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
and
channel potentials in the S-wave approximation are taken into account. In the sector of isospin
and strangeness
, a pole is detected in a reasonable region of the complex energy plane of
in the center-of-mass frame by analyzing the behavior of the scattering amplitude, which is higher than the
threshold and lies on the third Riemann sheet. Thus, it can be regarded as a resonance state and might correspond to the
particle of the Particle Data Group (PDG) review. The coupling constants of this resonance state to the
,
,
and
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 resonance states with strangeness zero. The interaction of the pseudoscalar meson and the baryon octet is repulsive in the sector of isospin
and strangeness
, so that no resonance state can be generated dynamically.
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
channel,
and
, 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 the
flavor space, as are these two particles. In Ref. [1], it is pointed out that the hidden strange channels of
and
might play an important role in the dynamical generation of the
particle.
The
particle, generated dynamically in the unitary coupled-channel approximation with a three-body
final state, was considered in [2]. However, the inclusion of
as a final state in the calculation is complex, especially as there are six arbitrary constants in the real part of the three-body
loop function, and thus it must be treated as a free function consistent with experimental data. This approach was studied again by including the
and
channels in a non-relativistic approximation, besides the pseudoscalar meson-baryon octet channels [3]. In fact, the elastic scattering process
gives the main contribution to the dynamical generation of the
resonance, as was shown in Ref. [4]. In the processes
and
, the Kroll-Ruderman term supplies a constant potential and plays a dominant role, while the
-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
and
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
and
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 the
partial wave amplitude with the experimental data up to the energy
GeV, the resonance state corresponding to the
particle is generated dynamically. In addition, it is amazing that the
resonance state can be produced at the same time at higher energies.
The properties of the
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 the
,
and
resonance states were investigated, and it was announced that the
,
,
and
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
and
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
and strangeness
, a pole at
MeV in the complex energy plane is detected, which might correspond to the
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
channel and
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 isospin
and
are discussed in Section 4 and Section 5, respectively. Finally, the summary is given in Section 6.
Framework
The effective Lagrangian of the pseudoscalar meson and baryon octet interaction can be written as
In the above equation, the symbol
denotes the trace of matrices in the
flavor space, and
with
and
, where D=0.80, F=0.46 and
=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
and
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
and
give a contribution to the
and
channel interactions, as shown in Fig. 1.
Feynman diagrams of the pseudoscalar meson-baryon octet interaction.
contact term,
channel and
channel.
According to the Feynmann rules, the Weinberg-Tomozawa contact potential of the pseudoscalar meson and baryon octet interaction can be written as
where
are the momenta of the initial and final baryons (mesons), and
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 as
As
and
are the wave functions of the initial and final baryons, the matrix
in Eq. (5) can be replaced by the unit matrix
in the low energy region, i.e.,
. Finally, the Weinberg-Tomozawa contact potential of the pseudoscalar meson and baryon octet interaction takes the form
where
is the total energy of the system,
and
denote the initial and final baryon masses, while
and
stand for the initial and final baryon energies in the center-of-mass frame. The coefficients
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 with
,
and
MeV, as given in Ref. [2, 7].
The coefficients
in the pseudoscalar meson and baryon octet interaction with strangeness
and charge
,
.
1
0
0
0
1
0
0
0
0
The second term in Eq. (1) represents the antibaryon-baryon-meson vertices, and can be rewritten as
with
and
.
The coefficient
in Eq. (7) takes the form
where
with
the matrix of the SU(3) generator and
Thus, the
and
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
channel potential of the pseudoscalar meson and baryon octet interaction can be written approximately as
where
denotes the mass of the intermediate baryon,
and
represent the coefficients given in Eq. (8).
Similarly, the
channel potential can be obtained as
with the Mandelstam variable
.
In the calculation of Eqs. (11) and (12), a physical baryon mass is adopted so as to obtain the
-channel and
-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
channel potential of the pseudoscalar meson and baryon octet interaction are only related to the Mandelstam variable
, 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.,
with
the
th Legendre polynomial and the coefficients
In the S-wave approximation, only the coefficient
is necessary.
The denominator
in Eq. (12) can be written as
where
is the angle between the three-momenta of incoming and outgoing mesons, and
and
are the three-momentum in the center-of-mass frame and the mass of the initial baryon (final meson). Taking
and
, we obtain
Thus, the
channel potential of the pseudoscalar meson and baryon octet interaction in the S-wave approximation can be easily calculated as
Therefore, the S-wave potential of the pseudoscalar meson and baryon octet interaction can be written as
Bethe-Salpeter equation
The Bethe-Salpeter equation can be expanded as
When the Bethe-Salpeter equation in Eq. (19) is solved, only the on-shell part of the potential
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 mesons
and
. A more detailed discussion can be found in Refs. [13, 14]. Therefore, if the potential in Eq. (5) is adopted, the second term
in Eq. (19) can be written as
If the relativistic kinetic correction of the loop function of the pseudoscalar meson and baryon octet interaction is taken into account, the loop function
can be written as
with
the total momentum of the system,
the meson mass, and
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
where
is the subtraction constant,
is the regularization scale, and
is the loop function in Ref. [15],
with
the three-momentum of the meson or the baryon in the center-of-mass frame.
Since the total three-momentum
in the center-of-mass frame, only the
part remains in Eq. (22). Similarly, the matrix
can be replaced by the unit matrix
since
and
are the wave functions of the initial and final baryons. Thus, the loop function of the intermediate pseudoscalar meson and baryon octet becomes
When the
channel and
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 the sector of isospin
and strangeness
, the wave function in the isospin space can be written as
and
Thus, the coefficients
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.
The coefficients
in the pseudoscalar meson and baryon octet interaction with isospin
and strangeness
,
.
2
0
0
0
0
2
The
channel,
channel and Weinberg-Tomozawa contact potentials in the S-wave approximation are depicted in Fig. 2. From Fig. 2, it is seen that the
channel potential is repulsive and the other
channel potentials are weaker than the
potential, while all
channel potentials in the S-wave approximation are attractive. Although the curves for
and
are not smooth when
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 the
channel potential and the S-wave
channel potential is not important.
Potentials of the pseudoscalar meson and baryon octet interaction as functions of the total energy of the system
in the sector of isospin
and strangeness
. (left):
channel. (middle):
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
and strangeness
are depicted in the right side of Fig. 2, which shows that the
and
potentials are attractive, while the
and
interactions are weak.
Although the
channel and
channel potentials are weaker than the Weinberg-Tomozawa contact interaction in the sector of isospin
and strangeness
, the subtraction constants must be readjusted in the calculation when the contribution of the
and
channel potentials are taken into account.
According to the PDG data, the
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
[12]. When the Bethe-Salpeter equation is solved in the unitary coupled-channel approximation, we set the regularization scale to
MeV, just as is done in most works using this method[3, 4, 11, 14]. Moreover, all subtraction constants change from
to
with a step of
, 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
, which is thought to be a natural value. We changed the subtraction constants in the neighborhood of
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
,
,
and
are also listed. A resonance state with a mass of about
MeV and a decay width of about
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 constant
changes from
to
in successive steps . Since the
threshold is far lower than the energy region where the
particle might be generated dynamically, it is understandable that the pole position is not sensitive to the value of the subtraction constant
. The changes of the other three subtraction constants
,
and
are not so large. Especially, the subtraction constant
in 38 sets of parameters, while for the eighth set it takes the value of
. The
threshold is close to the energy region we are interested in, and as the subtraction constant
is stable, it plays an important role in the generation of the
particle.
(color online) The subtraction constants
,
,
,
with the regularization scale
MeV in the loop function in Eq. (24).
A pole is generated dynamically at
MeV in the complex energy plane of
when the Bethe-Salpeter equation is solved in the unitary coupled-channel approximation with the 19th set of parameters, i.e.
,
,
and
. The amplitude squared
as a function of the total energy
for different channels with isospin
and strangeness
is depicted in Fig. 4. The real part of the pole position is higher than the
threshold, and is lower than the
threshold, so we assume that it might be a resonance state corresponding to the N(1535) particle of the PDG review.
The amplitude squared
as a function of the total energy
for different channels with isospin
and strangeness
. The
,
and
channels are labeled in the figure, while the
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
particle couples more strongly to the
channel, which is different from the results listed in Table3. 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.
The subtraction constants
,
,
,
, the pole positions in the complex energy plane of
and the couplings of the resonance states to different channels. The regularization scale is fixed to
MeV in the loop function in Eq. (24).
The wave functions with isospin
and strangeness
can be written as
and
From Eqs. (31) and (32), the coefficients
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 isospin
and strangeness
. Even when the correction from the
channel and
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.
The coefficients
in the pseudoscalar meson and baryon octet interaction with isospin
and strangeness
,
.
The potential of the pseudoscalar meson and baryon octet interaction as a function of the total energy of the system
in the sector of isospin
and strangeness
. The solid lines denote the contact interaction, while the dashed lines stand for the total S-wave potential from Eq. (18).
Summary
In this work, the interaction of the pseudoscalar meson and the baryon octet was studied within a nonlinear realized Lagrangian. The
,
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
and strangeness
, a resonance state is generated dynamically by solving the Bethe-Salpeter equation, which might be regarded as the
particle listed in the PDG review. We find that hidden strange channels, such as
,
and
, 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.