The $\varLambda (1405)$ resonance as a genuine three-quark or molecular state

Abstract

The mechanism for the formation of the $\varLambda (1405)$ resonance is studied in a chiral quark model that includes quark-meson as well as contact (four point) interactions. The negative-parity S-wave scattering amplitudes for strangeness $-1$ and 1 are calculated within a unified coupled-channel framework that includes the KN, $\bar{K}N$, $\pi \varSigma$, $\eta \varLambda$, $K\varXi$, $\pi \varLambda$, and $\eta \varSigma$ channels and possible genuine three-quark bare singlet and octet states corresponding to $\frac{1}{2}^-$ resonances. It is found that in order to reproduce the scattering amplitudes in the $S_{01}$ partial wave it is important to include the pertinent three-quark octet states as well as the singlet state, while the inclusion of the contact term is not mandatory. The Laurent-Pietarinen expansion is used to determine the S-matrix poles. Following their evolution as a function of increasing interaction strength, the mass of the singlet state is strongly reduced due to the attractive self-energy in the $\pi \varSigma$ and $\bar{K}N$ channels; when it drops below the KN threshold, the state acquires a dominant $\bar{K}N$ component which can be identified with a molecular state. The attraction between the kaon and the nucleon is generated through the $\bar{K}N\varLambda ^*$ interaction rather than by meson-nucleon forces.

Appendices

The Cloudy Bag Model meson-quark vertices and coupling constants

The s-wave quark-meson vertices $\hat{V}(k)$ in Eq. (11) are evaluated in the Cloudy Bag Model assuming that in the resonant state one of the three quarks is excited from the 1s state to the $1p_{1/2}$ state.

Table 4 Reduced $X^*\rightarrow MB$ matrix elements

Table 5 Reduced $B\rightarrow MX^*$ matrix elements

Table 6 Reduced $B\rightarrow MS^*$ matrix elements for non-strange s-wave resonances

Table 7 $g_{\alpha \beta }$ for isospin 0 and 1

For the quark part of the quark-pion, quark-eta meson and quark-kaon interaction we obtain

$$\begin{aligned}&\hat{V}^\pi _{l=0,t}(k) = V^\pi (k)\sum _{i=1}^3 \tau _t(i)\, \mathcal {P}_{sp}(i)\,, \end{aligned}$$

(19)

$$\begin{aligned}&\hat{V}^\eta (k) =V^\eta (k) \sum _{i=1}^3 \lambda _8(i)\,\mathcal {P}_{sp}(i)\,, \end{aligned}$$

(20)

$$\begin{aligned}&\hat{V}^K_{t}(k) = V^K(k) \end{aligned}$$

(21)

Here $\mathcal {P}_{sp} =\sum _{m_j}|sm_j\rangle \langle p_{1/2}m_j|$, $\omega _s=2.043$, $\omega _{p_{1/2}}=3.811$, $X(K^0)= -(\lambda _6- i\lambda _7)/\sqrt{2}$, $X(\bar{K}^0)= -(\lambda _6+i\lambda _7)/\sqrt{2}$, $X(K^+)= -(\lambda _4- i\lambda _5)/\sqrt{2}$, $X(K^-)= (\lambda _4+ i\lambda _5)/\sqrt{2}$. Assuming $f_\eta =f_K=f_\pi$, the form-factors of the surface part and of the volume part take the form

$$\begin{aligned} V^\pi _S(k)= & {} V^\eta _S(k) = V^K_S(k) \nonumber \\= & {} {1\over 2f_\pi }\sqrt{\omega _{p_{1/2}}\omega _s\over (\omega _{p_{1/2}}+1)(\omega _s-1)}\, {1\over 2\pi }\,{k^2\over \sqrt{\omega _k}}\,{j_0(kR)\over kR}\,, \nonumber \\ V_V(k)= & {} \left[ {\omega _s\over R} - {\omega _{p1/2}\over R} + \omega _M(k)\right] \nonumber \\\times & {} \int _0^R dr \,r^2[u_s(r)u_p(r)+v_s(r)v_p(r)]j_0(kr)\,. \end{aligned}$$

(22)

For the physical $\eta$ we assume $\eta = \cos \theta _P\eta _8-\sin \theta _P\eta _1$, for the singlet $\eta _1$ $\lambda _8$ is replaced by $\lambda _1$ in Eq. (20).

The coupling constants for the s-channel exchange potential are collected in Table 4, those for the u-channel exchange potential involving strange baryons in Table 5, and for exchange of nonstrange baryons in Table 6.

Contact interaction

For the s-wave mesons the contact interaction can be cast in the form

$$\begin{aligned} V^c_{\alpha \beta }(k,k')= & {} {g_{\alpha \beta }\over 2f^2_\pi } {kk'\over 2\pi ^2\sqrt{2\omega _\alpha (k)}\sqrt{2\omega _\beta (k')}} \nonumber \\&\times [\omega _\alpha (k) + \omega _\beta (k')] \nonumber \\&\quad \int _0^R dr \,r^2[u_s^2(r) + v_s^2(r)]j_0(kr)j_0(k'r)\,.\nonumber \\ \end{aligned}$$

(23)

Here $g_{\alpha \beta }$ can be identified with $-2f_\pi ^2\lambda ^I_{\alpha \beta }$ of [26] and ${\textstyle {\frac{1}{2}}}\mathcal {D}_{\alpha \beta }$ of [11] and are collected in Table 7.

Oset [11] has an opposite sign for $\bar{K}N\leftrightarrow \pi \varSigma$, which is compensated by changing the sign for $\varSigma ^*\rightarrow \pi \varSigma$.