,1,?, Xian-Wei Kang,2,?, Tao Luo,3,§, Ulf-G. Meiβner,4,5,? Corresponding authors: ? E-mail:dailingyun@hnu.edu.cn;? E-mail:kangxianwei1@gmail.com;§ E-mail:luot@fudan.edu.cn;? E-mail:meissner@hiskp.uni-bonn.de
Received:2019-05-31Online:2019-11-1
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Ling-Yun Dai, Xian-Wei Kang, Tao Luo, Ulf-G. Meiβner. A Study on the Correlation Between Poles and Cuts in ππ Scattering*. [J], 2019, 71(11): 1309-1318 doi:10.1088/0253-6102/71/11/1309
1 Introduction
In a two-body scattering system, for example two hadrons, the general principals that we know are unitarity, analyticity, crossing, the discrete symmetries, etc. The resonances that appear as the intermediate states in such system are important. Among them the lightest scalar mesons, related to $\pi\pi$ scattering, have the same quantum numbers as the QCD vacuum and are rather interesting, for some early references, see Refs. [1--3]. The $\pi\pi$ scattering amplitude is also crucial to clarify the hadronic contribution to the anomalous magnetic moment of the muon, see e.g. Refs. [4--5]. To study the resonances in a given scattering process, one needs dispersion relations to continue the amplitude from the real $s$-axis (the physical region) to the complex-$s$ plane (see e.g., Refs. [6--9]) where the pole locations and their couplings are extracted. Following this method, some work on the light scalars can be found in Refs. [10--15], where the accurate pole locations and residues of the $\sigma$ and $\kappa$ mesons are given.For the dispersive methods, a key problem is how to determine the left hand cut (l.h.c.) and the right hand cut (r.h.c.), with the unitarity kept at the same time. In Refs. [7--8] the l.h.c. is estimated by crossed-channel exchange of resonances, where chiral effective field theory ($\chi$EFT) is used to calculate the amplitude. And the contribution of r.h.c. is represented by an Omnés function, with unitarity kept. In the well-known Roy equations, crossing symmetry and analyticity are perfectly combined together as the l.h.c. is represented by the unitary cuts of the partial waves. The single channel unitarity is also well imposed by keeping the real part of the partial wave amplitudes the same as what is calculated by the phase shift directly, which could be obtained by fitting to the experimental data in some analyses. Until now, Roy and Roy-Steiner equations certainly give the most accurate description of the two-body scattering amplitude and the information of resonances appearing as the intermediate states, such as $\pi\pi$, $\pi K$ scattering and the pole locations and residues of the $\rho$, $\sigma$, $f_0(980)$ and $\kappa$, etc., see e.g. Refs. [13--15]. In addition, Ref. [14] shows that the l.h.c. can not be ignored for the determination of the pole location of the $\sigma$. By removing the parabola term of the l.h.c., the $\sigma$ pole location is changed by about 15% accordingly, while the unitarity is violated due to the removal of the l.h.c. And thus the method to get the poles on the second Riemann sheet, calculated from the zeros of the S-matrix, is not reliable any more, as the method is based on the continuation implemented by unitarity. Here, we focus on obtaining a quantitative relation between cuts and poles, with unitarity imposed and the l.h.c. and r.h.c. are correlated with each other.
This paper is organized as follows: In Sec. 2 we establish a dispersive method based on the phase. In the physical region we also represent the amplitudes by an Omnés function of the phase above threshold. In Sec. 3 we fit the $\pi\pi$ scattering amplitudes up to 1 GeV in a model-independent way, including the $IJ=00,02,11,20$ waves, where $I$ denotes the total isospin and $J$ the angular momentum. The fit results are the same as those given by the Omnés function representation and comparable with those of chiral perturbation theory ($\chi$PT) in the low-energy region. The poles and couplings are also extracted. In Sec. 4 we give the estimation of the relation between poles and cuts, including both the l.h.c. and the r.h.c. We end with a brief summary.
2 Scattering Amplitude Formalism
2.1 A Dispersive Representation
The two-body partial wave scattering amplitude can be written as:with $\varphi(s)$ the phase and $f(s)$ a real function. By writing a dispersion relation for $\ln T(s)$, one has:
Here, $s_0$ is chosen at a specific point where the amplitude is real, and "$L$" denotes the l.h.c. and "$R$" stands for the r.h.c. The amplitude turns into
On the other hand, unitarity is a general principal required for the scattering amplitude. In the single channel case one has
where $s$ is in the elastic region and $\rho(s)$ is the phase space factor. Substituting Eq. (3) into Eq. (4), we obtain a representation (in the elastic region) for a single channel scattering amplitude
Also, the Omnés function of the phase[16] for the l.h.c. is correlated with that of the r.h.c.
which is again valid in the elastic region. A simple way to get the two-body scattering amplitude proceeds in two steps: First, we follow Eq. (5) to fit the Omnés function of the r.h.c. to experimental data, and then use Eq. (6) and other constraints below the threshold to fit the Omnés function of the l.h.c. Note that Eq. (5) does not only work for the single channel case, but also for the coupled channel case in the physical region.
2.2 On $\pi\pi$ Scattering
In the equations above, the threshold factor is not included. Considering such factors, we need to change the amplitudes into:Here and in what follows, we take $\pi\pi$ scatering as an example. Thus one has $z^I_{ J}=4 M_\pi^2$ for the P-, D-, and higher partial waves, and $z^I_{ J}$ is the Adler zero for the S-waves. $n_{J}$ is one for S- and P-waves and two for D waves. We define a reduced amplitude
and again we can write a dispersion relation for $\ln\tilde{T}^I_J(s)$, so that we have
Here, $s_0$ could be chosen from the range $[0,4 M_\pi^2]$. For the r.h.c., we cut off the integration somewhere in the high energy region, see discussions in the next sections. We have
The $T^I_J(s_0)$ could be fixed by $\chi$PT or scattering lengths, or other low-energy constraints. For simplicity, we choose $s_0=0$. Combining unitarity, embodied by Eq. (5), we have a correlation between Omnés functions of l.h.c. and r.h.c. in the elastic region
This is similar to Eq. (6). Substituting Eq. (11) into Eq. (10), we still have Eq. (5). Since we know the $\pi\pi$ scattering amplitudes well in the region [$4M_\pi^2$, 2 GeV$^2$] and $\chi$PT describes the amplitudes well in the low-energy region, we have to fit the l.h.c. to both Eq. (6) and $\chi$PT.
3 Phenomenology
3.1 $\pi\pi$ Scattering Amplitudes
For the $\pi\pi$ scattering amplitude, we can parameterize the phase caused by the l.h.c. by a conformal mappingwith
Notice that ${\rm Im} \omega(s)$ behaves as $\sqrt{-s^3}$ around $s=0$, which is consistent with that of $\chi$PT, see Ref. [12] and references therein.
As concerns the r.h.c., it is less known in the high energy region. However, these distant r.h.c. should have less important effects in the low-energy region, especially in the region $s\leq 1$ GeV$^2$. We choose three kinds of $\Omega^{IJ}_R(s)$ to test the stability and uncertainty caused by the distant r.h.c. In Case A, the phases are cut off at $s=2.25$ GeV$^2$.[7] In Case B, the phases are given by Ref. [7], up to $s=22$ GeV$^2$. In Case C, the phases/Omnés functions of the r.h.c. are given by Refs. [17--19] and references therein, up to $s=22$ GeV$^2$. Here, the phases are fitted to the experimental data in Refs. [20--21] up to $\sqrt{s}=2$ GeV and constrained by unitarity up to $\sqrt{s}=4$ GeV. Notice that in Case A and B the phase of the isospin-one P-wave is given by CFDIV in Ref. [22], and we continue it to the higher energy region by means of the function
with
The function (and also its first derivative) is smooth at the point $s_{R}$. We set $k=1$, $n=2$, $s_{R}=1.4^2$ GeV$^2$ and $\varphi^{11}_{\infty}=160^\circ$, which is close to $\varphi^{11}_R((1.4 \text{GeV})^2)=170^\circ$ and ensures that the phase in the high energy region behaves smoothly. The upper limits of the integration of the r.h.c. of isospin-one P-wave are the same as the other partial waves. The three kinds of Omnes functions have different magnitudes. For example, in isospin zero S-wave the difference between Cases A/B and C is 17/6 percent at $s=0.5$ GeV$^2$. And that is 2/17 percent for isospin one P-wave.
The parameters of our fits for all the Cases are given in Table 1. The $c^{IJ}_n$ are determined by the following procedure. In the elastic region, we choose one or two "mesh points", equaling to the number of the coefficients $c^{IJ}_n$. Combining Eqs. (11) and (12), we can build a matrix and solve for $c^{IJ}_n$. We adjust the "mesh points" and $s^{IJ}_L$ to make the solutions to be consistent with that of Eq. (5) in the physical region. For instance, we choose one "mesh point" as $s_1=0.7$ GeV$^2$ in Case A, and two points as $s_1=0.2$ GeV$^2$ and $s_2=0.6$ GeV$^2$ in Cases B and C. Other partial waves are treated similarly. Note that we do not choose more mesh points, as the correlation between each terms of dispersion relation will become stronger and we will have larger number of $c^{IJ}_n$, resulting in more twisted amplitudes. This strategy gives a good description of the amplitudes, with unitarity kept. See the fit results shown in Fig. 1. Here all the partial waves refer to Case B, in which the phase is cut off at $s=2.25$ GeV$^2$. The amplitudes from the other Cases are quite close to this one, except for the inelastic region and the distant l.h.c. ($\tilde{E}\leq-0.4$ GeV$^2$). Our fit, both the real part (black solid line) and imaginary (black dotted line) part of the amplitudes shown in Fig. 1, is indistinguishable from that given by the K-Matrix[7] or CFDIV.[22] Note that the amplitudes given by Eq. (5) are exactly the same as those of the K-Matrix or CFDIV from $\pi\pi$ threshold to the inelastic threshold. This implies that the unitarity is respected. To test it quantitatively, we define
Table 1
Table 1The parameters for each fit. "-" means the absence of the corresponding quantity. For comparison, we also give the one-loop $\chi$PT results.
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Fig. 1
New window|Download| PPT slideFig. 1(Color online) Fit of the $\pi\pi$ scattering amplitudes for Case B. Notice that $\tilde{E}=\rm{sgn}(s)\sqrt{s}$. The solid lines denote the real part of the amplitudes and the dotted, dashed, dash-dotted and dash-dot-dotted lines denote the imaginary part. The black lines are from our fit. The red lines are from a K-matrix fit[7] for the isospin-zero S-wave, and the violet lines are from CFDIV[22] for other waves. The borders of the cyan and green bands in the low-energy region are from SU(2) and SU(3) $\chi$PT, respectively. The CERN-Munich data are from Ref. [20], and the OPE and OPE-DP data are from Ref. [21].
$\Delta T^I_J(s_n)$ is the difference between our amplitude and that of Eq. (5). We use this equation to refine our solutions, that is, the $s^{IJ}_L$ and "mesh points" are selected out to make $\mathcal{R}_{ T^I_J}$ small. Here we choose $s_n=(0.1$--0.9) GeV$^2$ for the S-waves, $s_n=(0.1$--0.8) GeV$^2$ for the P-wave, and $s_n=(0.1$-1.0) GeV$^2$ for the D-wave, with step of 0.1 GeV$^2$. These points are located between the $\pi\pi$ and the inelastic thresholds. From here on all the steps are chosen to be 0.1 GeV$^2$ (or 0.1 GeV for $\tilde{E}$). We find that $\mathcal{R}_{ T^0_S}=0.1%$, $\mathcal{R}_{ T^1_P}=0.1%$, $\mathcal{R}_{ T^2_S}=1.4%$, and $\mathcal{R}_{ T^0_D}=1.4%$. The violation of unitarity is rather small at elastic region. Notice that if we remove out the Omnes function of the right hand phase, only the l.h.c. part (times $T(0)$) remains and it is a smooth real function above $s>0$. Since the contribution of the l.h.c. is determined rather well in the elastic region, it would be natrual to keep working well not faraway from the inelastic threshld. This is why the amplitudes could be described well up to 1.2 GeV for S- and P-waves. These are enough for us to discuss the poles and cuts of $\sigma$, $\rho$ and $f_0(980)$.
For $T^I_J(0)$, $\chi$PT could be used to fix it. The analytical SU(3) 1-loop $\chi$PT amplitudes of each partial waves, are recalculated and given in Appendix A. The low-energy constants are given by Ref. [23]. Those of SU(2) 2-loop $\chi$PT amplitudes are given by Refs. [24--25] and references therein. All the values of $T^I_J(0)$ in Table 1 are chosen to be close to the prediction of $\chi$PT or our earlier analyses,[17-18] as well as to minimize $\mathcal{R}_{ T^I_J}$. In the isospin-zero S-wave, the magnitude of our $T^0_0(0)$ is a bit larger than that of $\chi$PT. This is consistent with what is known about this scattering length, where the one-loop $\chi$PT calculation gives a smaller result than what is obtained by dispersive methods, Roy equations or in experiment, see e.g. the review.[23] A better comparison would be given with the 2-loop $\chi$PT amplitudes.
In the isospin-zero D-wave, the $T^0_2(0)$ varies more in the different Cases. The reason is that some fine-tuning is needed as the inelastic r.h.c. is difficult to be implemented well. The amplitudes given by Eq. (5) are much different from that of CFDIV in the inelastic region where the $f_2(1270)$ appears. Notice further that the value of $T^0_D(0)$ is very small, one order smaller than that of the other waves.
3.2 Pole Locations and Couplings
With these amplitudes given by a dispersion relation, the information of the poles can be extracted. It is worth pointing out that the main goal of this section is to check the reliability of our representation, that is, whether our model is good on extracting the poles and residues in the (deep) complex plane or not. The pole $s_{R}$ and its coupling/residue $g_{f\pi\pi}$ on the second Riemann sheet are defined asNote that the continuation of the $T(s)$ amplitude to the second Riemann sheet is based on unitarity,
The poles and couplings/residues for Cases A, B, C are given in Table 2.
Table 2
Table 2The pole locations and residues given by our fits. The notation "$2S$ v.s." denotes the virtual state in the isospin-two $S$-wave.
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All the poles and residues of the different Cases are close to each other, and also to those given by Roy[14] or Roy-like[22] equations,?(? Here we stress that we do not intend to extract more accurate pole locations than those of Roy or Roy-like equations. Note we o give an explicit description on the l.h.c., while it is hidden in the Roy or Roy-like equations.) except for the pole location of the $f_2(1270)$. The reason is that the $f_2(1270)$ is located outside the elastic unitary cut of $T^0_D(s)$, while Eq. (11) only works in the elastic region. For this partial wave one needs a more dedicated method to study, including coupled-channel unitarity. For the poles of the $\sigma$, the differences between the different Cases is also a bit larger than those of other resonances such as the $\rho(770)$ and the $f_0(980)$. This is because the $\sigma$ is far away from the real axis. For the virtual state in the isospin-two S-wave, the poles and residues are a bit different from Cases A and B to Case C. This situation is comparable with that of $T^2_S(0)$, where in Cases A and B $T^2_S(0)$ is 0.055 and in Case C it is 0.060, respectively.
In addition, we also find that there exists a virtual state in the isospin-two S-wave very close to $s=0$.?(? It has already been discussed in Ref. [26], within a unitarized $\chi$PT method. Here we use dispersion approach and re-confirm it, but we do not have the extra poles caused by unitarization.) According to Eq. (18), the virtual state at the zero of the S-matrix below the threshold. This zero equals to the intersection point between two lines: $T^2_S(s)$ and $i/2\rho(s)$. As shown in Fig. 2 the line of the $T^2_S(s)$ and the line of $i/2\rho(s)$ will always intersect with each other and the crossing point always lies in the energy region of $[0,s_a]$, where $s_a$ is the Adler zero. This is the virtual state. Since the scattering length is negative and the Adler zero (only one) is below threshold, one would expect that the amplitude of $T^2_S(s)$, from $s=4 M_\pi^2$ to $s=0$, will always cross the real axis of $s$ and arrive at the positive vertical axis. In all events, it will intersect with that of the $i/2\rho(s)$. Thus the existence of the virtual state is confirmed. This inference is model-independent, only the sign of the scattering length,§(§ Recently, Lattice QCD gives negative scattering length as $-0.0412(08)(16)M_\pi^{-1}$,[27] $-0.04430$ $(25)(40)M_\pi^{-1}$.[28] $-0.04430$ $(2)(^{+4}_{-0})M_\pi^{-1}$.[29] These values are consistent with that of $\chi$PT,[23] the Roy equations matched to $\chi$PT[30] and a dispersive analysis.[22]) the Adler zero, and analyticity are relevant.
Fig. 2
New window|Download| PPT slideFig. 2(Color online) The lines of $T^2_S(s)$ for the different Cases and $i/2\rho(s)$. Note that all of them are real. The intersection point corresponds to the virtual state.
For a general discussion of the virtual state arising from a bare discrete state in the quantum mechanical scattering, we recommend readers to read[31-32] and references therein. We suggest that the isospin-two S-wave amplitude could be checked in the future measurement of $\Lambda_c^+\to\Sigma^-\pi^+\pi^+$. Its branching ratio[33] is large enough.
The average values of the poles and residues of all the Cases define our central values. The deviations of the different Cases to the central values are used to estimate the uncertainties. The results are shown in Table 3. These are very similar from those of previous analyses.[7,14-15, 34-35] The $f_2(1270)$ has a much larger uncertainty compared to the other resonances, just as discussed before. The residues of all resonances have roughly similar magnitude at the region [0.25,0.55] GeV, except for that of the virtual state in the isospin-two S-wave, which is much weaker. But their phases are quite different. The phases of $\rho(770)$ and $f_2(1270)$ are close to zero, while those of the $\sigma$ and $f_0(980)$ are close to $-90^\circ$, and the virtual state one is close to $90^\circ$. This may imply that $\rho(770)$ and $f_2(1270)$ are normal $\bar{q}q$ states but that the $\sigma$ and $f_0(980)$ have large molecular components.
Table 3
Table 3The pole locations and residues by taking the averages between Cases A, B and C as discussed in the text.
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3.3 The Correlation Between Poles and Cuts
It is interesting to find the correlation between the poles and cuts. We focus here on the isospin-zero S-wave and isospin-one P-wave, as the $f_2(1270)$ is far away from the l.h.c. and the virtual state is too close to the l.h.c. Also, the light scalars are more difficult to understand. All the fits of different Cases about these two partial waves are shown in Fig. 3. In our approach only unitarity is used to constrain the amplitudes, but the low-energy amplitudes are consistent with those of $\chi$PT. Only in Cases B and for the isospin-zero S-wave, the $T^0_S(s)$ amplitude is inconsistent with that of $\chi$PT at $\tilde{E}<-0.4$ GeV. This implies that unitarity has a strong constraint on the low-energy amplitudes below $\tilde{E}=0$. This could also be simply checked by using Eq. (16), with $\tilde{E}=-0.1$ GeV to $-0.4$ GeV and $s_n=-\tilde{E}_n^2$. Here, $\Delta T^I_J(s_n)$ is the difference between our amplitude and that of SU(2) $\chi$PT. Typically, in Case B, $\mathcal{R}^L_{ T^0_S}=21%$, $\mathcal{R}^L_{ T^1_P}=16%$, and they are quite close to those of other Cases. Note that the real parts of the amplitudes are more consistent with those of $\chi$PT, while the imaginary parts have a bit larger deviation (as is expected as imaginary parts start later in the chiral expansion).Fig. 3
New window|Download| PPT slideFig. 3(Color online) Comparison of different solutions of the $\pi\pi$ scattering amplitudes. The solid lines are the real part of the amplitudes and other lines are the imaginary part. The green lines are from Case A, the cyan lines are from Case B, and the black lines are from Case C. The red lines are from K-Matrix[7] for isospin 0 S-wave, and the violet lines are from CFDIV[22] for isospin 1 P-wave. Note that the lines of K-matrix and/or CDFIV are overlapped with our fits in the elastic region, or even a bit further in the inelastic region.
To see the variation of the l.h.c. in the different solutions, we apply Eq. (19) on ${\rm Im}T^I_J(s_n)$, with $\tilde{E}=-0.1$ GeV to $-0.6$ GeV. Notice that at $\tilde{E}=0$ all of the l.h.c. are zero and behave as $\sqrt{-s^3}$, this partly ensures the l.h.c. to be consistent with that of $\chi$PT in the low-energy region. We fix the average value of all Cases as the central value, and calculate the relative deviation for each point. At last we avarage these relative deviations to estimate the variation of the cuts. The variation of cuts and poles are defined as
with $s_p$ the pole on the second Riemann Sheet. Finally, we collect the uncertainties in Table 4. And we define the correlation between poles and cuts as
Table 4
Table 4The correlation between l.h.c. and poles (represented by superscript 'L'), and between r.h.c. and poles (represented by superscript"$R$").
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The simple meaning of the correlation $C_{\rm pole}$ is to answer the following question: When the cut is changed by 100%, how much would the pole location be changed?
To test the correlation between poles and the r.h.c., we simply set $\varphi_{\rm test}(s) = 1.04\varphi(s)$, and check the variation of poles and cuts, respectively. The relative uncertainty of the r.h.c. is also estimated by Eq. (19), with $s_n=(0.1$--0.9) GeV$^2$ for the isospin-zero S-wave and $s_n=(0.1$--0.8) GeV$^2$ for the isospin-one P-wave. The relative uncertainty of the poles and the correlation are calculated in the same way as that of the l.h.c., see Eqs. (19) and (20).
From Table 4, we find that $C_\sigma^R$ is roughly two orders larger than that of $C_\sigma^L$, though $\sigma$ is rather close to the l.h.c. Comparing to Ref. [14], which has roughly 15% contribution from l.h.c., we have a rather smaller contribution from the l.h.c., caused by the constraint of unitarity on the l.h.c. Also, $C_{f_0(980)}^R$ is roughly three orders larger than that of $C_{f_0(980)}^L$, and $C_{\rho(770)}^R$ is roughly one order larger than that of $C_{\rho(770)}^L$. These indicate that the correlation between the unitarity cut and the poles is much larger than that of the l.h.c. and poles. Note that in our case the l.h.c. is not arbitrary but correlated with the r.h.c., constrained by unitarity and analyticity, see Eq. (11). For each Case, $C_\sigma^{L}$ is larger than $C_{f_0(980)}^{L}$. This is not surprising as the $\sigma$ is much closer to the l.h.c. Also, $C_\sigma^{R}$ is larger than $C_{f_0(980)}^{R}$. The reason is that the $\sigma$ is farther away from the real axis, the uncertainty of the pole is larger as the amplitude is continued from the physical region deeper into the complex-$s$ plane. It is interesting to see that in average $C_\sigma^{L}$ is roughly two times larger than $C_{\rho(770)}^L$. And for the distance between these poles and l.h.c. (simply set $s=0$), $|s_\sigma|$ is one half of that of $|s_\rho|$, this tells us that the correlation between poles and l.h.c. is inversely proportional to their distance. In contrast, $C_\sigma^{R}$ is roughly one order larger than $C_{\rho(770)}^R$. For the distance between these poles and r.h.c. (simply set $s={\rm Re} s_{\rm pole}$), $|{\rm Im} s_\sigma|$ is two times larger than $|{\rm Im} s_\rho|$, this tells us that the correlation between poles and r.h.c. is proportional to their distance. These conclusions are still kept when comparing the $\sigma$ and the $f_0(980)$. The discussion of the correlation between poles and cuts reveals that our model, keeping analyticity and unitarity, give strong constrains on poles and l.h.c., even if the pole is a bit faraway from the real axis.
4 Summary
We proposed a dispersive method to calculate the two-body scattering amplitude. It is based on the Omnés function of the phase, including that of the left hand cut and the right hand cut. The input of the r.h.c. is given by three kinds of parametrizations, and the l.h.c. is solved by Eq. (11), with unitarity and analyticity respected. The pion-pion $IJ=00,02,11,20$ waves are fitted within our method and the poles and locations are extracted. They are stable except for that of the $f_2(1270)$, which lies in the inelastic region. The r.h.c. has much larger contribution to the poles comparing to that of the l.h.c. This method could be useful for the studies of strong interactions in two-body scattering. For instance, it could be applied in the $\pi\eta$ scattering, where the correlation between l.h.c. and r.h.c. could help us to constrain the phase shift. Also the $\pi\pi$ scattering amplitudes obtained here could be used for the future studies when one has $\pi\pi$ final state interactions, see e.g. Refs. [36--40], and/or to multi-pions, see e.g. Refs. [41--42].Appendix A: Analytical Amplitudes of Partial Waves Within $\chi$PT
The analytical 1-loop amplitudes of these partial waves within of SU(3) $\chi$PT are recalculated. For reader's convenience, they are given below.We have the IJ=00 waves up to $\mathcal{O}(p^4)$:Here the superscript of $T$ in the bracket means the chiral order, and subscripts represent for Isospin and spin, respectively. Note that for reader's convenience we also give the analytical forms of the imaginary part (r.h.c.) of the amplitudes.The $I=2$ S wave is
The $I=1$ P wave is
And the $I=0$ D wave is
It should be noted that in all these partial waves, $2L_4+L_5$ and $2L_6+L_8$ appear together.[43] The $A$, $B$, $H$ functions are given as below
with $\rho(t,m)=\sqrt{1-4 m^2/t}$. Notice that our amplitudes are calculated in the formalism of $\overline{MS}$, while that of Refs. [44--47] is done in $\overline{MS}-1$. The relation between our LECs ($L_i$) and that of the latter one ($\tilde{L}_i$) is $\tilde{L}_i=L_i+{\Gamma_i}/{32\pi^2}$.
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