Structure of the $\Lambda(1405)$ and the $K^{-}d\rightarrow\pi\Sigma n$ reaction

We begin by constructing the three-body amplitudes relevant to the $K^{-}d\rightarrow\pi\Sigma n$ reaction. Throughout this paper it is assumed that the three-body processes take place via separable two-body interactions given by the following forms in the two-body center-of-mass (c.m.) frame:

where $g_{\alpha}^{(I)}(\bm{q}_{i})$ is a vertex (cutoff) factor of the two-body channel $\alpha$ with relative momentum $\bm{q}_{i}$ and isospin $I$. The interaction matrix $\lambda_{\alpha\beta}^{(I)}(E)$ is a function of the total energy $E$ in the two-body system. In the three-body system, we define the two-body energy as $E=\sqrt{(W-E_{i}(\bm{p}_{i}))^{2}-p_{i}^{2}}$ with the three-body energy $W$ and the spectator particle energy $E_{i}(\bm{p}_{i})$, where $\bm{p}_{i}$ is the relative momentum of the spectator particle $i$. The explicit forms of the relevant two-body interactions are presented in detail in Sec. III.

The ansatz (1) specifies strongly interacting two-body subsystems in the three-body processes. We refer to these meson-baryon or dibaryon subsystems conveniently as “isobars”. The three-body dynamics can then be described as quasi-two-body scattering of an isobar and a spectator particle in all possible coupled isobar-spectator channels. The quasi-two-body amplitudes, $X_{\alpha\beta}^{(I)(I^{\prime})}(\bm{p}_{i},\bm{p}_{j};W)$, are determined by solving the AGS equations Amado (1963); Alt et al. (1967),

Here, $\alpha$ and $\beta$ denote two-particle subsystems forming “isobars” with isospins $I$ and $I^{\prime}$, respectively; the subscripts $i,j,n$ represent the spectator particles which include, respectively, $N$, $\Sigma$, $\Lambda$, $\bar{K}$, or $\pi$. The notations for the isobars are summarized in Table 1. As pointed out in Sec. III, in this work we include only the ${}^{3}S_{1}$ partial wave for the $NN$ interaction, thus only the isospin $I=0$ state appears for the $NN$ subsystem (the isobar denoted $d$). For later purposes the partial wave projections of the amplitudes $X$ of Eq. (2) are needed. They are given as:

with $\cos\theta=\hat{\bm{p}}_{i}\cdot\hat{\bm{p}}_{j}$ and the notation $\hat{\bm{p}}\equiv\bm{p}/|\bm{p}|$. Here, $P_{L}$ is the Legendre polynomial with orbital angular momentum $L$ between isobar and spectator particle. After the partial wave projections the AGS equations (2) are written as:

The driving term $Z_{\alpha\beta}^{(I)(I^{\prime})}({\bm{p}}_{i},{\bm{p}}_{j};W)$ describes a particle-exchange interaction process connecting two-body channels, $\beta\rightarrow\alpha$, and corresponding spectators as illustrated in Fig. 1(a). It is given by:

where $E_{i}(\bm{p}_{i})$ and $E_{j}(\bm{p}_{j})$ are the energies of the spectator particles $i$ and $j$, respectively; $E_{k}(\bm{p}_{k})$ with $\bm{p}_{k}=-\bm{p}_{i}-\bm{p}_{j}$ is the energy of the exchanged particle $k$; $\bm{q}_{i}$ ($\bm{q}_{j}$) is the relative momentum between the exchange-particle and the spectator-particle $j$ ($i$). Using relativistic kinematics we have $E_{n}(\bm{p}_{n})=\sqrt{m_{n}^{2}+\bm{p}_{n}^{2}}$ ($n=i,j,k$) and $q_{i}=|\bm{q}_{i}|$ is defined as

The isobar amplitudes, $\tau_{\alpha\beta}^{(I)}\left(W-E_{i}(\bm{p}_{i}),\bm{p}_{i}\right)$ as illustrated in Fig. 1 (b), are determined by solving the Lippmann-Schwinger equations with the two-body interaction (1),

Here, $E_{jk}({\bm{p}}_{i},{\bm{q}}_{i})$ is the energy of the interacting pair ($jk$), $E_{jk}({\bm{p}}_{i},{\bm{q}}_{i})=\sqrt{M_{jk}^{2}(\bm{q}_{i})+\bm{p}_{i}^{2}}$.

After antisymmetrization of the two-nucleon states in the three-body system, the coupled-channels AGS matrix integral equations (4) are written formally and symbolically (suppressing sums, integrals and all indices other than the isobar assignments) as:

In this subsection, following Ref. Ohnishi et al. (2013), we present formulas for computing cross sections of the two-body-to-three-body reaction, $K^{-}d\rightarrow\pi\Sigma n$. By using the anti-symmetrized AGS amplitudes (9), the breakup amplitudes for $K^{-}d\rightarrow\pi\Sigma n$ are given as

Here $R_{d}$ is the residue of the two-body $NN$ propagator, $\tau_{dd}^{(I=0)}$ at the deuteron pole, with its proper binding energy, i.e., $\sqrt{R_{d}}$ normalizes the initial state deuteron wave function. Note again that all two-body subsystems listed in Table 1, including both hyperons $Y=\Sigma$ and $\Lambda$, contribute to $T_{K^{-}d\rightarrow\pi\Sigma n}$ when permitted by selection rules. The following notations are used for the expressions appearing in Eq. (10):

$|ABC\rangle$: plane wave state of the three-body system;

$|[[A\otimes B]_{\alpha}\otimes C]_{\Gamma};I,L\rangle$: three-body system in the $LS$ coupling scheme, with $\alpha$, $\Gamma$, $I$, and $L$ being the isobar quantum number, the total quantum number, the isospin of the isobar and its angular momentum relative to the spectator, respectively.

The projection $\langle ABC|[[A\otimes B]_{\alpha}\otimes C]_{\Gamma};I,L\rangle$ involves the product of spherical harmonics and spin-isospin Clebsch-Gordan coefficients. The $T$ matrix calculated in the isospin basis is then decomposed into the $\pi^{+}\Sigma^{-}n$, $\pi^{0}\Sigma^{0}n$, and $\pi^{-}\Sigma^{+}n$ final states using isospin CG coefficients. The momenta $\bm{p}_{\pi}$ and $\bm{p}_{\Sigma}$ are related to the momenta $\bm{p}_{N}$ and $\bm{q}_{N}$ by a Lorentz boost

With the $T$ matrix Eq. (10) the cross sections of interest are derived as

with the initial relative velocity $v=\frac{W}{E_{d}E_{\bar{K}}}p_{\bar{K}}$, the $\pi\Sigma$ invariant/missing mass $M_{\pi\Sigma}=E_{\pi}(\bm{q}_{N})+E_{\Sigma}(\bm{q}_{N})=\sqrt{(W-E_{N}(\bm{p}_{N}))^{2}-\bm{p}_{N}^{2}}$, and the $K^{-}d$ total energy $W=E_{\bar{K}}(\bm{p}_{\bar{K}})+E_{d}(\bm{p}_{\bar{K}})=\sqrt{m_{\bar{K}}^{2}+\bm{p}_{\bar{K}}^{2}}+\sqrt{m_{d}^{2}+\bm{p}_{\bar{K}}^{2}}$. In the second line of Eq. (13), the momenta $p_{N}$ and $q_{N}$ are the on-shell momenta for given energies $W$ and $M_{\pi\Sigma}$. Angular integrations are denoted by $\int d\hat{\bm{p}}\equiv\int d\cos\theta_{p}\,d\phi_{p}$ . The differential cross sections are

where

The symbol $\sum_{\bar{i}f}$ stands as usual for averaging of initial states and sum of final states subject to conservation laws.

Two models for the $s$-wave meson-baryon interactions used in Refs. Ikeda and Sato (2007, 2009); Ikeda et al. (2010); Ohnishi et al. (2013) are employed in this work. Both are derived from the leading order chiral Lagrangian (the Weinberg-Tomozawa term Weinberg (1966); Tomozawa (1966)) but have different off-shell behavior. One of them is referred to as the energy dependent (E-dep.) model Ikeda et al. (2010),

Here, $\omega_{\alpha}=\sqrt{q_{\alpha}^{2}+m_{\alpha}^{2}}$ is the meson energy of the channel $\alpha=\bar{K}N$, $\pi\Sigma$, $\pi\Lambda$; $m_{\alpha}$ $(M_{\alpha})$ is the meson (baryon) mass; $f_{\pi}=92.4$ MeV is the pion decay constant; the coupling coefficients $C^{I}_{\alpha\beta}$ are determined by the flavor SU(3) structure constant (see Ref. Ohnishi et al. (2013)). The vertex factors $g^{(I)}_{\alpha}(q_{\alpha})$ are chosen as dipole form factors with cutoff scales $\Lambda^{(I)}_{\alpha}$,

The characteristic energy dependence of the meson-baryon interaction (17) is dictated by spontaneously broken chiral $SU(3)_{L}\times SU(3)_{R}$ symmetry. The corresponding Nambu-Goldstone bosons are identified with the pseudoscalar meson octet, and their leading $s$-wave couplings involve the time derivatives of the meson fields.

The other model, referred to here as the energy independent (E-indep.) model Ikeda and Sato (2007, 2009), is obtained by fixing the two-body energy at each threshold energy, $2E=m_{\alpha}+M_{\alpha}+m_{\beta}+M_{\beta}$:

While this restricted model with constant couplings is not consistent with Goldstone’s theorem for low-energy pseudoscalar meson interactions, it is nonetheless a prototype of phenomenological potentials that have been used in the literature, and so we discuss it here for comparison with the energy-dependent approach based on chiral $SU(3)_{L}\times SU(3)_{R}$ meson-baryon effective field theory.

The cutoff parameters for the $\bar{K}N$-$\pi\Sigma$-$\pi\Lambda$ systems are determined by fitting the $K^{-}p$ scattering cross sections Humphrey and Ross (1962); Sakitt et al. (1965); Kim (1965); Kittel et al. (1966); Evans et al. (1983). Results of the fit for the E-dep. and E-indep. models are presented in Fig. 2. The fitted cutoff values are listed in Table 2.

The different off-shell behaviors of the two types of models leads to different analytic structure of the $\bar{K}N$ amplitudes. We find that the E-dep. model has two poles on the $\bar{K}N$ physical and $\pi\Sigma$ unphysical sheet. The behavior of the subthreshold amplitudes is similar to that obtained with the chiral SU(3) dynamics Ikeda et al. (2011, 2012) (see Fig. 3), and the scattering length is consistent with SIDDHARTA measurement in the E-dep. model. On the other hand, the E-indep. model has a single pole corresponding to $\Lambda(1405)$. It shares this property with other phenomenological potential models. The behavior below $\bar{K}N$ threshold of the amplitudes in the E-indep. model is very different compared with that obtained from chiral SU(3) dynamics (Fig. 3). In the E-indep. model, it is difficult to reproduce the $K^{-}p$ scattering length in comparison with SIDDHARTA measurements although the cross sections are reproduced within experimental errors.

Table 3 lists the pole energies of the $\bar{K}N$ $s$-wave scattering amplitudes in the complex energy plane between the $\bar{K}N$ and $\pi\Sigma$ threshold energies and the $K^{-}p$ scattering length. The primary purpose of this study is to clarify the influence of the subthreshold behavior of the $\bar{K}N$ interaction in the $\pi\Sigma$ spectrum.

As for the cutoff parameters of $\pi N$ interactions, we have determined them by fitting the $S_{11}$ and $S_{31}$ $\pi N$ scattering lengths Schroder et al. (1999). The resulting values are $\Lambda^{(I=1/2)}_{N^{*}}=\Lambda^{(I=3/2)}_{N^{*}}=500$ MeV for both the E-dep. and E-indep. models.

Parameters of the two-body potential are the cutoffs $\Lambda^{(I)}_{\alpha}$, determined by fitting the $\bar{K}N$ reaction cross sections within experimental errors. Acceptable variations of these cutoffs are examined for the E-dep. model. The ranges of cutoff scales compatible with experimental errors are listed in Table 4 and the corresponding fits to data are presented in Fig. 4. The resulting $K^{-}p$ scattering length including uncertainties is $a_{K^{-}p}=-(0.72^{+0.06}_{-0.12})+i~(0.77^{+0.19}_{-0.15})$ fm, consistent with the scattering length deduced from the SIDDHARTA kaonic hydrogen measurements.

As seen in Fig. 4 one might have the impression that the $K^{-}p\rightarrow\pi^{-}\Sigma^{+}$ cross section is not optimally reproduced. On the other hand, this is a relatively small cross section with limited weight in the overall fitting procedure. By examining the dashed curves in Fig. 4, we have checked that optimizing the fit to this selected cross section does not have a significant influence on the other cross sections within uncertainties.

The following baryon-baryon interactions are commonly used for the E-dep. and E-indep. meson-baryon models. As for the $NN$ interaction in ${}^{3}S_{1}$, we take the following Yamaguchi-type two-term separable form:

Here, $C_{R}$ ($C_{A}$) is the coupling strength of the repulsive (attractive) potential. The form factors $g_{R,A}({q})$ are defined by $g_{R,A}({q})={\Lambda_{R,A}}^{2}/({q}^{2}+{\Lambda_{R,A}}^{2})$, with $\Lambda_{R,A}$ being the cutoff parameters of the $NN$ interactions. The coupling strengths $C_{R,A}$ and the cutoff parameters $\Lambda_{R,A}$ are determined by fitting the ${}^{3}S_{1}$ phase shifts of the Nijmegen 93 model Stoks et al. (1994) (see Fig. 5 for the result of the fit). The resulting values of the parameters are summarized in Table 5. The obtained deuteron binding energy is $2.23$ MeV.

As for the $s$-wave $YN$ interactions, we follow the form given in Ref. Torres et al. (1986):

Here, $\mu_{\alpha}$ is the reduced mass of the $YN$ system; the form factor $g^{(I)}_{\alpha}(q_{\alpha})$ is defined as $g^{(I)}_{\alpha}(q_{\alpha})=\Lambda^{(I)2}_{\alpha}/(q_{\alpha}^{2}+\Lambda^{(I)2}_{\alpha})$. The coupling constants $C^{(I)}_{\alpha\beta}$ and the cutoff parameters $\Lambda^{(I)}_{\alpha}$ are determined by fitting the ${}^{3}S_{1}$ phase shifts of the Jülich’04 model Haidenbauer and Meissner (2005). The resulting values of the coupling constants $C^{(I)}_{\alpha\beta}$ are summarized in Table 6. The cutoff parameters $\Lambda^{(I)}_{\alpha}$ are $\Lambda^{(I=1/2)}_{\Sigma N}=261$ MeV, $\Lambda^{(I=3/2)}_{\Sigma N}=540$ MeV, and $\Lambda^{(I=1/2)}_{\Lambda N}=285$ MeV.