The pseudoscalar meson and baryon octet interaction with strangeness zero in the unitary coupled-channel approximation

TL;DR

This paper investigates pseudoscalar meson and baryon octet interactions with strangeness zero using a coupled-channel approach, identifying a resonance likely corresponding to N(1535) and highlighting the importance of hidden strange channels.

Contribution

It introduces a detailed coupled-channel analysis including s- and u-channel potentials, revealing a resonance consistent with N(1535) and emphasizing the role of hidden strange channels.

Findings

Identified a resonance near the N(1535) in the complex energy plane.

Resonance couples strongly to hidden strange channels.

No resonance generated in the isospin 3/2 sector.

Abstract

The interaction of the pseudoscalar meson and the baryon octet is investigated by solving the Bethe-Salpeter equation in the unitary coupled-channel approximation, In addition to the Weinberg-Tomozawa term, the contribution of the $s-$ and $u-$ channel potentials in the S-wave approximation are taken into account. In the sector of isospin $I=1/2$ and strangeness $S=0$, a pole is detected in the reasonable region on the complex energy plane of $\sqrt{s}$ in the center of mass frame by analyzing the behavior of the scattering amplitude, which is higher than the $\eta N$ threshold and lies on the third Riemann sheet. Thus it can be regarded as a resonance state and might correspond to the $N(1535)$ particle in the review of the Particle Data Group(PDG). The coupling constants of this resonance state to the $\pi N$, $\eta N$, $K \Lambda$ and $K \Sigma$ channels are calculated, and it is found that this resonance state couples strongly to the hidden strange channels. Apparently, the hidden strange channels play an important role in the generation of the resonance state with strangeness zero. The interaction of the pseudoscalar meson and the baryon octet is repulsive in the sector of isospin $I=3/2$ and strangeness $S=0$, therefore, no resonance state can be generated dynamically.

Tables

Table 1. Table 1: The coefficients C i ​ j subscript 𝐶 𝑖 𝑗 C_{ij} in the pseudoscalar meson and baryon octet interaction with strangeness S = 0 𝑆 0 S=0 and charge Q = 0 𝑄 0 Q=0 , C j ​ i = C i ​ j subscript 𝐶 𝑗 𝑖 subscript 𝐶 𝑖 𝑗 C_{ji}=C_{ij} .

$C_{ij}$	$K^{+}{\Sigma }^{-}$	$K^0{\Sigma }^0$	$K^0\Lambda$	${\pi }^{-}p$	${\pi }^{0}n$	$\eta n$
$K^{+}{\Sigma }^{-}$	$1$	$-\sqrt{2}$	$0$	$0$	$-\frac{1}{\sqrt{2}}$	$-\sqrt{\frac{3}{2}}$
$K^0{\Sigma }^0$		$0$	$0$	$-\frac{1}{\sqrt{2}}$	$-\frac{1}{2}$	$\frac{\sqrt{3}}{2}$
$K^0\Lambda$			$0$	$-\sqrt{\frac{3}{2}}$	$\frac{\sqrt{3}}{2}$	$-\frac{3}{2}$
${\pi }^{-}p$				$1$	$-\sqrt{2}$	$0$
${\pi }^{0}n$					$0$	$0$
$\eta n$						$0$

Table 2. Table 2: The coefficients C i ​ j subscript 𝐶 𝑖 𝑗 C_{ij} in the pseudoscalar meson and baryon octet interaction with isospin I = 1 2 𝐼 1 2 I=\frac{1}{2} and strangeness S = 0 𝑆 0 S=0 , C j ​ i = C i ​ j subscript 𝐶 𝑗 𝑖 subscript 𝐶 𝑖 𝑗 C_{ji}=C_{ij} .

$C_{ij}$	$\pi N$	$\eta N$	$K\Lambda$	$K\Sigma$
$\pi N$	$2$	$0$	$\frac{3}{2}$	$-\frac{1}{2}$
$\eta N$		$0$	$-\frac{3}{2}$	$-\frac{3}{2}$
$K\Lambda$			$0$	$0$
$K\Sigma$				$2$

Table 3. Table 3: The coefficients C i ​ j subscript 𝐶 𝑖 𝑗 C_{ij} in the pseudoscalar meson and baryon octet interaction with isospin I = 3 2 𝐼 3 2 I=\frac{3}{2} and strangeness S = 0 𝑆 0 S=0 , C j ​ i = C i ​ j subscript 𝐶 𝑗 𝑖 subscript 𝐶 𝑖 𝑗 C_{ji}=C_{ij} .

$C_{ij}$	$\pi N$	$K\Sigma$
$\pi N$	$-1$	$-1$
$K\Sigma$		$-1$

Table 4. Table 4: The values of the subtraction constant a π ​ N subscript 𝑎 𝜋 𝑁 a_{\pi N} , a η ​ N subscript 𝑎 𝜂 𝑁 a_{\eta N} , a K ​ Λ subscript 𝑎 𝐾 Λ a_{K\Lambda} , a K ​ Σ subscript 𝑎 𝐾 Σ a_{K\Sigma} , the pole position on the complex energy plane of s 𝑠 \sqrt{s} and couplings of the resonance state to different channels. The regularization scale μ = 630 𝜇 630 \mu=630 MeV is fixed in the loop function in Eq. ( 32 ).

$n$	$a_{\pi N}$	$a_{\eta N}$	$a_{K\Lambda }$	$a_{K\Sigma }$	$Poleposition\left(MeV\right)$	$g_{\pi N}$	$\left|g_{\pi N}\right|$	$g_{\eta N}$	$\left|g_{\eta N}\right|$	$g_{K\Lambda }$	$\left|g_{K\Lambda }\right|$	$g_{K\Sigma }$	$\left|g_{K\Sigma }\right|$
1	-3.2	-1.7	-3.2	-3.2	1518-46i	-3+ 1i	4	-65+ 25i	70	41+ 0i	41	94-27i	98
2	-3.2	-1.4	-3.2	-3.2	1530-58i	-4+ 1i	4	-66+ 28i	72	40+ 3i	40	95-27i	99
3	-2.9	-2.3	-3.2	-2.9	1520-41i	-3+ 5i	7	-62+ 21i	66	47+ 0i	47	96-24i	99
4	-2.9	-2.0	-3.2	-2.9	1532-51i	-4+ 5i	7	-64+ 22i	68	45+ 3i	45	97-24i	100
5	-2.9	-1.7	-3.2	-3.2	1518-46i	-3+ 2i	4	-65+ 25i	70	42+ 0i	42	94-27i	98
6	-2.9	-1.4	-3.2	-3.2	1530-58i	-4+ 1i	4	-66+ 28i	72	40+ 4i	40	95-27i	99
7	-2.6	-2.3	-3.2	-2.9	1520-41i	-2+ 6i	7	-62+ 21i	66	47+ 1i	47	96-24i	99
8	-2.6	-2.3	-2.9	-2.9	1527-41i	-2+ 4i	5	-60+ 19i	64	48+ 1i	48	95-21i	98
9	-2.6	-2.0	-3.2	-2.9	1532-51i	-3+ 6i	7	-64+ 22i	68	45+ 4i	45	97-24i	100
10	-2.6	-1.7	-3.2	-3.2	1518-46i	-3+ 2i	4	-65+ 25i	70	42+ 0i	42	94-27i	98
11	-2.6	-1.4	-3.2	-3.2	1530-59i	-4+ 2i	5	-67+ 28i	73	40+ 4i	40	95-27i	99
12	-2.3	-2.3	-3.2	-2.9	1520-41i	-2+ 7i	8	-62+ 21i	66	47+ 1i	47	96-25i	99
13	-2.3	-2.0	-3.2	-2.9	1532-51i	-2+ 6i	8	-64+ 23i	68	45+ 4i	46	97-24i	100
14	-2.3	-1.7	-3.2	-3.2	1518-46i	-2+ 3i	4	-65+ 25i	70	42+ 1i	42	94-27i	98
15	-2.3	-1.4	-3.2	-3.2	1530-59i	-3+ 2i	5	-67+ 28i	73	40+ 4i	40	95-27i	99
16	-2.0	-2.6	-3.2	-2.6	1535-47i	0+ 11i	12	-62+ 18i	65	51+ 5i	52	97-22i	100
17	-2.0	-2.3	-3.2	-2.9	1520-41i	0+ 7i	8	-62+ 21i	66	48+ 1i	48	95-25i	99
18	-2.0	-2.0	-3.2	-2.9	1531-51i	-1+ 7i	8	-64+ 23i	68	46+ 4i	46	96-24i	100
19	-2.0	-1.7	-3.2	-3.2	1518-46i	-2+ 3i	5	-65+ 25i	70	42+ 1i	42	94-27i	98
20	-2.0	-1.4	-3.2	-3.2	1530-59i	-3+ 3i	5	-67+ 28i	73	40+ 4i	40	95-27i	99
21	-1.7	-2.6	-3.2	-2.6	1534-47i	-1-12i	12	-62+ 18i	65	52+ 5i	52	97-22i	100
22	-1.7	-2.3	-3.2	-2.9	1519-41i	-1 -8i	8	-62+ 21i	66	48+ 1i	48	95-25i	99
23	-1.7	-2.0	-3.2	-2.9	1531-51i	0+ 7i	8	-64+ 23i	69	46+ 4i	46	96-24i	99
24	-1.7	-1.7	-3.2	-3.2	1518-46i	-1+ 3i	5	-65+ 25i	70	42+ 1i	42	94-27i	98
25	-1.7	-1.4	-3.2	-3.2	1530-59i	-2+ 3i	5	-67+ 28i	73	40+ 5i	41	95-27i	99
26	-1.4	-2.6	-3.2	-2.6	1534-47i	-3-11i	11	-62+ 19i	65	53+ 4i	53	97-22i	99
27	-1.4	-2.3	-3.2	-2.9	1519-41i	-2 -8i	8	-62+ 21i	66	49+ 1i	49	95-25i	98
28	-1.4	-2.0	-3.2	-2.9	1531-51i	-1 -8i	8	-64+ 23i	69	47+ 4i	47	96-25i	99
29	-1.4	-1.4	-3.2	-3.2	1530-59i	-2+ 4i	5	-67+ 28i	73	41+ 5i	41	95-28i	99
30	-1.1	-2.6	-3.2	-2.6	1534-46i	-4-10i	11	-62+ 19i	65	53+ 4i	53	96-22i	99
31	-1.1	-2.3	-3.2	-2.9	1519-41i	-3 -7i	8	-62+ 21i	66	49+ 0i	49	95-25i	98
32	-1.1	-2.0	-3.2	-2.9	1531-51i	-2 -7i	7	-64+ 23i	69	47+ 4i	47	96-25i	99
33	-1.1	-1.4	-3.2	-3.2	1530-59i	-1+ 4i	5	-67+ 28i	73	41+ 5i	41	95-28i	99
34	-0.8	-2.6	-3.2	-2.6	1533-46i	-5 -9i	10	-62+ 19i	65	53+ 3i	53	96-22i	98
35	-0.8	-2.0	-3.2	-2.9	1531-51i	-2 -7i	7	-64+ 23i	69	47+ 3i	48	95-24i	99
36	-0.8	-1.4	-3.2	-3.2	1530-59i	0+ 3i	4	-67+ 28i	73	41+ 5i	41	95-28i	99
37	-0.5	-2.6	-3.2	-2.6	1533-46i	-6 -8i	10	-62+ 19i	65	54+ 3i	54	95-21i	98
38	-0.5	-2.0	-3.2	-2.9	1531-51i	-3 -6i	7	-64+ 23i	69	48+ 3i	48	95-24i	98
39	-0.5	-1.4	-3.2	-3.2	1530-59i	0 -4i	4	-67+ 28i	73	41+ 4i	42	94-28i	99

Equations

$$\text{sl}L=⟨\bar{B}(i{\gamma }_{\mu }{D}^{\mu }-M)B⟩+\frac{D/F}{2}⟨\bar{B}{\gamma }_{\mu }{\gamma }_5[u^{\mu },B{]}_{\pm }⟩.$$

$$\Phi =$$

$$B=$$

$$V_{ij}^{con}=-C_{ij}\frac{1}{4f_i{f}_j}\bar{U}(p_j,{\lambda }_j){\gamma }_{\mu }U(p_i,{\lambda }_i)(k_i^{\mu }+k_j^{\mu }),$$

$$V_{ij}^{con}=-C_{ij}\frac{1}{4f_i{f}_j}\bar{U}(p_j,{\lambda }_j){\gamma }_{0}U(p_i,{\lambda }_i)(k_i^0+k_j^0).$$

$$V_{ij}^{con}=$$

$$L=A_{lmn}{\bar{N}}_l{\gamma }_{\mu }{\gamma }_5{\partial }^{\mu }{M}_m{N}_n,$$

$$A_{lmn}=-\frac{1}{2f_0}\left[(D+F)C_{lmn}+(D-F)C_{lnm}\right]$$

$$C_{lmn}=\frac{1}{2}\sum _{i,j,k=1}^8{X}_{il}^{†}{X}_{jm}{X}_{kn}⟨{\lambda }_i{\lambda }_j{\lambda }_k⟩,$$

$$\displaystyle X=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cccccccc}1&1&0&0&0&0&0&0\\ i&-i&0&0&0&0&0&0\\ 0&0&\sqrt{2}&0&0&0&0&0\\ 0&0&0&1&1&0&0&0\\ 0&0&0&i&-i&0&0&0\\ 0&0&0&0&0&1&1&0\\ 0&0&0&0&0&i&-i&0\\ 0&0&0&0&0&0&0&\sqrt{2}\\ \end{array}\right).$$

$$V_{ij}^s$$

$$V_{ij}^u$$

$$f(x)=\sum _{n=0}^{+\infty }{c}_n{P}_n(x),$$

$$c_n=\frac{2n+1}{2}{\int }_{-1}^{1}f(x)P_n(x)dx.$$

$$u-M^2$$

$$\frac{1}{2}{\int }_{-1}^1\frac{1}{1-\alpha x}dx=-\frac{1}{2}\frac{1}{\alpha }\ln \left(\frac{1-\alpha }{1+\alpha }\right).$$

$$V_{ij}^u(S)$$

$$V_{ij}=V_{ij}^{con}+V_{ij}^s+V_{ij}^u(S).$$

$$T$$

$$V_{jl}{G}_l{V}_{li}\sim \bar{U}(p_j,{\lambda }_j)G_{l}U(p_i,{\lambda }_i)(k_i^0+k_j^0{)}^2.$$

$$G_l=$$

$$+\left(\frac{{\gamma }_{\mu }{P}^{\mu }[P^2+M_l^2-m_l^2]}{4P^2{M}_l}+\frac{1}{2}\right)G_l^{\prime },$$

$$G_l^{\prime }(s)$$

$$G_l=$$

$$+\left(\frac{s+M_l^2-m_l^2}{4M_l\sqrt{s}}+\frac{1}{2}\right)G_l^{\prime }.$$

$$\tilde{V}$$

$${\tilde{G}}_l$$

$$\tilde{T}=[1-\tilde{V}\tilde{G}{]}^{-1}\tilde{V}$$

$$|\pi N;\frac{1}{2},-\frac{1}{2}⟩=-\sqrt{\frac{2}{3}}|{\pi }^{-}p⟩+\sqrt{\frac{1}{3}}|{\pi }^{0}n⟩,$$

$$|\eta N;\frac{1}{2},-\frac{1}{2}⟩=|\eta n⟩,$$

$$|K\Lambda ;\frac{1}{2},-\frac{1}{2}⟩=|K^0\Lambda ⟩,$$

Full text

I Introduction

The pion-nucleon interaction is an interesting topic and has attracted more attentions of the nuclear society in the past decades. There are two very closed excited states of the nucleon in the $S_{11}$ channel, $N(1535)$ and $N(1650)$, which are difficult to be described within the framework of the constituent quark modelKaiser . However, in the unitary coupled-channel approximation of the Bethe-Salpeter equation, most of the excited states of the nucleon are treated as resonance states of the pseudoscalar meson and the baryon in the $SU(3)$ flavor space, so are these two particles. In Ref. Kaiser , it is pointed out that the hidden strange channels of $K\Lambda$ and $K\Sigma$ might play an important role in the dynamically generation of the $N(1535)$ particle.

The $N(1535)$ particle is generated dynamically in the unitary coupled-channel approximation with the final state interaction of a three-body $N\pi\pi$ channel consideredInoue . However, the inclusion of the $N\pi\pi$ as a final state in the calculation is complex, especially there are six arbitrary constants in the real part of the three-body $N\pi\pi$ loop function, and thus it must be treated as a free function consistent to the experiment. This work is studied again by including the $\rho N$ and $\pi\Delta$ channels in a non-relativistic approximation besides the pseudoscalar meson -baryon octet channelsGarzon . Actually the the elastic scattering process of $\rho N\rightarrow\rho N$ mainly gives a contribution to the generation of the $N(1650)$ resonance dynamically, just as done in Ref. Ramos . In the processes of $\rho N\rightarrow\pi N$ and $\rho N\rightarrow\pi\Delta$, the Kroll-Ruderman term supplies a constant potential and plays a dominant role, while the $\pi$-exchange potential is trivial and proportional to the square of the three-momentum of the final state in the center of mass frame. Moreover, the structure of the $N(1535)$ and $N(1650)$ particles are also studied in the unitary coupled-channel approximation in Nieves ; Doring , where a loop function of the intermediate pseudoscalar meson and the baryon in the on-shell approximation is taken into account when the Bethe-Salpeter equation is solved.

In Ref. Bruns , the $N(1535)$ and $N(1650)$ resonance states are studied in the unitary coupled-channel approximation with a Lagrangian of the pseudoscalar meson and the baryon octet up to the next-to-leading-order term. By fitting the $S_{11}$ partial wave amplitude with experimental data up to the energy $\sqrt{s}=1.56$GeV, the resonance state corresponding to the $N(1535)$ particle is generated dynamically. In addition, it is amazing that the $N(1650)$ resonance state can also be produced at higher energies at the same time.

The property of the $N(1535)$ particle has also been studied by solving the relativistic Lippmann-Schwinger equation, where the corresponding Hamiltonian is divided into two parts, a non-interacting part and an interacting part, and the couplings and the bare mass of the nucleon are determined by fitting the experimental data. This method is named as Hamiltonian effective field theory by the authorsLiuzw . Recently, the different partial wave phase shifts are analyzed by calculating the K-matrix of the pion-nucleon interactionZheng . It is more interesting that the internal wave functions of the $\Delta(1232)$, $N(1535)$ and $N(1650)$ resonance states are investigated, and it is announced that the $\pi N$, $\eta N$, $K\Lambda$ and $K\Sigma$ components are negligible in these resonance statesSekihara . It is apparent that the conclusion made in Ref. Sekihara is inconsistent with the previous principles due to chiral unitary models.

In this work, the interaction of the pseudoscalar meson and the baryon octet is studied in the unitary coupled-channel approximation, and the contribution of the $s-$ and $u-$ channel potentials in the S-wave approximation is taken into account besides the Weinberg-Tomozawa contact term. Furthermore, a revised loop function of the Bethe-Salpeter equation is used in the calculationDongsun , where the relativistic correction is included.

By adjusting the subtraction constants for different intermediate particles of the loop function in the sector of isospin $I=1/2$ and strangeness $S=0$, a pole at $1518-i46$MeV on the complex energy plane is detected, which might be a counterpart of the $N(1535)$ particle in the PDG dataPDG .

This article is organized as follows. In Section II, the potential of the pseudoscalar meson and baryon octet is constructed, where the Weinberg-Tomozawa contact term, the $s-$ channel and $u-$ channel interactions are all taken into account in the S-wave approximation. In Section III, a basic formula on how to solve the Bethe-Salpeter equation in the unitary coupled-channel approximation is shown. The cases of isospin $I=1/2$ and $I=3/2$ are discussed in Section IV and Section V, respectively. Finally, the summary is given in Section VI.

II Framework

The effective Lagrangian of the pseudoscalar meson and the baryon octet can be written as

[TABLE]

In the above equation, the symbol $\langle...\rangle$ denotes the trace of matrices in the $SU(3)$ flavor space, and $D^{\mu}B=\partial^{\mu}B+\frac{1}{2}\left[[u^{\dagger},\partial^{\mu}u],B\right]$ with $u^{2}=U=\exp\left(i\frac{\Phi}{f_{0}}\right)$ and $u^{\mu}=iu^{\dagger}\partial^{\mu}u-iu\partial^{\mu}u^{\dagger}$, where $f_{0}$ is the meson decay constant in the chiral limit.

The matrices of the pseudoscalar meson and the baryon octet are given as follows

[TABLE]

and

[TABLE]

The first term in the Lagrangian in Eq. (1) supplies the contact interaction of the pseudoscalar meson and the baryon octet, which is usually called as Weinberg-Tomozawa term, while the other terms which are relevant to the coefficients $D$ and $F$ give a contribution to the $s-$ and $u-$ channel interactions, as shown in Fig. 1.

According to Feynmann rules, The Weinberg-Tomozawa contact potential of the pseudoscalar meson and the baryon octet can be written as

[TABLE]

where $p_{i},p_{j}(k_{i},k_{j})$ are the momenta of the initial and final baryons(mesons), and $\lambda_{i},\lambda_{j}$ denote the spin orientations of the initial and final baryons, respectively. For low energies, the three-momenta of the incoming and outgoing mesons can be neglected, and thus the potential in Eq. (4) is simplified as

[TABLE]

Because $U(p_{i},\lambda_{i})$ and $\bar{U}(p_{j},\lambda_{j})$ stand for the wave functions of the initial and final baryons, respectively, the matrix $\gamma_{0}$ in Eq. (5) can be replaced by the unit matrix $I$ at the low energy region, i.e., $\gamma_{0}\rightarrow I$. Finally, the Weinberg-Tomozawa contact term of the pseudoscalar meson and the baryon octet takes the form of

[TABLE]

where $\sqrt{s}$ is the total energy of the system, $M_{i}$ and $M_{j}$ denote the initial and final baryon masses, respectively, while $E$ and $E^{\prime}$ stand for the initial and final baryon energies in the center of mass frame, respectively. The coefficient $C_{ij}$ for the sector of strangeness zero and charge zero is listed in Table 1, Moreover, we assume the values of the decay constants are only relevant to the pseudoscalar meson with $f_{\eta}=1.3f_{\pi}$, $f_{K}=1.22f_{\pi}$ and $f_{\pi}=92.4$MeV, as given in Ref. Inoue ; Bruns .

The second term in Eq. (1) supplies antibaryon-baryon-meson vertices, and can be rewritten as

[TABLE]

with $N=\{\Sigma^{+},\Sigma^{-},\Sigma^{0},p,\Xi^{-},n,\Xi^{0},\Lambda\}$ and $M=\{\pi^{+},\pi^{-},\pi^{0},K^{+},K^{-},K^{0},\bar{K}^{0},\eta\}$.

The coefficient $A_{lmn}$ in Eq. (7) takes the form of

[TABLE]

where

[TABLE]

with $\lambda$ the matrix of the SU(3) generator and

[TABLE]

Thus the $s-$ and $u-$ channel interaction of the pseudoscalar meson and the baryon octet can be constructed according to the vertices in Eq. (7).

If the three-momenta of the incoming and outgoing particles are neglected in the calculation, the $s-$ channel potential of the pseudoscalar meson and the baryon octet can be written as

[TABLE]

approximately, where $M$ denotes the mass of the intermediate baryon, $A$ and $A^{\prime}$ represent the coefficients depicted in Eq. (8), respectively.

Similarly, the $u-$ channel potential can be obtained as

[TABLE]

with the Mandelstam variable $u=(p_{i}-k_{j})^{2}$.

In the calculation of Eqs. (19) and (20), a physical baryon mass is adopted so as to obtain the $s$-channel and $u$-channel interaction potentials of the pseudoscalar meson and the baryon octet. The mass renormalization of baryons has be assumed to be accomplished before the tree-level diagrams in the interaction of the pseudoscalar meson and the baryon octet are studied. In the chiral unitary model, the loop function of the intermediate pseudoscalar meson and baryon is considered in an on-shell approximation when the Bethe-Salpeter equation is solved, which will be iterated in Sect. III, so that the whole interaction chain is taken into account without a cutoff. Therefore, we can examine whether a resonance state can be generated dynamically or not.

The Weinberg-Tomozawa term and the $s-$ channel potential of the pseudoscalar meson and the baryon octet are only related to the Mandelstam variable $s$, therefore, they only give a contribution to the S-wave amplitude in the scattering process of the pseudoscalar meson and the baryon octet.

Since a function can be expanded with the Legendre polynomials, i.e.,

[TABLE]

with $P_{n}(x)$ the $n$th Legendre polynomial and the coefficient

[TABLE]

In the S-wave approximation, only the coefficient $c_{0}$ is necessary to be considered.

the denominator $u-M^{2}$ in Eq. (20) can be written as

[TABLE]

where $\theta$ is the angle between the three-momenta of incoming and outgoing mesons, and $\vec{p}_{i}(\vec{k}_{j})$ and $M_{i}(m_{j})$ are the three-momentum in the center of mass frame and the mass of the initial baryon (final meson), respectively. Supposing $\alpha=\frac{2|\vec{p}_{i}|\vec{k}_{j}|}{M_{i}^{2}+m_{j}^{2}-M^{2}-2E(\sqrt{s}-E^{\prime})}$ and $x=\cos\theta$, we can obtain

[TABLE]

Thus the $u-$ channel potential of the pseudoscalar meson and the baryon octet in the S-wave approximation can be calculated easily

[TABLE]

Therefore, the S-wave potential of the pseudoscalar meson and the baryon octet can be written as

[TABLE]

III Bethe-Salpeter equation

The Bethe-Salpeter equation can be expanded as

[TABLE]

When the Bethe-Salpeter equation in Eq. (27) is solved, only the on-shell part of the potential $V_{ij}$ in Eq. (5) gives a contribution to the amplitude of the pseudoscalar meson and the baryon octet, and the off-shell part of the potential can be reabsorbed by a suitable renormalization of the decay constants of mesons $f_{i}$ and $f_{j}$. More detailed discussion can be found in Refs. Oller97 ; Ramos97 . Therefore, if the potential in Eq. (5) is adopted, the second term $VGV$ in Eq. (27) can be written as

[TABLE]

If the relativistic kinetic correction of the loop function of the pseudoscalar meson and the baryon octet is taken into account, the loop function $G_{l}$ can be written as

[TABLE]

with $P$ the total momentum of the system, $m_{l}$ the meson mass, and $M_{l}$ the baryon mass, respectively.

The loop function in Eq. (29) can be calculated in the dimensional regularization (See Appendix 1 of Ref. Dongsun for details), and thus the loop function takes the form of

[TABLE]

where $a_{l}$ is the subtraction constant and $\mu$ is the regularization scale, and $G_{l}^{\prime}$ is the loop function in Ref. Oller ,

[TABLE]

with $\bar{q}_{l}$ the three-momentum of the meson or the baryon in the center of mass frame.

Since the total three-momentum $\vec{P}=0$ in the center of mass frame, only the $\gamma_{0}P^{0}$ parts remain in Eq. (30). Similarly, This matrix $\gamma_{0}$ can be replaced by the unit matrix $I$ since the $U(p_{i},\lambda_{i})$ and $\bar{U}(p_{j},\lambda_{j})$ denote the wave functions of the initial and final baryons, respectively. Thus the loop function of the intermediate pseudoscalar meson and baryon octet becomes

[TABLE]

When the $s-$ channel and $u-$ channel interaction are supplemented, the loop function in Eq. (32) is still suitable. However, the off-shell part of the potential is reabsorbed by a renormalization, so the decay constants of mesons, the masses of intermediate baryons all take physical values when the Bethe-Salpeter equation is solved.

In the calculation of the present work, we make a transition of

[TABLE]

so the scattering amplitude

[TABLE]

becomes dimensionless.

IV $I=\frac{1}{2}$ and $S=0$

In the sector of isospin $I=\frac{1}{2}$ and strangeness $S=0$, the wave function in the isospin space can be written as

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

respectively. Thus the coefficients $C_{ij}$ in the Weinberg-Tomozawa contact potential of the pseudoscalar meson and the baryon octet can be obtained in the isospin space, which are summarized in Table 2.

The $s-$ channel, $u-$ channel and Weinberg-Tomozawa contact potentials of the pseudoscalar meson and baryon octet in the S-wave approximation are depicted in Figs. 2, respectively. In Fig. 2, it is found that the $\pi N$ $s-$ channel potential is repulsive and the other $s-$ channel potential are weaker than the $\pi N$ case, while the $u-$ channel potentials in the S-wave approximation are attractive. Although the curves for $\eta N$ and $K\Sigma$ cases are not smooth when $\sqrt{s}<1300$MeV, it is far away from the energy region which we are interested in, and we assume that it would not give an effect on the pole position of the amplitude in the calculation. However, the contact interaction originated from the Weinberg-Tomozawa term is dominant in the pseudoscalar meson and the baryon octet potential, and the correction from the $s-$ channel potential and the S-wave $u-$ channel potential is not important.

The total potentials for different pseudoscalar meson and baryon systems with isospin $I=1/2$ and strangeness $S=0$ are depicted in the right figure of Fig. 2. It shows that the $\pi N$ and $K\Sigma$ potentials are attractive, while the $\eta N$ and $K\Lambda$ interactions are weak.

Although the $s-$ channel and $u-$ channel potentials are weaker than the Weinberg-Tomozawa contact interaction in the sector of isospin $I=1/2$ and strangeness $S=0$, the subtraction constants must be readjusted in the calculation when the contribution of the $s-$ and $u-$ channel potentials are taken into account.

According to the PDG data, the $N(1535)$ particle is assume to lie in the region of $Re(poleposition)=1490\sim 1530$MeV, and $-2Im(poleposition)=90\sim 250$MeV on the complex energy plane of $\sqrt{s}$PDG . When the Bethe-Salpeter equation is solved in the unitary coupled-channel approximation, we set the regularization scale $\mu=630$MeV, just as done in most of works with this methodGarzon ; Ramos ; Dongsun ; Ramos97 . Moreover, all of subtraction constants change from $-3.2$ to $-0.5$ with a step of $0.3$, and we hope a resonance state can be generated dynamically in the reasonable energy region. In the previous works, the subtraction constant is usually chosen to be $-2$, which is thought to be a natural value. we change the subtraction constant values in the neighboring region of $-2$ in order to find the influence of different subtraction constant values on the mass and decay width of the resonance state.

Altogether we find 39 sets of subtraction constant values is suitable to produce a pole in the energy region constrained by the PDG data, which are listed in the Appendix part of this manuscript. At the same time, the pole position and the couplings to $\pi N$, $\eta N$, $K\Lambda$ and $K\Sigma$ are also listed. A resonance state with a mass about $1520$MeV and a decay width about $90$MeV is generated by using 12 sets of subtraction constants, while both the mass and the decay width of the resonance state increase slightly when the other 27 sets of parameters are used in the calculation, respectively. These 39 sets of subtraction constants are depicted in Fig. 3, and it is found that the subtraction constant $a_{\pi N}$ changes from $-3.2$ to $-0.5$ successively. Since the $\pi N$ threshold is far lower than the energy region where the $N(1535)$ particle might be generated dynamically, it is understandable that the pole position is not sensitive to the value of the subtraction constant $a_{\pi N}$. The changes of the other three subtraction constants $a_{\eta N}$, $a_{K\Lambda}$ and $a_{K\Sigma}$ are not so large. Especially, the subtraction constant $a_{K\Lambda}=-3.2$ in 38 sets of parameters except the eighth set, where it takes the value of $-2.9$. The $K\Lambda$ threshold is close to the energy region where we are interested, thus the subtraction constant $a_{K\Lambda}$ is stable and plays an important role in the generation of the $N(1535)$ particle.

A pole is generated dynamically at $1518-i46$MeV on the complex energy plane of $\sqrt{s}$ by solving the Bethe-Salpeter equation in the unitary coupled-channel approximation with the 19th set of parameters, i.e., $a_{\pi N}=-2.0$, $a_{\eta N}=-1.7$, $a_{K\Lambda}=-3.2$ and $a_{K\Sigma}=-3.2$. The squared amplitude $|T|^{2}$ as a function of the total energy $\sqrt{s}$ for different channels with isospin $I=1/2$ and strangeness $S=0$ are depicted in Fig. 4. The real part of the pole position is higher than the $\eta N$ threshold, and lower than the $K\Lambda$ threshold, so we assume it might be a resonance state and correspond to the N(1535) particle in the PDG data.

The couplings of the resonance state to the $\pi N$, $\eta N$, $K\Lambda$ and $K\Sigma$ channels are also calculated. and it is found that it couples strongly to the $\eta N$, $K\Lambda$ and $K\Sigma$ channels, which implies that these channels is important in the generation of the $N(1535)$ resonance state. If different sets of subtraction constants are chosen in the calculation, the changes of couplings are not significant, as shown in the table of the Appendix part.

In Ref. Bruns , Eq. (15) indicates that the $N(1535)$ particle couples more strongly to the $K^{+}\Lambda$ channel, which is different from the results listed in Table 4. The different values of the coupling constants might be relevant to the next-to-leading-order chiral Lagrangian used in Ref. Bruns , while it is not included in this manuscript.

V $I=\frac{3}{2}$ and $S=0$

The wave functions with isospin $I=3/2$ and strangeness $S=0$ can be written as

[TABLE]

and

[TABLE]

respectively. According to Eqs. (39) and (40), the coefficients $C_{ij}$ in the isospin space can be calculated and listed in Table 3. Since the coefficients are all negative, the Weinberg-Tomozawa contact interaction between the pseudoscalar meson and the baryon octet is repulsive in the sector of isospin $I=3/2$ and strangeness $S=0$. Even if the correction from the $s-$ channel and $u-$ channel interaction is taken into account, as is shown in Fig. 5, the total potential of the pseudoscalar meson and the baryon octet is still repulsive. Thus no resonance state could be generated in the S-wave approximation.

VI Summary

In this work, the interaction of the pseudoscalar meson and the baryon octet is studied within a nonlinear realized Lagrangian. The $s-$, $u-$ channel potentials and the Weinberg-Tomozawa contact interaction are obtained when the three-momenta of the particles in the initial and final states are neglected in the S-wave approximation.

In the sector of isospin $I=1/2$ and strangeness $S=0$, a resonance state is generated dynamically by solving the Bethe-Salpeter equation, which might be regarded as counterparts of the $N(1535)$ particle listed in the PDG data. We find the hidden strange channels, such as $\eta N$, $K\Lambda$ and $K\Sigma$, play an important role in the generation of the resonance state when the Bethe-Salpeter equation is solved in the unitary coupled-channel approximation. The coupling constants of this resonance state to different channels are calculated, and it is found that it couples strongly to the hidden strange channels.

Acknowledgements.

Bao-Xi Sun would like to thank Han-Qing Zheng and Yu-Fei Wang for useful discussions.

Appendix: Subtraction constants, pole position and couplings of the resonance state to different channels