the values are still fairly close to the expected values. For n ¼ 1 we have F 10 1 ð4M 2 π Þ ¼ 0.72 AE 0.04 and F 21 1 ð4M 2 π Þ ¼ 0.22 AE 0.15, and for n ¼ 3 we find F 10 3 ð4M 2 π Þ ¼ 0.59 AE 0.04 and F 21 3 ð4M 2 π Þ ¼ −0.38 AE 0.17. The F 21 n ð0; 4M 2 π Þ at large N C indicates that the f 2 ð1270Þ [also the f 0 ð980Þ and the f 0 ð1370Þ] will cancel the contribution of the ρð770Þ most at n ¼ 2. By increasing/decreasing n the cancellation is less precise, as the masses of these resonances are different. Divided by ν n , the contributions of the ρ and that of the f 2 ð1270Þ=f 0 ð980Þ are mismatched, especially around ν ¼ M 2 R . As discussed in the earlier sections, n ¼ 1-3, especially n ¼ 2, 3, are the most valuable cases to check the semilocal duality; one thus concludes that the results support that the semilocal duality is fulfilled well up to N C ¼ 180. There are some other points that could be interesting. Almost all the lines in Fig. 4 are increased/ decreased a bit strongly from N C ¼ 3 to N C ¼ 30. One of the reasons is that the physical amplitudes are not as simple as that just represented by one or two Breit-Wigner resonances. Other components, such as multiquark components and other background, will also contribute a lot when N C is not far away from 3. Such variation of the lines of F 21 n¼2;3 ðtÞ is more obvious than that of F 10 n¼2;3 ðtÞ, where the latter is within 1=3 level. This is because the former has a strong cancellation between isospin 0 and 1 waves in the s-channel, as shown in Eqs. (6) and (7). The complex N C relations of scalars enlarge the variation. We also notice that the lines are very flat in the region of N C ∈ ½100; 180. It is thus natural to infer that they will stay flat for larger N C . This suggests that the semilocal duality will hold in the large N C limit.
To estimate the contribution of each resonance at large N C , we perform the following calculations:  Table IV. For Cases A, D and E, the results by tuning N C are plotted as magenta, black, and cyan lines, respectively, in Fig. 5. For Cases B and C, we cannot extract the contributions of the relevant resonances except at large N C , as there is no obvious peak for f 0 ð980Þ and f 0 ð1370Þ around N C ¼ 3. Therefore, we only show the results at N C ¼ 180; see Table IV.
For case A, we have F 10 2 ð4M 2 π Þ ¼ 0.63 AE 0.07 and F 21 2 ð4M 2 π Þ ¼ −0.15 AE 0.20 at N C ¼ 180. These satisfy the semilocal duality well. At N C ¼ 3, we have F 21 2 ð4M 2 π Þ ¼ −0.36 AE 0.12. It changes a lot compared to the original result; cf. Table III. It confirms that the σ contribution is not ignorable at N C ¼ 3, while it is smaller and irrelevant in the large N C limit. The cancellation does not happen between the σ and ρð770Þ in the large N C limit. This also suggests that σ is dominated by the non-qq structure components.
It is worth pointing out that in Ref. [14], where UχPT based on the N/D method has been applied to the mesonmeson scattering of the Uð3Þ nonet, the f 0 ð980Þ contribution cannot be ignored to cancel the contribution of ρ. In contrast, in Ref. [6], where UχPT is realized by an inverse amplitude method, the f 0 ð980Þ is irrelevant and a subdominantqq component is needed for the cancellation. For the f 0 ð1370Þ, both of these two works agree as the resonance does not contribute a lot at large N C . In our case, the two resonances behave as "peaks" around 0.85 and 1.23 GeV at large N C , respectively. This supports their possible innerqq component, resulting in a possible contribution to cancel the ρ in F 21 n ðtÞ. We find numerically for case B, where the contribution of the f 0 ð980Þ has been removed at large N C , that F 10 2 ð4M 2 π Þ ¼ 0.59 AE 0.08 and F 21 2 ð4M 2 π Þ ¼ −0.36 AE 0.20 at N C ¼ 180. Compared to the original results (case O), the value of F 10 2 ð0; 4M 2 π Þ changes a bit, and that of F 21 2 ð0; 4M 2 π Þ changes a lot. Thus, the contribution of the f 0 ð980Þ at large N C cannot be ignored. For case C, by removing the f 0 ð1370Þ at large N C , the change is smaller than that of case B but still distinct; see Table IV. This implies that the f 0 ð980Þ and the f 0 ð1370Þ have a significantqq component and they will partly cancel the contribution of the ρ.
In case D the f 2 ð1270Þ has been removed. One finds F 10 2 ð4M 2 π Þ ¼ 0.59 AE 0.03 and F 21 2 ð4M 2 π Þ ¼ −0.37 AE 0.15 at N C ¼ 180. The f 2 ð1270Þ contribution is rather important to cancel that of the ρ, just as expected. The result of case D is rather close to that of the f 0 ð980Þ at large N C , implying the same important contribution of the f 0 ð980Þ as the f 2 ð1270Þ. In case E we consider the FESR with the upper limit ν max ¼ 4 GeV 2 . The result at N C ¼ 180 is almost the same as that of case O. This supports the view that the ρð1450Þ as well as other heavier resonances do not have a large effect. In Fig. 5 one clearly sees that in case D the F 10 2 ðtÞ and F 10 2 ðtÞ deviate much more from the expected values than those in cases A and E. This confirms that the f 2 ð1270Þ has a much larger effect than the f 0 ð500Þ and heavier resonances such as the ρð1450Þ.

IV. SUMMARY
In this paper we have studied the semilocal duality for large N C . The isospin-2 ππ scattering amplitudes with finalstate interactions are constructed in a model-independent way and fit to the data. Comparing with the amplitudes of χPT, we generate the N C dependence of the amplitudes. With this N C dependence the semilocal duality in terms of finite energy sum rules is tested. Our results show that the semilocal duality is satisfied well in the large N C limit, at least up to N C ¼ 180. This study confirms that the method of generating the N C dependence proposed in Ref. [29] is reliable. At large N C , the contributions of the f 2 ð1270Þ and the f 0 ð980Þ are important to cancel that of the ρð770Þ, the latter is consistent with what has been found in Ref. [14]. Also the f 0 ð1370Þ contributes significantly to the cancellation. In contrast, the f 0 ð500Þ (or σ) has a large effect for N C ¼ 3 and a small effect at large N C . These support theqq component in the f 0 ð980Þ and the f 0 ð1370Þ, but not for the σ as required in Ref. [6].