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Coupled Channel model for $\bar{K}N$ scattering in the Resonant Region

The full model, fitting procedure, and results are detailed in [Fer15a] . We report here only the main features of the model.

Observables and definition of partial waves

The differential cross section and polarization observable for the processes $\bar{K}N, \pi \Sigma, \ldots \to \bar{K}N,\pi\Sigma, \ldots$ are given by \begin{alignat}{2} \frac{d\sigma}{d\Omega} (s,\theta)=& \frac{1}{q^2} \left[ |f(s,\theta)|^2 + |g(s,\theta)|^2 \right] , \\ P(s,\theta) =& \frac{2 \: \text{Im} \left[ f (s,\theta)\: g^*(s,\theta) \right] }{|f(s,\theta)|^2 + |g(s,\theta)|^2} , \end{alignat} where $q$ is the center of mass momentum of the incoming kaon, $\theta$ is the scattering angle in the center of mass frame. The amplitudes $f(s,\theta)$ and $g(s,\theta)$ give the contribution from no spin-flip and spin-flip, respectively.

Specifically, in this work we consider the following cases which have been measured (dropping the $s$ and $\theta$ dependence) \begin{alignat}{5} f^{K^-p\to K^-p} &=& &\frac{1}{2}f_{\bar{K}N\to\bar{K}N}^{1} +\frac{1}{2} f_{\bar{K}N\to\bar{K}N}^{0} ,&\\ f^{K^-p\to\bar{K}^0n} &=& &\frac{1}{2}f_{\bar{K}N\to\bar{K}N}^{1} - \frac{1}{2} f_{\bar{K}N\to\bar{K}N}^{0} ,&\\ f^{K^-p\to\pi^- \Sigma^+} &=&-&\frac{1}{2}f_{\bar{K}N\to\pi\Sigma}^{1}- \frac{1}{\sqrt{6}}f_{\bar{K}N\to\pi\Sigma}^{0} ,& \\ f^{K^-p\to\pi^+ \Sigma^-} &=& &\frac{1}{2}f_{\bar{K}N\to\pi\Sigma}^{1} - \frac{1}{\sqrt{6}}f_{\bar{K}N\to\pi\Sigma}^{0}, & \\ f^{K^-p\to\pi^0 \Sigma^0} &=& &\frac{1}{\sqrt{6}}f_{\bar{K}N\to\pi\Sigma}^{0},& \\ f^{K^-p\to\pi^0 \Lambda} &=& &\frac{1}{\sqrt{2}}f_{\bar{K}N\to\pi \Lambda}^{1},& \end{alignat} and similarly for $g(s,\theta)$.

These amplitudes are related to the $s$-channel isospin $I=0$ and $I=1$ amplitudes through a general relation \begin{alignat}{1} f(s,\theta) = \alpha^0\: f^0_{kj}(s,\theta) + \alpha^1\: f^1_{kj}(s,\theta), \\ g(s,\theta) = \alpha^0\: g^0_{kj}(s,\theta) + \alpha^1\: g^1_{kj}(s,\theta), \end{alignat} where $f^I_{kj}(s,\theta)$ and $g^I_{kj}(s,\theta)$ are the isospin amplitudes. Here $\alpha^0$ and $\alpha^1$ are the corresponding Clebsch-Gordan coefficients for isospin zero and one, respectively, and $kj$ label the initial ($k$) and final ($j$) state, respectively.

Partial wave expansion of isospin amplitudes is given by \begin{alignat}{2} f^I_{kj}(s,\theta) =& \sum_{\ell=0}^{\infty} \left[ (\ell+1) R^{I,kj}_{\ell + }(s) + \ell R^{I,kj}_{\ell-}(s) \right] P_\ell \left( \theta\right), \\ g^I_{kj}(s,\theta)=& \sum_{\ell=1}^{\infty} \left[ R^{I,kj}_{\ell + }(s) - R^{I,kj}_{\ell-}(s) \right] P^1_\ell \left( \theta \right), \end{alignat} where $P_\ell \left( \theta \right)$ is the Legendre polynomial with $P^1_\ell \left( \theta \right)= \sin \theta d P_\ell \left( \theta \right) /d \cos \theta $, $R^{I,kj}_{\ell\tau}(s)$ ($\tau = \pm$) are the partial waves which are to be considered as $kj$ elements of the channel-space matrix $R_{\ell \tau}(s)$ as defined below, $\ell$ is the orbital angular momentum of the partial wave and $J=\ell + \tau/2$ is the total angular momentum for $R^{I,kj}_{\ell \tau}(s)$. The orbital angular momentum $\ell$ coincides with the orbital angular momentum of the initial $\bar{K}N$ state in $R^{I,kj}_{\ell \tau}(s)$ but it is not necessarily the orbital angular momentum of other possible initial states. For example, for the $I=1, \ell=0$ partial wave it is possible to have $\bar{K} \Delta(1232)$ in a $D$ wave state ($L=2$) as initial (final) state.

Finally, the total cross section can be expressed in terms of the partial waves \begin{equation} \begin{split} \sigma (s) = \frac{4\pi}{q^2} \sum_{\ell=0}^{\infty} \left[ (\ell+1) |R_{\ell +}(s)|^2 + \ell \: |R_{\ell -}(s)|^2\right], \end{split} \end{equation} where $R_{\ell \tau}(s)=\alpha^0 R^{0,kj}_{\ell \tau}(s) + \alpha^1 R^{1,kj}_{\ell \tau}(s)$.

Partial wave scattering matrix

For a given partial wave we write the scattering amplitude as a matrix in the channel-space \begin{equation} S_\ell=\mathbb{I}+2iR_\ell(s)=\mathbb{I}+2i \left[C_\ell (s) \right]^{1/2} T_\ell(s) \left[C_\ell (s) \right]^{1/2}, \end{equation} where $\mathbb{I}$ is the identity matrix, $C_\ell (s)$ is a diagonal matrix which accounts for the phase space and $T_\ell(s)$ is the analytical partial wave amplitude matrix. We write $T_\ell(s)$ in terms of a $K$ matrix to ensure unitarity \begin{equation} T_\ell (s)= \left[ K(s)^{-1} -i \rho(s,\ell) \: \right]^{-1}. \end{equation}

For real $s$, $K(s)$ is a real symmetric matrix and $\rho(s,\ell)$ is a diagonal matrix. To ensure that $\rho(s,\ell )$ is free from kinematical cuts and has only the square-root branch point demanded by unitarity, we write it as a dispersive integral over the phase space matrix $C_\ell (s)$, a.k.a. as the Chew-Mandelstam representation, \begin{equation} i \rho (s,\ell) =\frac{s-s_k}{\pi}\int_{s_k}^\infty\frac{ C_\ell (s') }{s'-s} \frac{ds'}{s'-s_k}. \end{equation}

Here $s_k$ is the threshold center of mass energy squared of the corresponding channel $k$ and we define \begin{equation} C_\ell (s) = \frac{q_k (s)}{q_0}\left[ \frac{r^2q^2_k(s)}{1+ r^2q^2_k(s) }\right]^{\ell}. \end{equation}

The first factor on the r.h.s is related to the breakup momentum near threshold. For a meson-baryon pair with masses $m_1$ and $m_2$, respectively, $s_k = (m_1 + m_2)^2$ , and \begin{equation} q_k(s) = \frac{\sqrt{(s - (m_1+m_2)^2)(s - (m_1 - m_2)^2)}}{2\sqrt{s}} \simeq \frac{\sqrt{m_1 m_2}}{(m_1+m_2)} \sqrt{ s-s_k }. \end{equation}

The remaining factor ensures the threshold behavior and introduces the effective interaction radius, $r=1 \: \text{fm}$. Finally, $q_0=2 \: \text{GeV}$ is a normalization factor for the momentum in the resonance region. Evaluation of the dispersive integral can be found in [Fer15a] ,

Construction of the $K(s)$ matrix

We define the $K(s)$ matrix as the addition of $K_a(s)$ matrices \begin{equation} \left[ K(s) \right]_{kj} = \sum_a x^a_k\:K_a(s)\: x^a_j \:, \end{equation} where $K_a(s)$ can be of two kinds, pole and background: \begin{alignat}{2} \left[ K_P (s)\right]_{kj} = x^P_k \: \frac{M_P}{M_P^2-s}\: x^P_j,\\ \left[ K_B (s)\right]_{kj} = x^B_k \: \frac{M_B}{M_B^2+s}\: x^B_j, \end{alignat} Each partial wave employs a different amount of pole and background $K$ matrices as well as a different amount of $n_C$ channels. This information is summarized in Table I of Ref. [Fer15a].

The $K(s)$ and $T(s)$ matrices are connected through \begin{equation} \left[ T(s) \right]_{kj} = \frac{1}{\mathcal{D}(s)} \sum_{a,b} x^a_k \: c_{ab} (s)\: x^b_j \:, \end{equation} where $\mathcal{D}(s)$ and $c_{ab}(s)$ for the combination of up to six $K$ matrices can be found in the Appendix in Ref. [Fer15a].

References

[Fer15a]
C. Fernandez-Ramirez, et al., ''Coupled-Channel Model for $\bar{K}N$ Scattering in the Resonant Region,'' arXiv:1510.07065 [hep-ph]