Application of a coupled-channel Complex Scaling Method with Feshbach projection to the $K^-pp$ system

Kaonic nuclei (nuclear system with anti-kaons) have been an interesting subject in hadron and strange nuclear physics, because the strong attraction between anti-kaon and nucleon might bring exotic properties to that system. In this article, we investigate $K^-pp$ as a prototype of kaonic nuclei. Here, $K^-pp$ is a three-body resonant state in the $\bar{K}NN$-$\pi YN$ coupled channels. ($Y$=$\Lambda$, $\Sigma$) To treat resonant states in a coupled-channel system properly, we propose newly a coupled-channel complex scaling method combined with the Feshbach projection (ccCSM+Feshbach method). In this method, the Feshbach projection is realized with help of so-called the extended closure relation held in the complex scaling method, and a complicated coupled-channel problem is reduced to a simple single-channel problem which one can treat easily. First, we confirm that the ccCSM+Feshbach method completely reproduces results of a full coupled-channel calculation in case of two-body $\bar{K}N$-$\pi Y$ system. We then proceed to study of three-body $\bar{K}NN$-$\pi YN$ system, and successfully find solutions of the $K^-pp$ resonance by imposing self-consistency for the complex $\bar{K}N$ energy. Obtained binding energy of $K^-pp$ is well converged around 27 MeV, with an energy-dependent $\bar{K}N$(-$\pi Y$) potential based on the chiral SU(3) theory, independently of ansatz for the self-consistency. This binding energy is small as ones reported in earlier studies based on chiral models. The decay width of $K^-pp$ strongly depends on the ansatz. We calculate also the correlation density of $NN$ and $\bar{K}N$ pairs by using the obtained complex-scaled wave function of the $K^-pp$ resonance. Effect of the repulsive core of $NN$ potential and survival of $\Lambda^*$ resonance are confirmed.


Introduction
On hadron and strange nuclear physics, nuclear system with anti-kaons (K mesons = (K − , K 0 )) has been a hot issue, since the anti-kaon is expected to cause several interesting phenomena in finite nuclear system due to a strong attraction between anti-kaon and nucleon. In particular, the anti-kaon could be a key to access dense nuclear matter, for which partial restoration of chiral symmetry [1,2] and kaon condensation [3,4] have been discussed for a long time.
Both theoretical and experimental studies indicate that theKN interaction in the isospin I = 0 channel is strongly attractive. While, it is known that mass of the excited hyperon Λ(1405) cannot be reproduced in a naive quark model with P -wave excitation [5], namely, mass of Λ(1405) is predicted about 100 MeV larger than the PDG value exceptionally. Consequently, Λ(1405) is considered to be a quasi-bound state of anti-kaon and nucleon since it exists at only ∼ 30 MeV belowKN threshold. For example, chiral unitary model [6], based on a meson-baryon dynamics, has successfully explained various properties of the Λ(1405). Thus, the Λ(1405) is getting recognized as aKN quasi-bound state, rather than a genuine three-quark state. Such the attractive nature of theKN interaction with a quasibound state, is consistent with the repulsive nature of the low-energyKN scattering data [7] and the 1s-level energy shift of kaonic hydrogen atom [8] which has been updated precisely [9].
For theKN interaction, there are two kinds of potential: one is phenomenological energyindependent potentials (e.g. Ref. [10]) and the other is chiral SU(3)-based energy-dependent potentials [11,12]. Both are fitted to observables of theKN system available at this moment, and are applied and discussed intensively. The former type of potentials is more strongly attractive than the latter type inKN sub-threshold region. One study based on a phe-nomenologicalKN potential [10] argues a possibility of so-called deeply bound kaonic nuclei where an anti-kaon is deeply bound in finite nuclei with a binding energy of more than 100 MeV, and such a deeply-bound state could exist as a quasi-stable state since the main decay channel πΣ is closed. In such a state, nucleons are drawn to the anti-kaon by the strongKN attraction, and hence a dense system is generated. In case of light p-shell nuclei, studies with antisymmetrized molecular dynamics (AMD) method have shown that the average density amounts to ∼ 4ρ 0 , when an anti-kaon is added [13]. (ρ 0 : normal nuclear density, around 0.17 fm −3 .) In case of medium and heavy nuclei, studies with the relativistic mean field (RMF) approach have been carried out, and it is shown that K − mesons can create a dense state inside the nucleus [14,15]. Therefore, we can expect kaonic nuclei to be a doorway to dense nuclear matter.
Thus, kaonic nuclei are considered to be an exotic system involving several interesting aspects from the viewpoint of hadron and nuclear physics. To reveal the nature of kaonic nuclei, a prototype system of kaonic nuclei "K − pp" has been studied extensively. Since K − pp is a three-body system composed of a single K − meson and two protons, various approaches are adopted. As summarized in Ref. [16], the resulting binding energy and decay width of K − pp become different depending on the combination of methods (variational approach or Faddeev-AGS) and potentials (phenomenological or chiral-theory based) [17][18][19][20]. From that time, further studies on K − pp have been carried out. A variational calculation with the hyperspherical harmonics basis function is reported in Ref. [21], where the result agrees with that of an earlier study of another variational calculation [20] when they use the sameKN potential. A Faddeev-AGS calculation with an energy-dependent type of chiral SU(3)-based KN potential is reported in Ref. [22], where a K − pp state with small binding energy and large decay width is obtained similarly to the the variational calculation [20]. On the other hand, there are several experiments to search for K − pp states. Actually, the experimental 2/23 result reported by FINUDA collaboration [23] triggered studies of K − pp, although several questions were casted to their interpretation of the result [25]. DISTO collaboration [24] reported a bump structure found in a Λp invariant-mass distribution from analysis of the past data on a p + p reaction. These two collaborations claim that if the observed state is a K − pp bound state the K − pp is strongly bound with the binding energy of more than 100 MeV, although its decay width is rather different between two. Thus, although much effort have been devoted to the study of K − pp, the definite conclusion has not been achieved yet in both the theoretical and experimental studies. However, in the theoretical side, we have one consensus that the K − pp exists as a resonance between theKN N and πΣN thresholds, as commonly reported in all those calculations.
From those theoretical studies, we believe that the following two ingredients are important in theoretical studies of the K − pp system: 1. coupled-channel problem and 2. resonant state. We employ a coupled-channel complex scaling method (ccCSM) to study the K − pp, since both the ingredients can be dealt with in this method simultaneously. Here, the complex scaling method (CSM) is an established powerful tool to investigate resonant states, which has already succeeded greatly in the studies of resonant states of stable/unstable nuclei [26,27]. The CSM has several advantages to investigate finite nuclear systems as follows: First, we can handle resonant states in the same way as bound states, since the resonant wave function in the CSM can be represented by using only the L 2 basis functions such as the Gaussian basis functions which have been often used in bound-state studies. Second, it is straightforward to increase the number of particles in the CSM, which means that we can apply the CSM to various many-body systems. In addition, detailed properties of resonant states can be investigated by analyzing the obtained CSM wave function as usually done for bound states.
As the first attempt, we have applied the ccCSM to the two-bodyKN -πY system in our previous paper [28]. (Y means Λ and Σ hyperons.) Through the study of scattering states as well as the resonant state Λ(1405), we have confirmed that the ccCSM is quite useful to look into such a hadronic system. In that work, we have constructed aKN -πY potential based on the chiral SU(3) theory, which has a Gaussian form factor in the coordinate space and the energy-dependence. It is shown in studies based on chiral models [6], that Λ(1405) should possess so-called a double-pole structure. We reconfirmed such a structure with our Gaussian-type potential. We have successfully identified the lower pole as a broad resonant state, in addition to the higher pole, by using an improved Gaussian basis function in the ccCSM [29].
Since we have confirmed that the ccCSM is quite effective to the two-body system of KN -πY , we tackle the three-body kaonic nucleus K − pp in this article. To study the K − pp, normally we have to solve an equation in the coupled-channelsKN N , πΣN and πΛN . But, in this paper we propose a convenient method to reduce such a coupled-channel problem to a single-channel problem, namely, we combine the ccCSM and the Feshbach projection method [30]. With this method, we can handle theKN N -πΣN -πΛN complicated system effectively as a simpleKN N system without loosing effect of the decay to two other open channels. Actually, we study the K − pp as a Gamow state and obtain the eigenstate as a definite pole on the complex-energy plane. Thus, the K − pp is treated as a resonant state correctly in this study. In contrast, in the earlier studies of K − pp with variational approaches 3/23 [20,21], the K − pp has been investigated within a bound-state approximation and the decay width is perturbatively estimated with the obtained wave function. This article is organized as follows. In the next section, we explain our new method of ccCSM+Feshbach projection in detail and give all tools for the present calculation of the K − pp. In the section 3, we examine our method by solving the two-bodyKN -πY system. Main results of this paper, i.e. application of our method to the three-body K − pp, are shown in the section 4. Section 5 is devoted to summary of the present study and discussion of our future plans.

Essence of complex scaling method
Here, we give a brief explanation on the usual Complex Scaling Method (CSM) on which the present study is based [26,27]. In the CSM, HamiltonianĤ and wave function |Φ are transformed with the complex scaling (complex rotation) operator U (θ) asĤ θ = U (θ)ĤU −1 (θ) and |Φ θ = U (θ)|Φ , respectively. With the complex scaling the coordinate r and the conjugate momentum p in the Hamiltonian and wave function are transformed as where the variable θ is called as the scaling angle.
In eigenvalues of the complex-scaled Hamiltonian, those of scattering continuum states appear along so-called 2θ line on the complex-energy plane, which satisfies a relation tan −1 (Im E/Re E) = −2θ. (The variable E means a complex eigen energy with H θ |Φ θ = E|Φ θ .) Namely, they are dependent on the scaling angle. On the other hand, eigenvalues of bound and resonant states are proven to be independent of the scaling angle. In addition, as is easily checked, if we choose appropriate values of the scaling angle θ, wave functions of resonant states are transformed to become square-integrable, which are originally not so. Therefore, the resonant-state wave function, which is complex-scaled, can be expanded with a square-integrable L 2 basis function such as Gaussian basis functions, similarly to the bound-state wave functions.
Due to those nature of the CSM, we can obtain the eigen energies and eigen wave functions of resonant states, by diagonalizing the complex-scaled Hamiltonian with Gaussian basis functions. Detailed explanation on the complex scaling method is summarized in Ref. [26].

Feshbach projection on the coupled-channel Complex Scaling Method
In the present study, we reduce a coupled-channel problem to a single-channel problem for an economical calculation, based on Feshbach projection method [30]. In the Feshbach method, a model space (P space) and outer space of the model space (Q space) are assigned with P + Q = 1 and P Q = 0. Schrödinger equation is given as a coupled equation of wave functions for P and Q spaces as which Φ P and Φ Q denote P -and Q-space wave functions, respectively. By the elimination of the Q-space wave function, an equation for the P -space wave function is derived from Eq.

4/23
(2): Since Hamiltonian for the P space H P P is composed of the kinetic energy term T P and the potential term V P , the above equation can be written as Here, the term U ef f P (E) is regarded as an effective potential for the P space. Thus, we obtain a single-channel Schrödinger equation for the P -space wave function in a formal way.
In application of Feshbach method to actual studies, the problem is how to represent the Q-space Green function, G Q (E) in Eq. (3). We realize the Feshbach method with help of a nature of the complex scaling method (CSM) as follows. It is proven that the closure relation holds in the CSM which includes explicitly resonant states as well as continuum scattering states and bound states [31]. (Extended Closure Relation, ECR, proposed by Berggren [32]) The ECR is shown to be useful to represent the Green function of a system [33]. In addition, the ECR is well described approximately with a set of finite number of the complex-scaled eigenstates {φ θ n } which are obtained by the diagonalization of a complex-scaled Hamiltonian H θ with Gaussian basis functions {G a } [34]: where n is the state index and complex parameters {C n,θ a } are determined by a diagonalization ofĤ θ .
We incorporate the ECR on the Q space into the Feshbach method. First, we consider the complex-scaled Green function for the Q-space, G θ Q (E) = U (θ)G Q (E)U −1 (θ). With the application of the ECR shown in Eq. (5), it is given approximately as where eigenenergies {e θ Q,n } and eigenstates {|φ θ Q,n } of the complex-scaled Hamiltonian H θ QQ are calculated with Gaussian basis functions {G a }. By the inverse transformation U −1 (θ), we obtain the non-scaled Green function G Q (E) from the complex-scaled one G θ Q (E); . Substituting the obtained G Q (E) to Eq. (4), we can represent the effective P -space potential as where V θ P Q (QP ) = U (θ)V P Q (QP ) U −1 (θ). Since the eigenstates {|φ θ Q,n } are expanded with Gaussian basis function, the effective potential is expressed with Gaussian functions when the original coupled-channel potential is given in a Gaussian form. Therefore, the effective potential U ef f P (E) derived in this way is easily handled in conventional many-body calculations with Gaussian basis functions.
Thus, we reduce a coupled-channel problem to a single-channel problem with Feshbach projection method which is assisted with a unique nature of the complex scaling method. 5/23 We call this method as a coupled-channel complex scaling method with Feshbach projection, which is hereafter denoted shortly as "ccCSM+Feshbach method".

Hamiltonian and trial wave function for the singleKNN channel
In theoretical studies, the K − pp system is treated as a coupled-channel system of KN N , πΣN and πΛN , involving quantum numbers J π = 0 − and I = 1/2. We apply the ccCSM+Feshbach projection to theKN N -πY N coupled-channel problem to reduce aKN N single-channel problem. (Y = Λ, Σ) At first, we consider the two-body system ofKN -πY . When we set theKN channel to P space and the πY channels to Q space, we can derive an effectiveKN potential for each isospin state (I = 0, 1) with the ccCSM+Feshbach method: where the channel indices α and β are πΣ for I = 0, and (πΛ, πΣ) for I = 1. The terms of ∆M α andT θQ α are a mass of the α channel measured from theKN threshold and the relative kinetic energy term of the α channel which is complex-scaled, respectively. The last termV (I),θQ αβ is a complex-scaled potential coupling between channels α and β with isospin I. Note that hereafter in this article the variable "θ Q " means the scaling angle which is used to construct an effective potential by the elimination of Q-space components as explained in the previous section.
With the effectiveKN potential U ef f KN (I) (E) plugged in, a three-body Hamiltonian for the singleKN N channel is constructed to bê The first two terms are kinetic-energy operators with respect to a Jacobi coordinate, The last term is the effectiveKN potential for aKN i pair with isospin I. Detailed explanation on the N N andKN potentials will be given at the beginning of the sections 3 and 4.
A trial wave function of theKN N system with quantum numbers (J π , T ) = (0 − , 1/2) is constructed in the similar way to an earlier study with a variational approach [20]. Since the spin of the two nucleons are assumed to be zero, the trial wave function consists of two components that satisfy the antisymmetrization for two nucleons. In one component, N N state has even-parity and isospin 1, and in the other component it has odd-parity and isospin 6/23 0; In the present study, the spatial part of the wave function Φ is expanded with correlated Gaussian basis functions [35], so that the N N parity is realized correctly in each part; The variables {C i } is complex-valued parameters which are determined by the diagonalization of the complex-scaled Hamiltonian. Here, the correlated Gaussian function is wherex T indicates a Jacobi coordinate (x 1 , x 2 ), A i is a real-symmetric 2 × 2 matrix and N i is a normalization factor. We comment that the basis functions, G , are even-and odd-parity functions for the exchange of two nucleons, respectively.
Resonant states of theKN N system are obtained in usual way with the complex scaling method. The Hamiltonian for theKN N system,ĤK N N given in Eq. (10), is complex-scaled with a scaling angle θ P . The complex-scaled Hamiltonian,Ĥ θP It is remarked that the scaling angle used to find resonant states ofKN N system is denoted as "θ P " hereafter, to distinguish from the scaling angle θ Q which is used for the construction of the effective potential.

Treatment of an energy dependence of the effective potential
The effectiveKN potential U ef f KN (I) (EK N ) which is constructed with the ccCSM+Feshbach has an energy dependence. As shown in Eq. (8), the potential depends on aKN energy (EK N ) which means the energy of aKN system included in a total system that we are considering. In other words, to determine the potential strength we need to know the energy of aKN two-body system in theKN N three-body system. However, such an energy of a subsystem in a total system cannot be determined uniquely in principle. We deal with the energy dependence of the effective potential, following a procedure proposed in a former study [20] in which the same issue was considered.
We calculate a so-called kaon's binding energy B K as an auxiliary quantity, which is obtained by subtracting the N N energy from theKN N energy: where the termĤ θP N N is a complex-scaled Hamiltonian for the N N system. The N N Hamiltonian is given asĤ N N =p 2 1 /2µ N N +V N N . Using the kaon's binding energy, we estimate theKN energy with two ansatz based on two extreme concepts; where M N and m K are nucleon and anti-kaon masses, respectively. On one ansatz, we consider the anti-kaon as a field which carries the kaon's binding energy. (See the left panel of Fig. 1.) Since the anti-kaon with the energy with static approximation applied to nucleons. On the other ansatz, we treat the anti-kaon as a particle. Since the anti-kaon is bound by two nucleons and the kaon's binding energy is provided by them, the binding energy per aKN bond should be a half of B K . (See the right panel of Fig.  1.) Therefore, the energy of aKN pair is equal to M N + m K − B K /2. Hereafter, we denote the first ansatz as Field picture and the latter ansatz as Particle picture. For the convenience, we refer theKN energy measured from theKN threshold; E(KN ) ≡ √ sK N − M N − m K . When bound and resonant states of theKN N system are considered with such energydependent potentials, the self-consistency for theKN energy has to be taken into account as explained in the former study [20]. TheKN energy set in the effectiveKN potential should finally coincide with that estimated with the obtained wave function by following the above-mentioned ansatz. It is noted that the self-consistency is realized for the complex KN energy in the current study. We, here, treat a resonantKN N state as a Gamow state with the correct boundary condition. Since the pole energy on the complex-energy plane is explicitly considered, theKN energy is treated as a complex value. On the other hand, such a self-consistency is considered only for the real energy in the former study with a variational approach [20], since theKN N is treated within a bound-state approximation having a real binding energy.

Nature of ccCSM+Feshbach method onKN -πY system
Before the investigation of the K − pp system, we study the nature of our method on the two-bodyKN -πY system. As theKN (-πY ) potential VK N,πY (I) shown in Eq. (8), which is the origin of the effectiveKN potential, we use a chiral SU(3)-based potential that was proposed in our previous study [28]. OurKN (-πY ) potential is energy-dependent and is 8/23 Table 1 Quantities of an I = 0KN -πΣ system calculated with ccCSM and ccCSM+Feshbach. The quantities of aK N (I=0) and a πΣ (I=0) areKN and πΣ scattering lengths, respectively. The values (−B(KN ), −Γ/2) indicate a resonance position of the system on the complex energy plane. r 2 K N and r 2 πΣ are a meson-baryon mean distance of each component of the resonant state. Here, energies and lengths are given in units of MeV and fm, respectively.
given with a single-range Gaussian form in the coordinate space. A non-relativistic version of this potential, called NRv2c, are employed in this section.

Test calculation of ccCSM+Feshbach projection on a two-bodyKN-πY system
First, we examine the ccCSM+Feshbach projection method on a two-bodyKN -πY system. Table 1 shows the results of scattering and resonance properties of I = 0 channel obtained with both methods of ccCSM and ccCSM+Feshbach. Here, the test calculation is performed with an energy-dependent potential (NRv2c). In the ccCSM both ofKN and πΣ components are explicitly treated as the model space (P space), while in the ccCSM+Feshbach one component is set to be P space and the other component is considered as Q space to be eliminated. In the two-body case, the calculation of the ccCSM+Feshbach via an effective two-body potential is completely equivalent to that of the ccCSM treating all channels explicitly. In principle, results of both calculations should agree with each other. However, the Q-space Green function used in ccCSM+Feshbach is approximately represented with a finite number of Gaussian basis as explained in the section 2.2. Under this approximation, the ccCSM+Feshbach is confirmed to reproduce quite well the ccCSM results of both scattering lengths and resonance properties. The scattering lengths are calculated with the CS-WF method which is a method to solve scattering problems with help of ccCSM as explained in the section 2.3 of our previous paper [28]. We plug the effective potential derived with ccCSM+Feshbach into the CS-WF method with a singleKN /πΣ channel. It is noted that the scattering amplitudes of these components are confirmed to be identical between the two methods in wide energy region of −200 MeV to 50 MeV measured from theKN threshold. Furthermore, in the I = 1 case that πΛ channel is additionally coupled withKN and πΣ channels, the ccCSM+Feshbach reproduces all theKN , πΣ and πΛ scattering amplitudes obtained with the ccCSM.
In the calculation of a resonance pole, the self-consistency for theKN energy is needed to be taken into account in both methods. As shown in Table 1, the pole position obtained self-consistently with the ccCSM+Feshbach is found to agree with that obtained with the 9/23 Table 2 Dependence of the Λ * eigen energy on the scaling angles θ P and θ Q . In the upper (lower) table, the scaling angle θ Q (θ P ) is varied with θ P (θ Q ) fixed to 30 • . "NF" means that no solutions are found below theKN threshold. NRv2 potential (f π = 110) is employed. Energies (B(KN ) and Γ/2) are given in unit of MeV. The unit of scaling angles is degree. ccCSM, whichever ofKN and πΣ channels is chosen as the P space. The meson-baryon mean distance inKN and πΣ components also coincides in both methods, when the normalization of each component in the ccCSM is appropriately considered; where φ θ M B is a complex-scaled wave function of the M B component of the resonant state andr 2 θ indicates the complex-scaled operator of the meson-baryon distance. The resonance pole given in Table 1 is the higher pole of the double pole obtained with our energy-dependent potential. It should be noted that the ccCSM+Feshbach reproduces well the result of the ccCSM also for the other pole, namely the lower pole [29]. 3.2. Dependence of two kinds of the scaling angles, θ P and θ Q We have made further investigation on properties of the ccCSM+Feshbach method. As explained in Section 2, there are two kinds of scaling angles, θ P and θ Q , in the method. The scaling angle θ Q is introduced when we construct an effective potential by eliminating Q-space components with Feshbach method in Eqs. (6) and (7). The other scaling angle θ P is used to find resonance states by means of the complex scaling method for the P -space Hamiltonian which involves the effective potential. Table 2 shows the dependence of the energy of the Λ * resonant state on those angles, where Λ * means the I = 0 resonance of theKN -πΣ system. In the upper table, we investigate the θ Q dependence by fixing the angle θ P . In principle, the resonance energy should be independent of the angle θ Q , since the complex-scaled Green function for the Q space G  (7)) Certainly, the resonance energy is confirmed to be stable for θ Q > 15 • . However, around θ Q = 15 • the resonance energy becomes unstable, and then 10/23 Table 3 Perturbative treatment of the imaginary part of the effective potential. "(Full)" indicates the full treatment of the complex potential, while "(Perturb)" indicates the perturbative treatment of the imaginary part of the complex potential. The two potentials, NRv1 and NRv2, are examined with a parameter f π varied from 90 to 120 MeV. Energies are given in unit of MeV. cannot be obtained for small angles θ Q < 5 • . We consider that this is due to insufficient description of the Q-space Green function. At such small scaling angles, since the extended closure relation is not well approximated with finite numbers of Gaussian basis functions, the Q-space Green function is not correctly represented [27]. Also in the former study of the complex scaling method where the level density was analyzed [37], it is shown that the Green function is stably described with a basis function expansion when the scaling angle is chosen to be sufficiently large. On the other hand, we check the θ P dependence in the lower table where θ Q is fixed. It is confirmed that the resonant energy is completely stable for the scaling angle θ P . Even at θ P = 0 • , namely no scaling, the same resonance energy is obtained. Thus, once the Green function for the Q space is well represented with a sufficiently large scaling angle θ Q , we can obtain resonant states within P space correctly, using any scaling angle θ P .

Treatment of a complex effective potential
Here, we examine a treatment of an effective potential. When the channels energetically lower than the specified state are eliminated as Q space in Feshbach method, the effective potential for P space is in general a complex potential. In our case, we consider the state located between theKN and πY threshold energies. Therefore, the effective potential for theKN channel should be a complex potential, when the lower channels, πY , are treated as Q space. In this article, such a complex potential is treated directly as it is, and the self-consistency for the energy is considered also with a complex energy. Table 3 shows the binding energy and decay width of the Λ * resonant state obtained with such a full treatment of the complex potential (denoted as "(Full)"), where two versions of our energy-dependent potential and several f π values are examined. On the other hand, so far, the imaginary part of the complex potential has often been treated perturbatively [13,20,21,36].
We estimate the effect of the perturbative treatment of the imaginary potential within our method as follows. First, we construct the effective potential U ef f KN (I=0) (EK N ) by the Feshbach method as explained in Section 2.2. With only the real part of the effective potential, we construct a Hamiltonian for theKN system;Ĥ ′K N =TK N + ReÛ ef f KN (I=0) (EK N ), where 11/23 TK N means the operator ofKN relative kinetic energy. We diagonalize the Hamiltonian H ′K N with Gaussian basis functions as usual, namely without the complex scaling. Since eigen energies of the HamiltonianĤ ′K N are real value, we consider the self-consistency for theKN energy within real values. In other words, we set a real-valued energy in the effective potential. After we obtain a self-consistent solution with a binding energy B(KN ), we estimate the half decay width Γ/2 by calculating the expectation value of the imaginary potential ImÛ ef f KN (I=0) with the eigen wave function. The result of the perturbative treatment of the imaginary potential is shown in Table 3 (denoted as "(Perturb)"). Compared with the result of full treatment as mentioned before, it is found that when the decay width is calculated to be small with the full treatment, the perturbative treatment gives a binding energy similar to that of the full treatment. However, in case that the large decay width is obtained in the full treatment, there is large difference among two treatments. In particular, in such a case the perturbative treatment tends to give a large binding energy compared with the full treatment. Therefore, the imaginary potential is found to give repulsive contribution to the binding energy when it is included explicitly in the calculation.

Results of K − pp system with ccCSM+Feshbach method
In the section, results of the three-body system K − pp with the ccCSM+Feshbach method are shown. In the three-body calculation, the central part of Av18 potential [38] is employed as a N N potentialV N N which appears in Eq. (10). As theKN (-πY ) potential VK N,πY (I) in Eq. (8), an energy-independent potential [10] is used for a test calculation, and two kinds of non-relativistic version of our energy-dependent potential, called NRv1c and NRv2c [28], are examined.
We remark on the scaling angles θ P and θ Q . As shown in the previous section, the result is found to be independent of the angle θ P in the two-body case, if the angle θ Q is sufficiently large to represent the Q-space Green function. Therefore, in the three-body calculation mentioned hereafter, we take a common angle θ for the θ P and θ Q ; θ P = θ Q ≡ θ. 1 Similarly to the two-body calculation, results of the three-body calculation are confirmed to be independent of the scaling angle θ.

Comparison of "Field picture" and "Particle picture" in an energy-independent potential case
To consider the three-body system of K − pp, we investigate how the self-consistency for theKN energy is accomplished with Field and Particle pictures which are explained in 1 As shown in Eq. (8), the complex-scaling operator U (θ Q ) remains in the effectiveKN potential U ef f KN (I) . When the Hamiltonian for theKN N ,ĤK N N given in Eq. (10), is complex-scaled with the operator U (θ P ), the effective potential is also transformed as U (θ P ) U ef f KN (I) U −1 (θ P ). If both the scaling angles θ P and θ Q are chosen to be equal as mentioned above, the operator U (θ Q ) included in the effective potential is cancelled out by the operator U (θ P ). With such a choice of the scaling angles, we can calculate matrix elements of complex-scaled potential with Gaussian basis functions as usual. Otherwise, since the operator like U (θ Q )U −1 (θ P ) = U (θ Q − θ P ) remains in the effective potential, we need to consider the transformation of the basis functions with that operator. In other words, setting θ Q = θ P , we are free from such transformation of the basis functions. Λ* Fig. 2 Eigenvalue distribution ofKN N system on the complex energy plane which is calculated with ccCSM+Feshbach method using a phenomenological potential [10]. TheKN energy is fixed at Λ(1405). "Λ * " means the Λ(1405) resonance. Horizontal and vertical axes correspond to the real and imaginary parts of the complexKN N energy "E(KN N )" which is measured fromK + N + N threshold energy, respectively. The unit of energy is given in MeV. The scaling angle θ is taken to be 25 • .
First, Fig. 2 shows the distribution of complex eigenvalues obtained with ccCSM+Feshbach method, when theKN energy is fixed at that of Λ(1405). In this condition, theKN energy set in theKN effective potential is not consistent for the three-bodyKN N system, but it is consistent for the I = 0KN two-body system of Λ(1405). In the figure, the origin corresponds to theK-N -N three-body threshold. Since eigenvalues of scattering continuum states are known to appear along the so-called 2θ line in the complex scaling method, the eigenvalues along a line running from the origin indicate theK-N -N scattering continuum states. There is another line which starts from E(KN N ) = (−27.7, −20.4) MeV (marked with blue-dashed circle in the figure). This energy is almost equal to the Λ(1405) energy of E(KN ) = (−28.2, −20.1) MeV which is set in the effectiveKN potential. Therefore, eigenvalues along this line indicate Λ(1405)-N scattering continuum states. There is an eigenvalue isolated from two energy lines mentioned above (marked with red-dashed circle in the figure). This state corresponds to aKN N resonance. In the case that theKN energy is fixed to the Λ(1405), theKN N resonance is obtained to be E(KN N ) = (−44.5, −28.7) MeV. Thus, theKN N resonance can be identified with the ccCSM+Feshbach method.
Next, we calculate the pole energy of theKN N resonance, taking into account the selfconsistency for theKN energy in theKN N three-body system. As explained in the section 2.4, we consider that a self-consistent solution is obtained when theKN energy set in the effective potential (E(KN ) In ) coincides with that calculated with the obtained wave function (E(KN ) Cal ). Fig. 3 figure) and it is confirmed to be stable for the θ variation. Details of the self-consistent solutions with ccCSM+Feshbach method using two pictures are given in Table 4. The binding energy ofKN N (B (KN N )), which is a real part of the resonance-pole energy, is not so different between two pictures. However, the half decay width (Γ/2), which is a imaginary part of the resonance-pole energy, is much different between them. Field picture gives half times smaller decay width than Particle picture. The spatial configuration is almost the same in those pictures, as indicated by the mean distance between two nucleons (R(N N )) and that between anti-kaon and center-of-mass of two nucleons (R (K-[N N ])).
In the last column of the table, the result of an earlier study with ATMS method using the sameKN potential [19] is listed, for comparison with the present result. Among the two ansatz, Particle picture provides apparently the decay width close to that of the ATMS result. Note that they use a different N N potential from Av18 central potential that we use in the current study. In addition, they apply the 1 E channel of the N N potential commonly to the 1 O state. Since the 1 E potential is more attractive than the 1 O potential, their calculation is expected to give slightly deeper binding than our calculation. Taking into account such a difference on N N potentials, Particle picture is considered to give the binding energy consistent to the ATMS result rather than Field picture. Thus, in a case of an energyindependent potential [10], our calculation of ccCSM+Feshbach method with Particle picture is found to give similar result of the former study with ATMS method. 14/23 Table 4 Self-consistent solutions with two ansatz, "Field picture" and "Particle picture". TheKN energy of the obtained resonant state is given as E(KN ), and "Re E(KN )" and "Im E(KN )" are its real and imaginary parts, respectively. Values with parentheses mean theKN energy set in the effectiveKN potential. The binding energy and half decay width are given as "B(KN N )" and "Γ/2", respectively. All energies are given in unit of MeV. "R(N N )" means a mean distance between two nucleons, and "R(K-[N N ])" means that between anti-kaon and center-of-mass of two nucleons. Unit of these lengths is fm. The last column lists a result of a different method "ATMS", which is quoted from Ref. [19].

The three-body K − pp system calculated with chiral SU(3)-based potentials
In this section, we investigate the K − pp system with the present method using a chiral SU(3)-basedKN (-πY ) potential. Here, two versions of non-relativisticKN potentials, NRv1c and NRv2c, are employed. It should be noted that those potentials themselves have an energy dependence due to the chiral dynamics. In other words, the original coupled-channel potentials are already an energy-dependent potential, before they are converted to effective single-channel potentials which involve an energy dependence due to the channel elimination by the Feshbach projection. We comment on the difference of the energy dependence between those potentials. In the NRv2c potential, the energy dependence is completely attributed to the chiral dynamics. On the other hand, in the NRv1c potential another energy dependence 15/23 is additionally involved which comes from so-called the flux factor that gives weak energy dependence. (See Eqs. (7) and (8) in Ref. [28].) Table 5 is the summary of the present calculation, which gives the binding energy and half decay width of theKN N resonance. In Fig. 4 the pole position of theKN N resonance, (−B (KN N ), −Γ/2), is depicted on the complex energy plane, when a parameter f π in the potentials is varied from 90 to 120 MeV. The left panel is the result obtained with NRv2c potential. From the figure, it is found that theKN N is bound more deeply and the decay width becomes wider, as the parameter f π decreases. However, the binding energy is not so large and it is 32 MeV at most. The binding energy is not so dependent on pictures for theKN -energy self-consistency. On the other hand, the decay width depends strongly on the pictures. Similarly to the case of an energy-independent potential as mentioned in the previous section, the Field picture gives nearly half of the decay width, compared with the Particle picture. The result of NRv1c potential has a similar tendency to that of the NRv2c potential. In this potential, the binding energy is slightly larger compared with the NRv2c case, and it amounts to 42 MeV at f π = 100 MeV with Field picture. We consider that this is because the NRv1c potential is more attractive than the NRv2c potential, since the scattering amplitudes of the NRv1c potential show more attractive nature in theKN subthreshold region than those of the NRv2c potential as shown in Ref. [28]. It should be noted that in both potentials we cannot find any self-consistent solution which is stable for the θ variation at f π = 90 MeV with Field picture. As a reference, we show also the result that theKN energy set in the potential is fixed to the energy of I = 0KN resonance. ("Λ * fixed" in the figure) The result with this ansatz is quite similar to that with Particle picture. The binding energy B(KN N ) and half decay width Γ/2 are summarized as (B (KN N ), Γ/2) = (26.7 ± 5.5, 12.6 ± 3.6) · · · Field picture (27.6 ± 2.9, 23.5 ± 8.3) · · · Particle picture (17) 16/23 Table 6 Self-consistent solutions of theKN N system with a chiral SU(3)-based potential, NRv2c (f π = 110 MeV). Distances "R(KN, I = 0)" and "R(KN, I = 1)" are a mean distance of aKN pair with isospin I in the obtainedKN N resonance. "Λ * vac. " means a I = 0KN -πΣ resonance in vacuum. "E(Λ * vac. )" and "R(Λ * vac. )" are a pole energy of the Λ * vac. and meson-baryon mean distance in the resonance, respectively. As for the meaning of other quantities, refer to the caption of Table 4. The last column "Variational" shows a result of a variational approach, which is quoted from Ref. [20].  (18) in case of NRv1c. We have investigated the structure of the obtainedKN N resonance. Table 6 shows details of the self-consistent solution of theKN N calculated with the NRv2c potential with f π = 110 MeV, as a typical result. For the spatial configuration, several kinds of mean distances are given in the table. The mean distance between two nucleons (R(N N )) and that between anti-kaon and center-of-mass of two nucleons (R(K-[N N ])) are calculated with the obtained complex-scaled wave function of theKN N resonance |Φ θK N N as wherex 1,θ andx 2,θ indicate Jacobi-coordinate operators that are complex-scaled. Mean distance of aKN pair with isospin I (R(KN, I)) is calculated as wherePK N (I) is anKN isospin projector. It is noted that two mean distances are obtained independently of θ [26][27][28][29]. In the complex scaling method, expectation values of distances are necessarily to be complex-valued since resonance states are treated as Gamow states. However, we refer to such complex-valued distances because we expect that they are useful 17/23 guide for the spatial configuration of the resonant states, especially when the imaginary part of them is small compared with the real part. As a result of calculation of these mean distances, it is found that there is not so large difference between the results of two potentials. In both cases, the imaginary part of all mean distances is small compared to the real part. When we see the real part, the N N mean distance is about 2.2 fm. For the comparison, the result of an earlier study with a variational calculation using a chiral SU(3)-basedKN potential [20] is shown on the last column in the table. The N N mean distance of the present study is found to be equal to that of the variational calculation. As mentioned in Ref. [20], this N N distance is almost identical to the mean distance between two nucleons in nuclear matter with normal density. As for theKN distance, the mean distance for the I = 0 component is smaller than that for the I = 1 component. This is due to the strongKN attraction in the I = 0 channel. The variational calculation gives the similar result. However, the present values of theKN distance are smaller than those of the variational calculation. The same tendency has been found also in our previous study of the Λ(1405) resonance treated as a two-body system of I = 0KN -πΣ [28]. We consider that such a difference is caused by the treatment of thē KN N resonance: In the present study it is treated as a Gamow state, whereas it is treated as a bound state approximately in the variational study. By the way, compared with the case of in-vacuum where the I = 0KN pair forms a Λ * resonance, the mean distance of the I = 0KN pair is slightly larger in case of theKN N resonance. We consider that such a small elongation is due to the attraction from the other nucleon.

NN/KN correlation density in the K − pp with the complex scaling method
We investigate the spatial configuration of theKN N resonance in more detail. Similarly to a former study with a variational approach [20], we calculate correlation density for N N and KN pairs with the CSM wave function of theKN N resonance to visualize its structure.
We give a brief explanation on the calculation of such densities, since they have to be calculated carefully in the complex scaling method. Here, we consider the case of the N N correlation density, as an example. In the usual quantum mechanics, the expectation value of the N N -correlation-density operator,ρ N N (d) ≡ δ 3 (x 1 − d), is calculated. In the complex scaling method, a wave function is complex-scaled with the scaling operator U (θ). At the same time, an operatorÔ is also complex-scaled asÔ θ = U (θ)ÔU −1 (θ). With this transformation, the coordinatex n in the operatorÔ is complex-scaled to bex n exp(iθ). Thus, the N N correlation density ρ N N (d) should be calculated with the CSM wave function as In the same way, the isospin-separatedKN correlation density ρK N (I) (d) can be calculated, to begin with the operatorρK N (I) (d) ≡ δ 3 (rK N − d)PK N (I) whererK N is aKN relative coordinate operator andPK N (I) is aKN isospin projector.
As explained above, we calculate the N N andKN correlation densities with the obtained CSM wave function of theKN N resonance. Here, a typical result of the NRv2c potential 18 NRv2c (f π =110) θ=15 deg.  Fig. 5 shows the N N correlation density. In a short distance, both the real and imaginary parts of the density is confirmed to be suppressed due to the strong repulsive core of the 19 NRv2c (f π =110) θ=15 deg. Fig. 7 Comparison of I = 0KN correlation density in theKN N resonance and that of Λ * resonance. The I = 0KN correlation density of the Λ * resonance is shown with green diamonds, together with that in theKN N resonance shown in Fig. 6. The densities of both resonances are normalized to unity to be displayed.
Av18 N N potential. In Fig. 6, theKN correlation densities for the isospin 0 and 1 components are displayed. Both densities are normalized to unity for the comparison. In the figure, we can confirm directly the consequence of the strongKN attraction in the I = 0 channel, as mentioned in the previous section. In other words, the I = 0KN component is found to distribute more compactly compared with the I = 1 component. Furthermore, the I = 0KN correlation density is compared with the density of the Λ * resonance. The Λ * density is theKN density of two-bodyKN system with I = 0 that is calculated with the ccCSM+Feshbach method using the same potential and the same scaling angle. As displayed in Fig. 7, both densities look rather similar to each other. Therefore, the present study with the ccCSM+Feshbach method also indicates that the Λ * resonance still survives in the K − pp resonance, as pointed out in the former study with a variational approach [20].

Summary and future plan
We have proposed a new method where the coupled-channel complex scaling method and the Feshbach projection are combined, and applied it (ccCSM+Feshbach method) to a threebody kaonic nucleus K − pp. Originally K − pp is aKN N -πΣN -πΛN coupled-channel system, but has been effectively reduced to a single-channel problem ofKN N by a channel elimination in the method. Recall that, since the ccCSM+Feshbach method is based on the complex scaling method, theKN N resonance has been regarded as a Gamow state with the correct boundary condition as a resonance.
In ccCSM+Feshbach method, the extended closure relation (ECR), which is held in the complex scaling method (CSM), is essentially important. In fact, Green's function for Qspace (outer space of the model space P ) needed to eliminate the Q space, has been obtained easily from ECR, since the ECR is well realized with the L 2 Gaussian basis function only. First, we have tested the ccCSM+Feshbach method in a two-bodyKN -πY coupledchannel system, and confirmed that it reproduces completely results of calculation treating all channels explicitly for both scattering and resonant problems. 20/23 At the first application of the ccCSM+Feshbach method to the three-body system of K − pp, we have examined an energy-independentKN (-πY ) potential derived phenomenologically. Even when the original potential is energy-independent, the effectiveKN potential appeared in the ccCSM+Feshbach method has an energy dependence as a result of channel elimination by the Feshbach projection, and the self-consistency for the complexKN energy has to be considered when we search for resonance states. We have tested two ansatz for the self-consistency, Field picture and Particle picture, and obtained a self-consistent solution successfully for each ansatz. We have found that the binding energy ofKN N does not depend on the ansatz, while the decay width depends on the ansatz strongly: Field picture gives decay width so small as half of Particle picture. Compared with the result of an earlier study with ATMS method using the same potential [19], the result of Particle picture is found to be close to the past result; (B(KN N ), Γ/2) = (45.8, 27.2) MeV.
For more theoretical investigation of the K − pp system, we have used an energy-dependent KN (-πY ) potential which was proposed in our previous work based on the chiral SU(3) theory. Two versions of non-relativistic potentials, NRv1c and NRv2c [28], have been examined. Also for those energy-dependent potentials, self-consistent solutions of theKN N resonance with the ccCSM+Feshbach method have been found. Similarly to the case of the energyindependent potential, the decay width is rather small in Field picture compared with that in Particle picture. In case of NRv2c potential, the binding energy and half decay width of theKN N resonance are obtained to be (26.7 ± 5.5, 12.6 ± 3.6) MeV with Field picture and (27.6 ± 2.9, 23.5 ± 8.3) MeV with Particle picture. The NRv1c potential gives slightly larger binding energy. As for the spatial configuration ofKN N resonance, we have calculated N N andKN correlation densities from the wave function, carefully following the procedure of the complex scaling method. Those densities are useful tool for intuitive understanding of the structure ofKN N although they are given as complex values in the complex scaling method. The N N correlation density is strongly suppressed at short distances as a result of the N N repulsive core. The N N mean distance is found to be about 2.2 fm with a small imaginary part. This distance is almost equal to the N N distance of a nuclear matter with the normal density. Concerning theKN correlation density, we have reconfirmed the survival of the Λ * resonance (I = 0KN resonance) in theKN N resonance, which was pointed out in a previous study with a variational approach [20].
Thus, through the present study with the ccCSM+Feshbach method, we have confirmed that the K − pp system is shallowly bound with a chiral SU(3)-based energy-dependent potential, as reported in earlier studies employing the same type ofKN potential [20][21][22]. However, in case of NRv2c potential with Particle picture, we have always obtained another quasi self-consistent solution around theKN energy E(KN ) ∼ (−60, −60) MeV. ("quasi selfconsistent" means a local minimum for the quantity |E(KN ) In − E(KN ) Cal | which is an indicator for the self-consistency as explained in the section 4.1.) In such quasi self-consistent solutions, aKN N resonance appears near the πΣN threshold with large decay width. Since Λ(1405) has the double pole structure with the NRv2c potential as many studies with chiral SU(3) models [29], we think that those quasi-consistent solutions are probably related to the lower pole of Λ(1405). While, the solutions reported in the previous section must be related to the higher pole. In other words, K − pp is supposed to have the double pole structure as same as the Λ(1405), as suggested in an earlier work with Faddeev-AGS approach [39]. In 21/23 order to have definite conclusion on this issue, we need more delicate calculation for the deeper pole, since it has large imaginary part.
In this article, we have successfully obtained the solution of the K − pp resonance with the ccCSM+Feshbach method. However, since we have eliminated πY N channel by the Feshbach projection, some of πY N dynamics might be lost in the present study. Toward more decisive conclusion on the K − pp problem, we will carry out a coupled-channel three-body calculation with explicit πY N channels. On the experimental side, results of the K − pp search are going to be reported from two experimental groups at J-PARC (E15 [40] and E27 [41]). We hope that these experimental results will provide us useful information of the K − pp.
Since extending application of the ccCSM+Feshbach method to four-body systems is straightforward, we can investigate rather easily four-body systems such as a kaonic nucleus K − ppn and a double kaonic nucleus K − K − pp, which have been investigated with a variational method [21] and Faddeev-Yakubovsky approach [42]. Generally, the method can be applied to various kinds of mesic nuclei which involve some decay modes, for example, mesic nuclei with η [43], η ′ [44], ω mesons and D meson in charm sector [45]. Those are interesting systems and in the scope of our study with the ccCSM+Feshbach method.