Strange dibaryon resonance in the K̄NN-πY N system
Y. Ikeda and T. Sato∗
Department of Physics, Graduate School of Science,
Osaka University, Toyonaka, Osaka 560-0043, Japan
(Dated: November 4, 2018)
Abstract
Three-body resonances in the K̄NN system have been studied within a framework of the
K̄NN − πY N coupled-channel Faddeev equation. By solving the three-body equation, the en-
ergy dependence of the resonant K̄N amplitude is fully taken into account. The S-matrix pole has
been investigated from the eigenvalue of the kernel with the analytic continuation of the scattering
amplitude on the unphysical Riemann sheet. The K̄N interaction is constructed from the leading
order term of the chiral Lagrangian using relativistic kinematics. The Λ(1405) resonance is dy-
namically generated in this model, where the K̄N interaction parameters are fitted to the data of
scattering length. As a result we find a three-body resonance of the strange dibaryon system with
binding energy, B ∼ 79 MeV, and width, Γ ∼ 74 MeV. The energy of the three-body resonance is
found to be sensitive to the model of the I = 0 K̄N interaction.
PACS numbers: 11.30.Rd, 11.80.Jy, 13.75.Jz
∗Electronic address: ikeda@kern.phys.sci.osaka-u.ac.jp, tsato@phys.sci.osaka-u.ac.jp
http://arxiv.org/abs/0704.1978v2
mailto:ikeda@kern.phys.sci.osaka-u.ac.jp, tsato@phys.sci.osaka-u.ac.jp
I. INTRODUCTION
The analysis of the kaonic atoms [1] revealed an attractive K̄-nucleus interaction. Al-
though the strength of the attraction depends on the parametrization of the density de-
pendence of the optical potential [1] and the theoretical study of the K̄ optical potential
suggests a rather shallow potential [2], there has been a great interest in the possibilities
of K̄-nucleus bound states in recent years. Akaishi and Yamazaki [3, 4] studied the kaon
bound states in light nuclei and found deeply bound kaonic states, for example, B ∼ 100
MeV for 3
H. In their study, the kaonic nuclear states were investigated by using the K̄
optical potential, which is constructed by folding the g matrix with a trial nuclear density.
The potential model of K̄N − πΣ interaction is determined to reproduce the Λ(1405) and
the scattering length. The kaonic nuclear states are further studied by using a method of
antisymmetrised molecular dynamics [5] using the K̄N g matrix.
Among the simplest K̄ nucleus state, theK−pp state, which has strangeness S = −1, total
angular momentum and parity Jπ = 0−, and isospin I = 1/2 dibaryon state, is expected to
have largest component of the I = 0 K̄N . An experimental signal of theK−pp bound state is
reported by the FINUDA Collaboration from the analysis of the invariant mass distribution
of Λ − p in the K− absorption reaction on nuclei [6]. The reported central value of the
binding energy, B, and the width, Γ, are (B,Γ) = (115, 67) MeV, which is below the πΣN
threshold energy. This data may be compared with the predicted values (B,Γ) = (48, 61)
MeV in Ref. [4]. However it was pointed out that the data can be understood by the two-
nucleon absorption of K− in nuclei together with the final state interaction of the outgoing
baryons [7].
In the attractive interaction of kaon in nuclei, the resonance Λ(1405) in the s-wave and
I = 0 channel K̄N scattering state plays an essential role. The energy of the Λ(1405) is
below the K̄N threshold and strongly couples with the πΣ state. Although the kaonic nuclear
states have been studied so far by using the K̄N g matrix or optical potential, it might be
very important to examine the full dynamical calculation of K̄N−πΣ system by taking into
account the energy dependence of the resonance t matrix and the coupling with the K̄N−πΣ
channel explicitly. Such a theoretical study may be possible in the simplest kaonic nuclei
with baryon number B = 2 system. In this work, we study the strange dibaryon system by
taking into account the three-body dynamics using the K̄NN−πΣN−πΛN (K̄NN−πY N)
coupled-channel Faddeev equation with relativistic and non-relativistic kinematics.
Methods to investigate resonances in the three-body system have been developed in the
studies of the three-neutron [8, 9], πNN dibaryon [10, 11] and ΣNN hypernuclei [10, 12, 13].
In this work, we employ a method started by Glöckle [8] and Möller [9] and developed by
Matsuyama and Yazaki, Afnan, Pearce and Gibson [10, 12, 13] to find a pole of the S matrix
in the unphysical energy plane from the eigenvalue of the kernel of the Faddeev equation.
To analytically continue the scattering amplitude into the unphysical sheet, the path of
the momentum integral must be carefully deformed in the complex plane to avoid possible
singularities.
The most important interaction for the study of the strange dibaryon system is for the
I = 0 K̄N states. The internal structure of the Λ(1405) has been a long standing is-
sue. The chiral Lagrangian [14, 15, 16] approach can describe well the low energy K̄N
reaction with the meson-baryon dynamics. A genuine q3 picture of the Λ(1405) coupled
with meson-baryon [17] may not yet be excluded. Though previous studies of the K̄NN
system used phenomenological models of the K̄N potentials, we use s-wave meson-baryon
coupled-channel potentials guided by the lowest order chiral Lagrangian. With this model,
the strength of the potentials and the relative strength of the potentials among various
meson-baryon channels are not parameters but are determined from the SU(3) structure of
the chiral Lagrangian. In this model, the Λ(1405) is an ’unstable bound state’, whose pole
on the unphysical sheet will become the bound state of K̄N when the coupling between the
K̄N and the πY is turned off. We examine a relativistic model as well as a nonrelativistic
model to account for the relativistic energy of pion in the πY N state.
We briefly explain our K̄NN − πY N coupled-channel equations and the procedure to
search for the three-body resonance in section 2. The model of the two-body interactions
used in this work is explained in section 3. We then report our results on the K̄NN
dibaryon resonance in section 4. This work is the extension of the early version of our analysis
reported in Ref. [18]. Recently Shevchenko et al. [19] performed a similar study of the K̄NN
system using Faddeev equation starting form the phenomenological K̄N interaction within
a nonrelativistic framework. The comparison of our results with theirs will be discussed in
section 4.
II. COUPLED CHANNEL FADDEEV EQUATION AND RESONANCE POLE
We start from the Alt-Grassberger-Sandhas(AGS) equation [20] for the three-body scat-
tering problem. The operators Ui,j of the three-body scattering satisfy the AGS equation
Ui,j = (1− δi,j)G−10 +
n 6=i
tnG0Un,j . (1)
Here we label the pair of particles j, k by the spectator particle i = 1, 2, 3. The two-body
t matrix ti of particles j, k with the spectator particle i is given by the solution of the
Lippmann-Schwinger equation:
ti = vi + viG0ti. (2)
Here G0 = 1/(W −H0 + iǫ) is the free Green’s function of the three particles, and W is the
total energy of the three-body system.
When the two-body interactions vi are given in separable form with the vertex form factor
|gi > and the coupling constant γi as
vi = |gi > γi < gi|, (3)
the AGS-equation of Eq. (1) is written in the form
Xi,j(~pi, ~pj,W ) = (1− δi,j)Zi,j(~pi, ~pj,W ) +
n 6=i
d~pnZi,n(~pi, ~pn,W )τn(W )Xn,j(~pn, ~pj,W ).
The amplitude Xi,j is defined by the matrix element of Ui,j between state vectors G0|~pi, gi >
Xi,j(~pi, ~pj,W ) = < ~pi, gi|G0Ui,jG0|~pj, gj > . (5)
The state vector |~pi, gi > represents a plane wave state of the spectator i and the state vector
|gi > of the interacting pair.
The driving term Zi,j of Eq. (4) shown in Fig. 1(a) is given by the particle exchange
mechanism defined as
Zi,j(~pi, ~pj,W ) = < ~pi, gi|G0|~pj, gj > (6)
g∗(~qi)g(~qj)
W −Ei(~pi)−Ej(~pj)−Ek(~pk)
. (7)
FIG. 1: Graphical representation of (a) one particle exchange interaction Zi,j(~pi, ~pj ,W ) and (b)
two-body t-matrix τi(W ). The relative momentum of the interacting particles is given by ~qi for
spectator particle i.
Here the momentum of the exchanged particle k( 6= i, j) is given as ~pk = −~pi − ~pj and g(~qi)
is the vertex form factor of the two-body interaction g(~qi) =< gi|~qi >. The energy Ei(~pi) is
given by Ei(~pi) = mi + ~p
i /2mi for the nonrelativistic model and Ei(~pi) =
m2i + ~p
i for the
relativistic model. The relative momentum is given by ~qi = (mk~pj −mj~pk)/(mj +mk) for
the nonrelativistic model, while we define qi = |~qi| for the relativistic model as
W 2i +m
j −m2k
)2 −m2j , (8)
(Ej(~pj) + Ek(~pk))2 − ~p2i . (9)
The two-body t matrix can be solved for the separable interaction as
ti = |gi > τi(W ) < gi|. (10)
Here the ’isobar’ propagator τi, illustrated in Fig. 1(b), is given as
τi(W ) = [1/γi −
|gi(~qi)|2
W −Ei(~pi)− Ejk(~pi, ~qi)
]−1. (11)
The two-body t matrix depends on the energy Ei(~pi) of the spectator particle. Here Ejk is
the energy of the interacting pair given as Ejk(~pi, ~qi) = mj +mk + ~p
i /(mj +mk) + ~q
i /µi for
the non-relativistic model, while Ejk(~pi, ~qi) =
(Ej(~qi) + Ek(~qi))2 + ~p
i for the relativistic
model. The reduced mass is defined as µi = mjmk/(mj +mk).
Following the standard method of angular momentum expansion [21], the AGS equation
reduces to following coupled integral equations by keeping only s-wave states:
Xi,j(pi, pj,W ) = Zi,j(pi, pj,W ) +
× Ki,n(pi, pn,W )Xn,j(pn, pj ,W ). (12)
Here we used a simplified notation for the kernel K = Zτ , which can be written as
Ki,n(pi, pn,W ) = 2π
d(p̂i · p̂n)
g∗(qi)g(qn)
W −Ei(pi)− Ej(pn)− Ek(~pi + ~pn)
τn(W ). (13)
The formulas given above are valid for the spinless and distinguishable particles without
channel coupling among the Fock-space vectors. In our K̄NN resonance problem, we have
included the following K̄NN and πY N states:
|a > = |N1, N2, K̄3 >, (14)
|b > = |N1, Y2, π3 >, (15)
|c > = |Y1, N2, π3 >, (16)
with Yi is Σi or Λi. After anti-symmetrizing the amplitude for identical particles of nucle-
ons [11], we obtain the following forms of the coupled AGS equations,
XYK ,YK
XYπ,YK
Xd,YK
XN∗,YK
ZYK ,YK
Zd,YK
−ZYK ,YKτYK ,YK −ZYK ,YKτYK ,Yπ 2ZYK ,dτd,d 0
0 0 0 −ZYπ ,N∗τN∗,N∗
Zd,YKτYK ,YK Zd,YKτYK ,Yπ 0 0
−ZN∗,YπτYπ ,YK −ZN∗,YπτYπ,Yπ 0 0
XYK ,YK
XYπ,YK
Xd,YK
XN∗,YK
. (17)
Here we have suppressed the spin-isospin quantum numbers, the spectator momentum pj
and the total energy of the three-body system W in Z, X and τ for simplicity. The concise
notation of YK , Yπ, d and N
∗ represents the ’isobars’ and their decay channels. The decay
channels of isobars YK , Yπ, d and N
∗ are K̄N(I = 0, 1), πΣ(I = 0, 1) and πΛ(I = 1),
NN(I = 1) and πN(I = 1/2, 3/2), respectively. Here I is isobar isospin. Those indices
uniquely specify the three-body states of X and Z except N∗ showing ΣN∗ and ΛN∗.
Therefore we have a nine-channel coupled equation of Eq. (17) for spin singlet, s-wave three-
body system. The explicit form of Eq. (7) when we include spin-isospin is summarized in
the Appendix.
The dominant Fock space component is expected to be |K̄NN >, and therefore the
most important amplitudes are XYK ,YK and Xd,YK . They couple to each other through
the kaon exchange ZYK ,YK and nucleon exchange ZYK ,d mechanisms. Notice, however, the
πY N component is also implicitly included in τYK ,YK when we solve the two-body K̄N −
πY coupled-channel equations. The πY N components, XY π,YK and XN∗,YK , couple with
the K̄NN components through the pion exchange mechanism ZN∗,Yπ and the πN and πY
’isobars’ τN∗,N∗ , τYπ,Y . The pion exchange mechanism may play an important role in the
width of the resonance. In this work, we have not included weak Y N interaction. It was
found in Ref. [19] that the Y N interaction plays rather minor role in this strange dibaryon
system.
To find the resonance energy of the three-body system using the AGS equation of Eq.
(17), we follow the method used in Refs. [8, 9, 10, 12, 13]. The AGS equation of Eq. (13)
is a Fredholm-type integral equation with the kernel K = Zτ . Using the eigenvalue ηa(W )
and the eigenfunction |φa(W ) > of the kernel for given energy W ,
Zτ |φa(W ) > = ηa(W )|φa(W ) >, (18)
the scattering amplitude X can be written as
|φa(W ) >< φa(W )|Z
1− ηa(W )
. (19)
At the energy W = Wp where ηa(Wp) = 1, the amplitude has a pole, and therefore Wp gives
the bound state or resonance energy.
Since a resonance pole appears on the unphysical energy Riemann sheet, we need analytic
continuation of the scattering amplitude. We use here the nonrelativistic model to explain
a method of analytic continuation, which is based on Refs. [9, 10]. At first we examine the
singularities of the kernel of Eq. (13). Above the threshold energy of the three-body break
up W > mi+mj+mk, Z(pi, pn,W ) has logarithmic singularities. The branch points appear
FIG. 2: The singularities of the one particle exchange interaction Z(pi, pn,W ) in the complex pn
plane at W = E + iǫ and the real pi.
at pn = ±pZ1,2 , where
pZ1 = −
2µjWth −
p2i , (20)
pZ2 = +
2µjWth −
p2i , (21)
mi +mk
mj(mi +mk)
mi +mj +mk
Wth = W −mi −mj −mk.
For given pi > 0, the cuts run from pZ1 to pZ2 above the positive real axis of complex pn
plane and from −pZ1 to −pZ2 below the negative real axis as shown in Fig. 2, while the
integration of momentum pn in Eq. (13) is along the real positive axis.
Let us consider the case when W has a negative imaginary part. For given pi > 0, the
cut from pZ1 to pZ2 moves into the fourth quadrant across the integration contour of pn.
Assuming the integrand of Eq. (12) is an analytic function around real positive pn, one can
perform an analytic continuation of the amplitudes by deforming the integration contour
along the logarithmic singularity as shown in Fig. 3 and then we obtain amplitudes on the
unphysical Riemann sheet.
FIG. 3: The integration contour C and the singularity of Z at W = E − iΓ/2 and real value of pi.
In principle it might be possible to solve the AGS equation keeping the momentum vari-
ables real and taking into account the discontinuity across the cut. The moving logarithmic
singularities depending on pi make it difficult to solve the integral equation. To overcome
this problem we deform the integration contour of pi, pn, into the fourth quadrant of the
complex momentum plane so that we take into account the contribution of the cuts. As an
example of our K̄NN − πY N problem, we choose the integration contour of pn as shown in
solid line in Fig. 4. Here we take the energy W = 10 − i35 +mπ +mΣ +mN MeV, which
is below the mass of K̄NN and above the πY N . The shaded region in Fig. 4 shows the
cuts of ’Z’ for the pion exchange mechanism. The cuts become ’forbidden regions’ because
the position of the cuts depends on pi, which runs the same integration contour as pn. In
our numerical calculation, we studied all the ’forbidden regions’ for π,N and K exchange
mechanism and determined the integration contour. With the integration contour C in Fig.
4, we choose the physical sheet of K̄NN .
The singularities of the isobar propagator τ(W ) arises from the three-body Green’s func-
tion in the integrand of τ . The poles are at qn = ±
2µjWth − µjηj p
i . Since qs = (pZ1+pZ2)/2,
we can analytically continue it into the same unphysical sheet as the case in Z as long as
we keep the same deformed contour as the one used in Z. Another singularity we have to
worry about is the singularity due to the two-body resonance. Since our K̄N − πΣ system
has the two-body resonance Λ(1405), the cut starts from the two-body resonance energy in
the complex energy plane. To examine this, we write the approximate energy dependence
FIG. 4: The logarithmic singularities of the π exchange mechanism Z(pi, pn, Z) at Wth = 10− i35
MeV in the complex pn plane. C is integration contour of pn and pi.
of the τ as
τi(W ) ∼
W − p
− EΛ∗ −mN
. (22)
Here pN and mN are the momentum and mass of the spectator nucleon. The reduced mass
of the spectator nucleon with the isobar pair K̄N or πΣ is denoted as ηN and EΛ∗ is the
pole energy of Λ(1405). At W =
+ EΛ∗ +mN with pN on the contour C in Figs. 5 (b),
the two-body t-matrix has a singularity, which is plotted as a solid line in Fig. 5(a). We
illustrate the typical trajectories of the three-body resonance pole W = Wp as curves A and
B in Fig. 5. If the pole trajectories A and B intercept the two-body NΛ(1405) cut, then
the analytic continuation to the NΛ(1405) unphysical energy sheet must be examined. The
same situation from the pN plane is shown in Fig. 5 (b). The momentum p
∗ corresponding
to the energy Wp of the three-body resonance is determined by
p∗ = ±
2ηN(Wp − EΛ∗ −mN). (23)
If p∗ intercepts the contour C, we have to take care of the analytic continuation of the
NΛ(1405) energy sheet. As will be seen in section 4, the trajectories of the three-body
resonance in our calculation follow line A of Fig. 5(a) and do not intercept the singularity
of the two-body resonance.
FIG. 5: The singularities due to the three-body resonance and the Λ(1405) in (a) the complex
energy plane and (b) the momentum plane.
III. MODEL OF THE TWO-BODY INTERACTIONS
We take into account the K̄N interactions in Jπ = 1/2−, I = 0 and I = 1 states, the
πN interactions in Jπ = 1/2−, I = 1/2 and 3/2 states and the NN interaction in I = 0,1 S0
state. Our s-wave meson-baryon interaction is guided by the leading order effective chiral
Lagrangian for the octet baryon ψB and the pseudoscalar meson φ fields given as
Lint =
8F 2π
tr(ψ̄Bγ
µ[[φ, ∂µφ], ψB]). (24)
The meson-baryon potential derived from the chiral Lagrangian can be written as
< ~p′, β|VBM |~p, α > = −Cβ,α
(2π)38F 2π
mβ +mα
4Eβ(~p′)Eα(~p)
× gβ(~p′)gα(~p). (25)
Here ~p and ~p′ are the momentum of the meson in the initial state α and the final state
β. The strength of the potential at zero momentum is not an arbitrary constant but is
determined by the pion decay constant Fπ. The relative strength between the meson-baryon
states is controlled by the constants Cβ,α which are basically determined by the SU(3) flavor
structure of the chiral Lagrangian. The parameter of our model is the cutoff Λ of the
phenomenologically introduced vertex function gα(~p) = Λ
α/(~p
2 + Λ2α)
The most important interaction for the study of the K̄NN system is the I = 0 K̄N
interaction. We describe the K̄N interaction by the coupled-channel model of the K̄N and
the πΣ states. The constants Cβ,α for this channel are given as CK̄N−K̄N = 6, CK̄N−πΣ =
6 and CπΣ−πΣ = 8. The cutoff Λ is determined by fitting the scattering length a
−1.70 + i0.68 fm of Ref. [22]. The values of Λ are around 1 GeV and are given as model
(a) in tables I and II for the nonrelativistic and the relativistic models. In general, the form
factors of the relativistic models are hard compared with those of the non-relativistic models
because of the weak relativistic kinetic energy. We found a resonance pole atW = 1420−i30
MeV for the non-relativistic and the relativistic models. The relativistic kinematics might
be important in describing πY channel because of the small pion mass. We choose this
model (a) as a standard parameter of the K̄N interaction.
The K̄N scattering lengths are not very well constrained from the data. The ranges of the
K̄N scattering lengths are studied within the chiral unitary model in Ref. [23]. In this work,
we simply examined models with the scattering length aI=0
= (−1.70±0.10)+ i(0.68±0.10)
fm in order to examine the sensitivity of the energy of the three-body resonance on the input
model of the two-body interaction. The cutoff Λ’s for those models are given as models (b)-
(e) of Tables I and II. The values of the resonance energy are about 1415 ∼ 1425 MeV, and
the width 50 ∼ 70 MeV, which are close to the values of the chiral model in Ref. [16]. One
can notice that there is a correlation between the real (imaginary) part of the pole energy
of the Λ(1405) and the imaginary (real) part of the scattering length. Those resonance
energies are slightly larger than the pole energy reported in Ref. [24]. Therefore as a last
model, model (f) reproduces the deeper resonance energy 1406− i25 MeV of Ref. [24]. The
scattering length of this model is −1.72 + i0.44 fm, which is, however, somewhat different
from the value −1.54 + i0.74 fm in Ref. [24].
The I = 1 K̄N interaction is described by the K̄N − πΣ − πΛ coupled-channel model.
The coupling constants Cβ,α are CK̄N−K̄N = 2, CK̄N−πΣ = −2, CK̄N−πΛ = −
6, CπΣ−πΣ = 4
and CπΣ−πΛ = CπΛ−πΛ = 0. The cutoff Λ’s are determined to fit the imaginary part of the
scattering length of Ref. [22], which are given as model (A) in Tables III and IV for the non-
relativistic and the relativistic models. The real part of the scattering length of those models
is larger than aI=1
= 0.37+ i0.60 fm of Ref. [22]. The K−p scattering length predicted from
model (aA) , which is model (a) for I = 0 and model (A) for I = 1 interactions, is between
the central values of the two kaonic hydrogen data [25, 26, 27]. To study the sensitivity of
the models of I = 1 K̄N interaction to the resonance energy ofK−pp system, we constructed
model (B) given in Tables III and IV. A similar model of K̄N interaction is developed to
study K−d scattering[28]. The range of the vertex form factor found in in Ref. [28], which
is monopole form factor with 880MeV cutoff mass, is comparable to ours.
The total cross sections of K−p reactions predicted from our models (aA), (aB) and (fA)
are shown in Fig. 6 together with the data [29, 30, 31, 32, 33]. The models (aA) and (aB)
describe well the K−p → K−p(Fig. 6a), K−p → π+Σ−(Fig. 6b) and K−p → π−Σ+(Fig.
6c) reactions, where both I = 0 and I = 1 interactions contribute to the cross section. The
models of I = 0 (I = 1) can be tested from K−p → π0Σ0(Fig. 6d) (K−p → π0Λ(Fig.
6e)) reactions, where models ’a’ and ’A/B’ describe the cross sections well. The model (fA)
tends to give smaller cross sections. It is noticed, however, as we will see, that the resonance
energy of the K−pp system is more sensitive to the I = 0 K̄N interaction and less sensitive
to I = 1 interactions, while both I = 0 and I = 1 interactions are equally important to
describe K−p cross sections and kaonic hydrogen data.
K̄N(MeV) πΣ(MeV) Scattering Length(fm) Resonance energy(MeV)
(a) 1095 1450 −1.70 + i0.68 1419.8 − i29.4
(b) 1105 1550 −1.60 + i0.68 1422.2 − i33.7
(c) 1085 1350 −1.80 + i0.68 1418.5 − i25.0
(d) 1120 1340 −1.70 + i0.59 1414.6 − i29.4
(e) 1070 1540 −1.70 + i0.78 1424.3 − i28.3
(f) 1160 1100 −1.72 + i0.44 1405.8 − i25.2
TABLE I: The cutoff parameters, scattering length and the resonance pole of the relativistic models
of I=0 K̄N − πΣ interaction.
The form of the s-wave πN interactions is taken as Eq. (25). The constant Cα,β is 4
for I = 1/2 and −2 for I = 3/2 states. The parameters of the potentials are determined
by fitting the scattering length and the low energy phase shifts. For I = 1/2 state, the
strength of the potential is modified as λCβ,α by introducing a phenomenological factor λ to
describe the data of the scattering length (0.1788±0.0050)m−1π [34] and the phase shifts [35].
The fitted parameters λ and Λ are shown in Table V together with the scattering length
K̄N(MeV) πΣ(MeV) Scattering Length(fm) Resonance energy(MeV)
(a) 946 988 −1.70 + i0.68 1420.1 − i30.1
(b) 954 1035 −1.60 + i0.68 1422.4 − i34.7
(c) 940 944 −1.80 + i0.68 1418.7 − i26.0
(d) 968 933 −1.70 + i0.58 1414.3 − i30.5
(e) 927 1031 −1.70 + i0.78 1424.7 − i29.0
(f) 1000 800 −1.72 + i0.43 1404.8 − i25.5
TABLE II: The cutoff parameters, scattering length and the resonance pole of the nonrelativistic
models of I=0 K̄N − πΣ interaction.
K̄N(MeV) πΣ(MeV) πΛ(MeV) Scattering Length(fm)
(A) 1100 850 1250 0.68 + i0.60
(B) 950 800 1250 0.65 + i0.46
TABLE III: The cutoff parameters, scattering length of the relativistic models of I=1 K̄N − πY
interaction.
calculated using the models. The models describe well the S11 phase shifts up to 1.2 GeV
as shown in Fig. 7.
For the I = 3/2 πN scattering, the πN potential is constructed so as to reproduce the
scattering length (−0.0927 ± 0.0093)m−1π [34] and the S31 partial wave phase shifts data.
Here we introduced a modified dipole form factor as
g(~p) =
(~p2 + Λ2)2
× (1 + a~p2). (26)
K̄N(MeV) πΣ(MeV) πΛ(MeV) Scattering Length(fm)
(A) 920 960 640 0.72 + i0.59
(B) 800 940 660 0.68 + i0.45
TABLE IV: The cutoff parameters, scattering length of the nonrelativistic models of I=1 K̄N−πY
interaction.
FIG. 6: The total cross section of (a) K−p → K−p, (b) K−p → π+Σ− (c) K−p → π−Σ+ (d)
K−p → π0Σ0 and (e) K−p → π0Λ reactions in the relativistic model. The solid (dashed, dotted)
curve shows the result using model (aA) ((aB), (fA)). Data are taken from Ref. [29, 30, 31, 32, 33].
λ Λ(MeV) scattering length
Relativistic 0.90 800 0.175m−1π
Nonrelativistic 0.85 800 0.177m−1π
TABLE V: Parameters and scattering length for the relativistic and nonrelativistic model of I=1/2
πN interaction.
The parameters of the model are Λ and a for the form factor and the strength parameter λ.
The obtained parameters are summarized in Table VI. The relativistic model can describe
well the phase shifts up to 1.2 GeV as shown in Fig. 7; however, the non-relativistic model
starts to deviate from the data at around 1.1 GeV.
We used a Yamaguchi-type separable interaction for the nucleon-nucleon potential. To
take into account the long range attractive interaction and the short range repulsion of the
two-nucleon interaction, we used a two-term separable potential,
< ~p′|VBB|~p >= CRgR(~p′)gR(~p)− CAgA(~p′)gA(~p). (27)
λ Λ(MeV ) a(fm)2 scattering length
Relativistic 2.7 618 0.50 −0.095m−1π
Nonrelativistic 3.0 628 0.30 −0.101m−1π
TABLE VI: Parameters and scattering length for the relativistic and nonrelativistic model of I=3/2
πN interaction.
FIG. 7: The phase shift of the πN scattering for S11 (a), and S31 (b) partial waves. The solid
curve shows the relativistic model and the dashed curve shows the nonrelativistic model. Data are
taken from Ref. [35].
Here CR (CA) is the coupling strength of the repulsive (attractive) potential. gR(~p) (gA(~p))
is the form factor, whose form is given as gR(~p) = Λ
R/(~p
2 + Λ2R) (gA(~p) = Λ
A/(~p
2 + Λ2A)),
where Λ is a cutoff of the nucleon-nucleon potential. The adjustable parameters in our
nucleon-nucleon potential are determined by fits to the data of the 1S0 phase shifts [36].
The best-fit parameters are summarized in Table VI. The low energy phase shifts of the 1S0
state is shown in Fig. 8.
ΛR(MeV ) ΛA(MeV ) CR(MeV fm
3) CA(MeV fm
Relativistic 1144 333 5.33 5.61
Nonrelativistic 1215 352 5.05 5.84
TABLE VII: Our parameters of the relativistic and nonrelativistic model for NN scattering.
FIG. 8: Phase shifts of the NN scattering for 1S0 state. Solid curve shows the relativistic model
and dashed curve shows the nonrelativistic model. The phase shifts calculated from the model of
Ref. [36] are shown in triangles.
IV. RESULTS AND DISCUSSION
The dibaryon resonance with Jπ = 0−, S = −1, I = 1/2 is studied using a formalism
of the Faddeev equation as explained in section 2. We assume all the angular momentum
to be in an s-wave state and the spin singlet state SBB = 0 for the two baryon states. We
have included the dominant K̄NN , πΣN and πΛN Fock-space components, whose isospin
wave functions are [K̄⊗ [NN ]I=1]I=1/2, [π⊗ [ΣN ]I=1/2,3/2]I=1/2 and [π⊗ [ΛN ]I=1/2]I=1/2. An
approximation within this model is that the weak Y N interaction is not included.
Let us start to examine the three-body resonance energy by taking into account only
the K̄N interactions v
I=0,1
K̄N−K̄N
neglecting the πY N Fock space. In this case, the bound
state pole is expected to lie on the physical Riemann sheet below mK + 2mN if the K̄N
attraction is strong enough. Therefore it is not necessary to use the analytic continuation of
the amplitude with the deformed contour discussed in section 2, so we simply use the integral
over the momentum pi in the real axis. The results are shown in Fig. 9 marked by a and a
for the the ’relativistic’ and ’non-relativistic’ models. Here we use the ’standard’ parameters
(aA) of the K̄N interaction with nonrelativistic and relativistic kinematics. The binding
energies are about 18 MeV. The K̄N interaction included in τ and Z is strong enough to
bind the K̄NN system, where the I = 0 K̄N interaction plays a dominant role. We then
take into account the NN interaction. Then the binding energy increased furthermore to
25.1 MeV (22.8 MeV) shown as b (b′) for ’relativistic’ (’nonrelativistic’) model. Notice that
FIG. 9: The pole trajectories of the K̄NN − πY N scattering amplitude for the Jπ = 0− and
I = 1/2 state. The solid curve and the filled circles (dashed curve and filled triangles) show the
results of the relativistic (nonrelativistic) model (aA). Here WKNN = mK + 2mN .
if we neglect the repulsive component of the NN interaction, we obtain a much more deeply
bound state.
In the next step, we gradually include the πY N interactions, while the pion-exchange
Z diagram is not yet included. To do this, we multiply by factor x the coupling constants
Cα,β of the K̄N − πY and πY − πY interactions as xCα,β . When the parameter is zero,
x = 0, the πY is disconnected from K̄N and when it takes the value 1, x = 1, we recover
the full model. By varying the parameter x from 0 to 1, we can follow the trajectory of
the resonance pole from the bound state pole. Now the K̄NN bound state decays into
the πY N channel and the bound state pole moves into the unphysical sheet. Since the
K̄NN bound state was found above the πΣN threshold, the resonance pole may be on the
πY N unphysical and K̄NN physical Riemann sheet, which we have discussed in section
2. The results of the pole trajectories are shown by the solid and dashed curves in Fig. 9
corresponding to the relativistic and the nonrelativistic models. Increasing the coupling to
the πY N channel causes the width as well as the binding energy of the resonance increases.
For larger binding energy Re(Wpole − WKNN) < −60 MeV the width starts to decrease
because of the decreasing phase space for the decay into the πΣN state. The pole position
is at −82− i29 MeV (−91−28i) for the relativistic (nonrelativistic) model shown as c(c′). It
is noticed that the numerical method to follow the pole trajectories helps us to find whether
FIG. 10: The pole trajectories of the three-body resonance (solid curve) and Λ(1405) (dashed
curve) using the nonrelativistic model of (aA).
we encounter singularities or not. As an example, the pole of the three-body resonance is
shown by the solid line in Fig. 10 for 0 < x < 1. The pole of the Λ(1405) is also shown by
the dashed curve. The trajectory of the K̄NN resonance is similar to the the case A in Fig.
5, and the integration contour does not intercept the singularity arising from the two-body
resonance Λ(1405).
Finally we include the π exchange mechanism in Z and πN two-body scattering terms in
τ , which adds another mechanism for the decay of the K̄NN into πY N and is important for
the width of the three-body resonance. The final results of the K̄NN−πY N resonance poles
are denoted by d and d′ in Fig. 9. This mechanism increases the width of the three-body
resonance by about 14 MeV, while the effect on the real part is small. The cancellation
between the attractive I = 1/2 πN interaction and the repulsive I = 3/2 πN interaction
may lead to the small effects on the real part of the resonance energy. The effects of πΛN
channel is small which increases binding energy and half-width at most by 1MeV. The pole
position of the three-body resonance is W = M − iΓ/2 = 2mN +mK − 79.3 − i37.1 MeV
(2mN +mK − 92.2− i35.4 MeV) for relativistic (nonrelativistic) model shown as d and d′.
The model dependence of our results on the three-body resonance is summarized in Tables
VIII and IX. The K̄NN − πY N resonance pole is located on the K̄NN physical and πY N
unphysical sheet with the binding energy, B ∼ 60 − 95 MeV, and the width, Γ ∼ 45 − 80
MeV, using relativistic models. All of our models predict resonance energies above the πΣN
threshold. The relatively large model dependence of our results is due to the uncertainty
in the models of I = 0 K̄N − πΣ interaction. Comparing the results of model (A) with
Model (A) Model (B)
(a) −79.3− i37.1 −79.3− i37.3
(b) −93.3− i27.4 −93.3− i27.6
(c) −57.2− i38.6 −56.9− i38.6
(d) −72.4− i31.7 −72.2− i31.9
(e) −87.1− i40.8 −87.1− i41.0
(f) −63.3− i22.2 −63.2− i22.3
TABLE VIII: The pole energy (Wpole −mK − 2mN ) of the three-body resonance using relativistic
models. The listed pole energies in MeV can be related to binding energy B and the width Γ as
Wpole −mK − 2mN = −B − iΓ/2.
Model (A) Model (B)
(a) −92.2 − i35.4 −92.3− i35.6
(b) −101.6 − i20.7 −101.6 − i20.7
(c) −72.7 − i53.9 −72.5− i54.9
(d) −83.0 − i33.3 −83.0− i33.6
(e) −98.1 − i33.2 −98.2− i33.3
(f) −66.5 − i24.4 −66.3− i24.4
TABLE IX: The pole energy of the three-body resonance. The same as Table VIII but for the
nonrelativistic models.
model (B), we can see the three-body pole position is almost independent of the parameters
of the I = 1 K̄N − πY interaction. By varying the real(imaginary) part of the fitted
scattering length by ±0.1 fm, the binding energy of the three-body resonance is affected by
∼ ±14(8) MeV. Following another way to construct the model, the parameters of the model
(f) are fitted to the pole energy of the Λ(1405). This model predicts the scattering length
−1.72+ i0.44 fm. The energy of the three-body resonance is found to be B = 63 MeV with
a rather small width, Γ = 44 MeV, compared with the models (a-e), which can already be
seen in the small imaginary part of the scattering length in model (f).
Let us briefly compare our results with those of the other theoretical studies of the K−pp
resonance, which use a nonrelativistic approach. Our resonance has a deeper binding energy
and a similar width compared with those in Ref. [4]. However, it is not straightforward
to compare with the pole energy of Ref. [4] because of the differences in the method to
obtain the three-body resonance energy and on the model for the K̄N interaction. Their
K̄N potential is stronger and has a short range than ours. Recently Shevchenko, Gal and
Mares [19] studied K−pp system using the nonrelativistic coupled-channel Faddeev equation.
Though the details of their method is not described in Ref. [19], it seems their approach is
quite similar to our present study. They employed a phenomenological K̄N potential model,
and reported B ∼ 55− 70 MeV and Γ ∼ 95− 110 MeV. Their result is consistent with our
results of the nonrelativistic model. Specially our result using the model (c) gives a quite
similar resonance energy and width.
In summary we have studied the existence and properties of a strange dibaryon resonance
using the K̄NN − πY N coupled channel Faddeev equation. By solving the three-body
equation the energy dependence of the resonant K̄N amplitude is fully taken into account.
The resonance pole has been investigated from the eigenvalue of the kernel with the analytic
continuation of the scattering amplitude on the unphysical Riemann sheet. The model of
the K̄N−πY interaction is constructed from the leading order term of the chiral Lagrangian
takes into account the relativistic kinematics. The K̄N interaction parameters are fitted to
the scattering length given by Martin. We found a resonance pole at B ∼ 79 MeV and
Γ ∼ 74 MeV in the relativistic model (aA). However, as the K̄N interaction is not very well
constrained by the data, we studied a possible range of the resonance energies by considering
different parameter sets of the K̄N −πY interaction. The binding energy and the full width
can be in the range of B ∼ 60 − 95 MeV and Γ ∼ 45 − 80 MeV when computed in the
relativistic model. In order to connect the resonance found in this work to the experimental
signal, further theoretical studies on the production mechanism and further decay of the
resonance especially to the Λ− p channel are necessary.
Acknowledgments
The authors are grateful to Prof. A. Matsuyama for very useful discussions on the three-
body resonance. We also thank Drs. B. Juliá-Dı́az, T.-S. H. Lee and Prof. A. Gal for
discussions. This work is supported by a Grant-in-Aid for Scientific Research on Priority
Areas(MEXT), Japan with No. 18042003.
APPENDIX
The spin-isospin recoupling coefficient of the particle exchange interaction Z is briefly
explained. The coefficient given in Eq. (1.181) of Ref. [21] can be simplified for s-wave
states. The three-body state with total spin and isospin (Stot, Itot), which couples with
baryon ’isobar’ with spin and isospin (S, I) and the spectator baryon Bi(Si,Ii), is given as
|[[M3(S3,I3) ⊗ Bj(Sj ,Ij)](S,I) ⊗ Bi(Si,Ii)](Stot,Itot) > . (A.1)
Here baryon i, j represents particle 1 or 2, and the meson is always assigned as the third
particle. The wave function of the three-body state, which couples with dibaryon ’isobar’
and the spectator meson M3, is given as
|[[B1(S1,I1) ⊗B2(S2,I2)](S,I) ⊗M3(S3,I3)](Stot,Itot) > . (A.2)
Then Eq. (7) is extended to include spin-isospin degrees of freedom. The particle exchange
interaction for the spectators l, m, the isobars f ′, f with spin-isospin (S ′, I ′) and (S, I) and
the exchanged particle n can be expressed as follows
Zl,f ′(S′,I′),m,f(S,I)(pl, pm,W ) = Rl,f ′(S′,I′),m,f(S,I)
d(p̂l · p̂m)
2πgf ′(S′,I′)(ql)gf(S,I)(qm)
W −El(pl)−Em(pm)−En(~pl + ~pm)
(A.3)
where f represents isobar YK , Yπ, d and N
Rl,f ′(S′,I′),m,f(S,I) is given by the overlap of the initial and final spin-isospin wave functions.
For the meson(M3) exchange mechanism, Ri,f ′(S′,I′),j,f(S,I) is given as,
Ri,f ′(S′,I′),j,f(S,I) = < [[M3(S3,I3) ⊗ Bj(Sj ,Ij)](S′,I′) ⊗ Bi(Si,Ii)](Stot,Itot)|
×|[[M3(S3,I3) ⊗Bi(Si,Ii)](S,I) ⊗Bj(Sj ,Ij)](Stot,Itot) >
= (−1)S+S′−S3−StotW (Si, S3, Stot, Sj;S, S ′)
(2S + 1)(2S ′ + 1)
×(−1)I+I′−I3−ItotW (Ii, I3, Itot, Ij ; I, I ′)
(2I + 1)(2I ′ + 1). (A.4)
For the baryon(Bj) exchange mechanism, Ri,f ′(S′,I′),3,f(S,I) is given as,
Ri,f ′(S′,I′),3,f(S,I) = < [[M3(S3,I3) ⊗Bj(Sj ,Ij)](S′,I′) ⊗Bi(Si,Ii)](Stot,Itot)|
×|[[B1(S1,I1) ⊗ B2(S2,I2)](S,I) ⊗M3(S3,I3)](Stot,Itot) >
= (−1)S3+S−Stot+I3+I−ItotW (S3, Sj , Stot, Si;S ′, S)
(2S ′ + 1)(2S + 1)
×W (I3, Ij, Itot, Ii; I ′, I)
(2I ′ + 1)(2I + 1)
×(δi,2δj,1 + δi,1δj,2(−1)Si+Sj−S+Ii+Ij−I). (A.5)
When we anti-symmetrize the AGS equation, the last factor in the bracket in Eq. (A.5)
projects the anti-symmetric two nucleon states. This can be explicitly seen by comparing the
exchange of nucleon 2 Z2,YK(S′,I′),3,d(S,I) and nucleon 1 Z1,YK(S′,I′),3,d(S,I) interactions. Using
Eq. (A.5), those interactions are related as
R1,YK(S′,I′),3,d(S,I) = (−1)S+IR2,YK(S′,I′),3,d(S,I), (A.6)
which leads to Eq. (17) as
XYK ,YK = (1− (−1)S+I)ZYK ,dτd,dXd,YK + · · · . (A.7)
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http://arxiv.org/abs/nucl-th/0701001
	Introduction
	Coupled channel Faddeev equation and resonance pole
	Model of the two-body interactions
	Results and Discussion
	Acknowledgments
	References

  Three-body resonances in the \bar{K}NN system have been studied within a
framework of the \bar{K}NN-\pi YN coupled hannel Faddeev equation. By solving
the three-body equation the energy dependence of the resonant \bar{K}N
amplitude is fully taken into account. The S-matrix pole has been investigated
from the eigenvalue of the kernel with the analytic continuation of the
scattering amplitude on the unphysical Riemann sheet. The \barKN interaction is
constructed from the leading order term of the chiral Lagrangian using
relativistic kinematics. The \Lambda(1405) resonance is dynamically generated
in this model, where the \bar{K}N interaction parameters are fitted to the data
of scattering length. As a result we find a three-body resonance of the strange
dibaryon system with binding energy, B~79 MeV, and width, \Gamma~74 MeV. The
energy of the three-body resonance is found to be sensitive to the model of the
I=0 \barKN interaction.

Introduction
	Coupled channel Faddeev equation and resonance pole
	Model of the two-body interactions
	Results and Discussion
	Acknowledgments
	References
