-
The pion-nucleon interaction is an interesting topic and has attracted considerable attention of the nuclear physics community in the past decades. There are two very close excited states of the nucleon in the
S11 channel,N(1535) andN(1650) , which are difficult to describe within the framework of the constituent quark model [1]. 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 theSU(3) flavor space, as are these two particles. In Ref. [1], it is pointed out that the hidden strange channels ofKΛ andKΣ might play an important role in the dynamical generation of theN(1535) particle.The
N(1535) particle, generated dynamically in the unitary coupled-channel approximation with a three-bodyNππ final state, was considered in [2]. However, the inclusion ofNππ as a final state in the calculation is complex, especially as there are six arbitrary constants in the real part of the three-bodyNππ loop function, and thus it must be treated as a free function consistent with experimental data. This approach was studied again by including theρN andπΔ channels in a non-relativistic approximation, besides the pseudoscalar meson-baryon octet channels [3]. In fact, the elastic scattering processρN→ρN gives the main contribution to the dynamical generation of theN(1650) resonance, as was shown in Ref. [4]. In the processesρN→πN andρN→πΔ , the Kroll-Ruderman term supplies a constant potential and plays a dominant role, while theπ -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 ofN(1535) andN(1650) particles was also studied in the unitary coupled-channel approximation in [5, 6], 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. [7], the
N(1535) andN(1650) resonance states were studied in the unitary coupled-channel approximation with the Lagrangian of the pseudoscalar meson and the baryon octet up to the next-to-leading-order. By fitting theS11 partial wave amplitude with the experimental data up to the energy√s=1.56 GeV, the resonance state corresponding to theN(1535) particle is generated dynamically. In addition, it is amazing that theN(1650) resonance state can be produced at the same time at higher energies.The properties of the
N(1535) particle have 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 called the Hamiltonian effective field theory by the authors [8]. Recently, the different partial wave phase shifts were analyzed by calculating the K-matrix of the pion-nucleon interaction [9]. It is interesting that the internal wave functions of theΔ(1232) ,N(1535) andN(1650) resonance states were investigated, and it was announced that theπN ,ηN ,KΛ andKΣ components are negligible in these resonance states [10]. It is apparent that the conclusion made in Ref. [10] is inconsistent with the previous results due to the 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− andu− 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 calculation [11], 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 strangenessS=0 , a pole at1518−i46 MeV in the complex energy plane is detected, which might correspond to theN(1535) particle of the PDG review [12].This article is organized as follows. In Section 2, the potential of the interaction between the pseudoscalar meson and the baryon octet is constructed, where the Weinberg-Tomozawa contact term, and the
s− channel andu− channel interactions are all taken into account in the S-wave approximation. In Section 3, the fundamental formula for solving the Bethe-Salpeter equation in the unitary coupled-channel approximation is presented. The cases of isospinI=1/2 andI=3/2 are discussed in Section 4 and Section 5, respectively. Finally, the summary is given in Section 6. -
The effective Lagrangian of the pseudoscalar meson and baryon octet interaction can be written as
L=⟨ˉB(iγμDμ−M)B⟩+D/F2⟨ˉBγμγ5[uμ,B]±⟩.
(1) In the above equation, the symbol
⟨...⟩ denotes the trace of matrices in theSU(3) flavor space, andDμB= ∂μB+12[[u†,∂μu],B] withu2=U=exp(iΦf0) anduμ= iu†∂μu−iu∂μu† , where D=0.80, F=0.46 andf0 =92.40 MeV is the meson decay constant in the chiral limit.The matrices of the pseudoscalar meson and the baryon octet are given as follows
Φ=√2(1√2π0+1√6ηπ+K+π−−1√2π0+1√6ηK0K−ˉK0−2√6η)
(2) and
B=(1√2Σ0+1√6ΛΣ+pΣ−−1√2Σ0+1√6ΛnΞ−Ξ0−2√6Λ).
(3) The first term in the Lagrangian in Eq. (1) represents the contact interaction of the pseudoscalar meson and the baryon octet, which is usually called the Weinberg-Tomozawa term, while the other terms which are relevant to the coefficients
D andF give a contribution to thes− andu− channel interactions, as shown in Fig. 1.Figure 1. Feynman diagrams of the pseudoscalar meson-baryon octet interaction.
(a) contact term,(b) u− channel and(c) s− channel.According to the Feynmann rules, the Weinberg-Tomozawa contact potential of the pseudoscalar meson and baryon octet interaction can be written as
Vconij=−Cij14fifjˉU(pj,λj)γμU(pi,λi)(kμi+kμj),
(4) where
pi,pj(ki,kj) are the momenta of the initial and final baryons (mesons), andλi,λj denote the spin orientations of the initial and final baryons. For low energies, the three-momenta of the incoming and outgoing mesons can be neglected, and thus the potential in Eq. (4) is simplified asVconij=−Cij14fifjˉU(pj,λj)γ0U(pi,λi)(k0i+k0j).
(5) As
U(pi,λi) andˉU(pj,λj) are the wave functions of the initial and final baryons, the matrixγ0 in Eq. (5) can be replaced by the unit matrixI in the low energy region, i.e.,γ0→I . Finally, the Weinberg-Tomozawa contact potential of the pseudoscalar meson and baryon octet interaction takes the formVconij=−Cij14fifj(2√s−Mi−Mj)(Mi+E2Mi)12(Mj+E′2Mj)12,
(6) where
√s is the total energy of the system,Mi andMj denote the initial and final baryon masses, whileE andE′ stand for the initial and final baryon energies in the center-of-mass frame. The coefficientsCij for the sector of strangeness zero and charge zero are listed in Table 1. Moreover, we assume that the values of the decay constants are only relevant for the pseudoscalar meson withfη=1.3fπ ,fK=1.22fπ andfπ=92.4 MeV, as given in Ref. [2, 7].Cij K+Σ− K0Σ0 K0Λ π−p π0n ηn K+Σ− 1 −√2 0 0 −1√2 −√32 K0Σ0 0 0 −1√2 −12 √32 K0Λ 0 −√32 √32 −32 π−p 1 −√2 0 π0n 0 0 ηn 0 Table 1. The coefficients
Cij in the pseudoscalar meson and baryon octet interaction with strangenessS=0 and chargeQ=0 ,Cji=Cij .The second term in Eq. (1) represents the antibaryon-baryon-meson vertices, and can be rewritten as
L=AlmnˉNlγμγ5∂μMmNn,
(7) with
N={Σ+,Σ−,Σ0,p,Ξ−,n,Ξ0,Λ} andM={π+,π−,π0,K+, K−,K0,ˉK0,η} .The coefficient
Almn in Eq. (7) takes the formAlmn=−12f0[(D+F)Clmn+(D−F)Clnm]
(8) where
Clmn=128∑i,j,k=1X†ilXjmXkn⟨λiλjλk⟩,
(9) with
λ the matrix of the SU(3) generator andX=1√2(11000000i−i00000000√20000000011000000i−i0000000011000000i−i00000000√2).
(10) Thus, the
s− andu− 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 baryon octet interaction can be written approximately asVsij≈AA′(√s−E)(√s−E′)√s+M,
(11) where
M denotes the mass of the intermediate baryon,A andA′ represent the coefficients given in Eq. (8).Similarly, the
u− channel potential can be obtained asVuij≈AA′(√s−E)(E+E′−√s−M)(√s−E′)u−M2,
(12) with the Mandelstam variable
u=(pi−kj)2 .In the calculation of Eqs. (11) and (12), a physical baryon mass is adopted so as to obtain the
s -channel andu -channel interaction potentials. The mass renormalization of baryons has 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 the on-shell approximation when the Bethe-Salpeter equation is solved, which will be discussed in Sect. 3, 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 baryon octet interaction are only related to the Mandelstam variables , therefore, they only give a contribution to the S-wave amplitude in the scattering process of the pseudoscalar meson and the baryon octet.As is well known, a function can be expanded with the Legendre polynomials, i.e.,
f(x)=+∞∑n=0cnPn(x),
(13) with
Pn(x) then th Legendre polynomial and the coefficientscn=2n+12∫1−1f(x)Pn(x)dx.
(14) In the S-wave approximation, only the coefficient
c0 is necessary.The denominator
u−M2 in Eq. (12) can be written asu−M2=M2i+m2j−M2−2(p0ik0j−→pi⋅→kj)=(M2i+m2j−M2−2p0ik0j)×(1−2|→pi||→kj|cosθM2i+m2j−M2−2p0ik0j),
(15) where
θ is the angle between the three-momenta of incoming and outgoing mesons, and→pi(→kj) andMi(mj) are the three-momentum in the center-of-mass frame and the mass of the initial baryon (final meson). Takingα=2|→pi|→kj|M2i+m2j−M2−2E(√s−E′) andx=cosθ , we obtain12∫1−111−αxdx=−121αln(1−α1+α).
(16) Thus, the
u− channel potential of the pseudoscalar meson and baryon octet interaction in the S-wave approximation can be easily calculated asVuij(S)=AA′(√s−E)(E+E′−√s−M)(√s−E′)M2i+m2j−M2−2E(√s−E′)×−12αln(1−α1+α).
(17) Therefore, the S-wave potential of the pseudoscalar meson and baryon octet interaction can be written as
Vij=Vconij+Vsij+Vuij(S).
(18) -
The Bethe-Salpeter equation can be expanded as
T=V+VGT=V+VGV+VGVGV+....
(19) When the Bethe-Salpeter equation in Eq. (19) is solved, only the on-shell part of the potential
Vij in Eq. (5) gives a contribution to the amplitude of the pseudoscalar meson and baryon octet interaction, and the off-shell part of the potential can be reabsorbed by a suitable renormalization of the decay constants of mesonsfi andfj . A more detailed discussion can be found in Refs. [13, 14]. Therefore, if the potential in Eq. (5) is adopted, the second termVGV in Eq. (19) can be written asVjlGlVli∼ˉU(pj,λj)GlU(pi,λi)(k0i+k0j)2.
(20) If the relativistic kinetic correction of the loop function of the pseudoscalar meson and baryon octet interaction is taken into account, the loop function
Gl can be written asGl=i∫ddq(2π)4/q+Mlq2−M2l+iϵ1(P−q)2−m2l+iϵ,
(21) with
P the total momentum of the system,ml the meson mass, andMl the baryon mass.The loop function in Eq. (21) can be calculated in the dimensional regularization (See Appendix 1 of Ref. [11] for details), and thus the loop function takes the form
Gl=γμPμ32P2π2[(al+1)(m2l−M2l)+(m2llnm2lμ2−M2llnM2lμ2)]+(γμPμ[P2+M2l−m2l]4P2Ml+12)G′l,
(22) where
al is the subtraction constant,μ is the regularization scale, andG′l is the loop function in Ref. [15],G′l(s)=2Ml16π2{al(μ)+lnm2lμ2+M2l−m2l+s2slnM2lm2l+ˉql√s[ln(s−(M2l−m2l)+2ˉql√s)+ln(s+(M2l−m2l)+2ˉql√s)−ln(−s+(M2l−m2l)+2ˉql√s)−ln(−s−(M2l−m2l)+2ˉql√s)]},
(23) with
ˉql the three-momentum of the meson or the baryon in the center-of-mass frame.Since the total three-momentum
→P=0 in the center-of-mass frame, only theγ0P0 part remains in Eq. (22). Similarly, the matrixγ0 can be replaced by the unit matrixI sinceU(pi,λi) andˉU(pj,λj) are the wave functions of the initial and final baryons. Thus, the loop function of the intermediate pseudoscalar meson and baryon octet becomesGl=√s32π2s[(al+1)(m2l−M2l)+(m2llnm2lμ2−M2llnM2lμ2)]+(s+M2l−m2l4Ml√s+12)G′l.
(24) When the
s− channel andu− channel interactions are considered, the loop function in Eq. (24) is still suitable. However, the off-shell part of the potential is reabsorbed by the renormalization, so that the decay constants of mesons and the masses of intermediate baryons all take physical values when the Bethe-Salpeter equation is solved.In our calculations, we make a transition
˜V=V√MiMj,˜Gl=Gl/Ml,
(25) so that the scattering amplitude
˜T=[1−˜V˜G]−1˜V
(26) becomes dimensionless.
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In the sector of isospin
I=12 and strangenessS=0 , the wave function in the isospin space can be written as|πN;12,−12⟩=−√23|π−p⟩+√13|π0n⟩,
(27) |ηN;12,−12⟩=|ηn⟩,
(28) |KΛ;12,−12⟩=|K0Λ⟩,
(29) and
|KΣ;12,−12⟩=−√13|K0Σ0⟩+√23|K+Σ−⟩.
(30) Thus, the coefficients
Cij in the Weinberg-Tomozawa contact potential of the pseudoscalar meson and baryon octet interaction can be obtained in the isospin space, which are summarized in Table 2.Cij πN ηN KΛ KΣ πN 2 0 32 −12 ηN 0 −32 −32 KΛ 0 0 KΣ 2 Table 2. The coefficients
Cij in the pseudoscalar meson and baryon octet interaction with isospinI=12 and strangenessS=0 ,Cji=Cij .The
s− channel,u− channel and Weinberg-Tomozawa contact potentials in the S-wave approximation are depicted in Fig. 2. From Fig. 2, it is seen that theπN s− channel potential is repulsive and the others− channel potentials are weaker than theπN potential, while allu− channel potentials in the S-wave approximation are attractive. Although the curves forηN andKΣ are not smooth when√s<1300 MeV, this occurs far away from the energy region we are interested in, and we assume that there is no effect on the pole position of the amplitude. As the Weinberg-Tomozawa term is dominant, the correction from thes− channel potential and the S-waveu− channel potential is not important.Figure 2. Potentials of the pseudoscalar meson and baryon octet interaction as functions of the total energy of the system
√s in the sector of isospinI=1/2 and strangenessS=0 . (left):s− channel. (middle):u− channel in the S-wave approximation. (right): The solid lines denote the Weinberg-Tomozawa contact interaction, while the dashed lines stand for the total S-wave potential from Eq. (18).The total potential for the different pseudoscalar meson and baryon systems with isospin
I=1/2 and strangenessS=0 are depicted in the right side of Fig. 2, which shows that theπN andKΣ potentials are attractive, while theηN andKΛ interactions are weak.Although the
s− channel andu− channel potentials are weaker than the Weinberg-Tomozawa contact interaction in the sector of isospinI=1/2 and strangenessS=0 , the subtraction constants must be readjusted in the calculation when the contribution of thes− andu− channel potentials are taken into account.According to the PDG data, the
N(1535) particle is assumed to lie in the region of Re (pole position) = 1490~1530 MeV, and −2Im (pole position) = 90~250 MeV in the complex energy plane of√s [12]. When the Bethe-Salpeter equation is solved in the unitary coupled-channel approximation, we set the regularization scale toμ=630 MeV, just as is done in most works using this method[3, 4, 11, 14]. Moreover, all subtraction constants change from−3.2 to−0.5 with a step of0.3 , and we hope a resonance state can be generated dynamically in a reasonable energy region. In the previous works, the subtraction constant was usually chosen to be−2 , which is thought to be a natural value. We changed the subtraction constants in the neighborhood of−2 in order to find the influence of different values on the mass and decay width of the resonance state.Altogether, we found 39 sets of subtraction constants suitable for producing a pole in the energy region constrained by the PDG data, listed in Table 3. The pole positions and the couplings to
πN ,ηN ,KΛ andKΣ are also listed. A resonance state with a mass of about1520 MeV and a decay width of about90 MeV is generated using 12 sets of subtraction constants, while both the mass and the decay width increase slightly when the other 27 sets are used in the calculation. The 39 sets of subtraction constants are depicted in Fig. 3. It is found that the subtraction constantaπN changes from−3.2 to−0.5 in successive steps . Since theπN threshold is far lower than the energy region where theN(1535) particle might be generated dynamically, it is understandable that the pole position is not sensitive to the value of the subtraction constantaπN . The changes of the other three subtraction constantsaηN ,aKΛ andaKΣ are not so large. Especially, the subtraction constantaKΛ=−3.2 in 38 sets of parameters, while for the eighth set it takes the value of−2.9 . TheKΛ threshold is close to the energy region we are interested in, and as the subtraction constantaKΛ is stable, it plays an important role in the generation of theN(1535) particle.Figure 3. (color online) The subtraction constants
aπN ,aηN ,aKΛ ,aKΣ with the regularization scaleμ=630 MeV in the loop function in Eq. (24).A pole is generated dynamically at
1518−i46 MeV in the complex energy plane of√s when the Bethe-Salpeter equation is solved in the unitary coupled-channel approximation with the 19th set of parameters, i.e.aπN=−2.0 ,aηN=−1.7 ,aKΛ=−3.2 andaKΣ=−3.2 . The amplitude squared|T|2 as a function of the total energy√s for different channels with isospinI=1/2 and strangenessS=0 is depicted in Fig. 4. The real part of the pole position is higher than theηN threshold, and is lower than theKΛ threshold, so we assume that it might be a resonance state corresponding to the N(1535) particle of the PDG review.Figure 4. The amplitude squared
|T|2 as a function of the total energy√s for different channels with isospinI=1/2 and strangenessS=0 . TheπN ,ηN andKΣ channels are labeled in the figure, while theKΛ channel is drawn with the dashed line.The couplings of the N(1535) resonance state to different channels are listed in Table 3. If different sets of subtraction constants are used, the changes of couplings are not significant, as shown in Table 3.
In Ref. [7], the
N(1535) particle couples more strongly to theK+Λ channel, which is different from the results listed in Table 3. The different values of the coupling constants might be relevant to the next-to-leading-order chiral Lagrangian used in Ref. [7], while it is not included in this work.n aπN aηN aKΛ aKΣ Pole position(MeV) gπN |gπN| gηN |gηN| gKΛ |gKΛ| gKΣ |gKΣ| 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 Table 3. The subtraction constants
aπN ,aηN ,aKΛ ,aKΣ , the pole positions in the complex energy plane of√s and the couplings of the resonance states to different channels. The regularization scale is fixed toμ=630 MeV in the loop function in Eq. (24).Table 3-continued from previous page n aπN aηN aKΛ aKΣ Pole position(MeV) gπN |gπN| gηN |gηN| gKΛ |gKΛ| gKΣ |gKΣ| 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 -
The wave functions with isospin
I=3/2 and strangenessS=0 can be written as|πN;32,−12⟩=√23|π0n⟩+√13|π−p⟩,
(31) and
|KΣ;32,−12⟩=√23|K0Σ0⟩+√13|K+Σ−⟩.
(32) From Eqs. (31) and (32), the coefficients
Cij in the isospin space can be calculated and are listed in Table 4. Since the coefficients are all negative, the Weinberg-Tomozawa contact interaction between the pseudoscalar meson and the baryon octet is repulsive for isospinI=3/2 and strangenessS=0 . Even when the correction from thes− channel andu− channel interaction is taken into account, the total potential is still repulsive, as shown in Fig. 5. Thus, no resonance state can be generated in the S-wave approximation.Cij πN KΣ πN −1 −1 KΣ −1 Table 4. The coefficients
Cij in the pseudoscalar meson and baryon octet interaction with isospinI=32 and strangenessS=0 ,Cji=Cij .Figure 5. The potential of the pseudoscalar meson and baryon octet interaction as a function of the total energy of the system
√s in the sector of isospinI=3/2 and strangenessS=0 . The solid lines denote the contact interaction, while the dashed lines stand for the total S-wave potential from Eq. (18). -
In this work, the interaction of the pseudoscalar meson and the baryon octet was 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 strangenessS=0 , a resonance state is generated dynamically by solving the Bethe-Salpeter equation, which might be regarded as theN(1535) particle listed in the PDG review. We find that hidden strange channels, such asηN ,KΛ andKΣ , play an important role in the generation of this 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 were calculated, and it was found that it couples strongly to the hidden strange channels.Bao-Xi Sun would like to thank Han-Qing Zheng and Yu-Fei Wang for useful discussions.
Pseudoscalar meson and baryon octet interaction with strangeness zero in the unitary coupled-channel approximation
- Received Date: 2019-01-21
- Available Online: 2019-06-01
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