# molecule state with and studied with three-body calculation

###### Abstract

A system with and is investigated with non-relativistic three-body calculations by using effective , and interactions. The interaction describes the as a molecule, and the interaction is adjusted to give and states as molecules. The present investigation suggests that a bound state can be formed below the threshold (1930 MeV) with a MeV width of three-hadron decays, which are dominated by and . It is found that the state is a weakly bound hadron molecular state with a size larger than an particle because of the repulsive interactions.

###### pacs:

14.20.Gk, 13.75.Jz, 13.30.Eg, 21.45.-v^{†}

^{†}preprint: YITP-08-53

## I Introduction

Exploring composite systems of mesons and baryons is a challenging issue both in theoretical and experimental hadron-nuclear physics. One of the historical examples in two-hadron systems is as a quasi-bound state of Dalitz:1959dn . For mesonic resonances, the scalar mesons, and , are also the candidates of the hadronic molecular states Weinstein:1982gc . Baryon resonances as three-hadron systems have been also investigated theoretically for systems of Bicudo:2003rw ; LlanesEstrada:2003us ; Kishimoto:2003xy , MartinezTorres:2007sr and ikeda07-jps ; KanadaEn'yo:2008wm . Based on the idea to regard as a quasi-bound state akaishi02 ; yamazaki02 , bound systems of a few nucleons with anti-kaon were investigated in Refs. akaishi02 ; yamazaki02 ; Yamazaki:2003hs ; Yamazaki:2007hj ; shevchenko07 ; ikeda07 ; yamazaki07 ; dote08 .

Recently a baryonic resonance with and composed by has been studied in details by the authors in Ref. KanadaEn'yo:2008wm based on three-body calculation with attractive interactions given by Refs. akaishi02 ; yamazaki07 ; hyodo07 . In this system, the anti-kaons play unique roles, because they have enough attraction with the nucleon to form a quasi-bound state as and possess so heavy mass to provide small kinetic energy in the system. The quasi-bound state of has a characteristic structure that one of the anti-kaons forms with the nucleon (-cluster) as seen also in system Yamazaki:2007hj , and the other anti-kaon spreads for long distance. This structure is caused by strong attraction with .

In this paper, we explore quasi-bound states of the system with and , assuming that the and systems have enough attractions to form quasi-bound states of in and () in (), respectively. We use the effective interactions of extracted by Akaishi-Yamazaki (AY) akaishi02 ; yamazaki07 and Hyodo-Weise (HW) hyodo07 in phenomenological way and chiral dynamics, respectively. These interaction provide as a quasi-bound state with and also weak attraction in the channel. The effective interactions are adjusted to reproduce the masses and the widths of and as the molecular states. The interactions are known to have strong repulsion in channel. We use the potential fitted by observed scattering lengths.

The “fate” of the molecular state strongly depends on its binding energy. If the energy of the system is above the lowest threshold of the subcomponents, the states can decay to the subcomponents and the width gets very large. If the state is bound with moderate binding energy below all the thresholds of +, + and +, the state is quasi-stable against these decay modes and has comparable decay width with those of the two-particle subsystems. For deeply bound system, since the constituents largely overlap each other, the molecular picture may be broken down and two-body decays are enhanced.

Having strong attractions in and subsystems, it is naturally expected that forms a hadron molecule below the thresholds of - and (-. The question arising here is whether or not the attractions are so strong that the hadronic molecular picture breaks down in deeply bound state and the quasi-bound state has large width, or in opposite direction, whether or not the repulsion of is too strong for spoiling the bound state.

In Sec. II, we describe the framework of the present calculations. We apply a variational approach with a Gaussian expansion method Hiyama03 to solve the Schrödinger equation of the three-body system. By treating the imaginary potentials perturbatively, we find the quasi-bound state. In Sec. III, we present our results of the three-body calculation. In analysis of the wave functions, we discuss the structure of the state. Section IV is devoted to summary of this work.

## Ii Formulation

We apply a non-relativistic three-body potential model for the system. The effective two-body interactions are given in local potential forms. The wave function is calculated by solving Schrödinger equation with a Gaussian expansion method for the three-body system. In this section, we briefly explain the formulation and interactions used in the present work. The details of the formulation and the interaction are discussed in Ref. KanadaEn'yo:2008wm .

### ii.1 Hamiltonian

In the present work, the Hamiltonian for the system is given by

(1) |

with the kinetic energy , the effective interaction , the interaction and the interaction . These interactions are given by -independent local potentials as functions of -, - and - distances, , and defined by , and , respectively, with spatial coordinates , , for the kaon, the anti-kaon and the nucleon. For convenience, we introduce Jacobian coordinates, and , in three rearrangement channels as shown in Fig. 1. We assume isospin symmetry in the effective interactions, and we also neglect the mass differences among , and , and that between proton and neutron by using the averaged masses, MeV and MeV. We do not consider three-body forces nor transitions to two-hadron decays, which will be important if the constituent hadrons are localized in a small region.

The kinetic energy is simply given by the Jacobian coordinates with one of the rearrangement channels as

(2) |

with the reduced masses and for the corresponding configuration, for instance, and for the rearrangement channel .

The effective interactions, , and , are obtained by -wave two-body scattering with isospin symmetry. The explicit expression of the effective interactions will be given in Sec. II.2. Open channels of and ( and for , and and for ) are implemented effectively to the imaginary parts of the interactions and . Consequently, the Hamiltonian (1) is not hermitian. In solving Schrödinger equation for , we first take only the real part of the potentials and obtain wavefunctions in a variational approach. Then we calculate bound state energies as expectation values of the total Hamiltonian (1) with respect to the obtained wave functions. The widths of the bound states are evaluated by the imaginary part of the complex energies as .

### ii.2 Effective interactions

In this subsection, we explain the details of the effective interactions of the , and two-body subsystems in our formulation. The interaction parameters and the properties of the two-body subsystems are listed in table 1.

#### ii.2.1 interaction

In this work, we use the same potential as used in Ref.KanadaEn'yo:2008wm for the calculations. We consider two different effective interactions to estimate theoretical uncertainties. These two interactions were derived in different ways. One of the interaction that we use is given by Hyodo and Weise in Ref. hyodo07 and was derived based on the chiral unitary approach for -wave scattering amplitude with strangeness . The other interaction is Akaishi-Yamazaki (AY) potential derived phenomenologically by using scattering and kaonic hydrogen data and reproducing the resonance as a bound state at 1405 MeV akaishi02 ; yamazaki07 . Both interactions have so strong attraction in as to provide the as a quasi-bound state of the system, and have weak attraction in . Hereafter we refer the quasi-bound state as .

The potential is written in a one-range Gaussian form as

(3) | |||||

with the isospin projection operator and the range parameter . The potential depth are given in a complex number reflecting the effects of the open channel of and . The numbers for and are given in Table 1. For the Hyodo-Weise potential, we use the parameter set referred as HNJH in Ref. hyodo07 , which was obtained by the chiral unitary model with the parameters of Ref. Hyodo03 . We refer this potential as “HW-HNJH potential”. The energy of the HW-HNJH potential is fixed at MeV which is the resonance position of , since the energy dependence in the potential is small in the region of interest and it was found in Ref. KanadaEn'yo:2008wm that the results were not sensitive to the choice of the energy for the system.

The important difference between the two interactions is the binding energy of the system. In chiral unitary approaches for the meson-baryon interactions, the resonance is described as a quasi-bound state Hyodo:2008xr located at MeV in scattering amplitude Jido:2003cb . This is a consequence of the double pole nature that is described by superposition of two poles as found in Refs. Oller:2000fj ; Jido:2002yz ; Jido:2003cb . For the AY potential, the resonance was reproduced at 1405 MeV as PDG reported. Thus, the AY potential has stronger attraction in than the HW-HNJH potential. The properties of the two-body system obtained by these potentials are summarized in Table 1.

#### ii.2.2 interaction

The interaction is derived in the present work under the assumption that forms quasi-bound states in and , which correspond to and , respectively. Thus, we use strong effective single-channel interactions which reproduce the masses and widths of and as the quasi-bound states. We refer the quasi-bound states as .

We take the one-range Gaussian form,

(4) |

where the range parameter is chosen to be the same value as that of the interaction. We adjust the strength to fit the and masses and the widths with the energies of two-body calculations of the system. The particle data group (PDG) reports PDG the and have MeV and MeV masses with the MeV and MeV widths, respectively, in average of the compilation of the experimental data. The dominant decay modes are for and for . We take the mass 980 MeV and the width 60 MeV as the inputs to determine the interactions in both the and channels. Then we get MeV for fm and MeV for fm, by fitting the energy of the bound state to the meson mass and the width. We refer the former potential as “KK(A)” and the latter as “KK(B)”. In this phenomenological single-channel interaction, the effect of the two-meson decays such as and decays is incorporated in the imaginary part of the effective interaction.

In the present parametrization of the potential, we have fitted the potential strengths to reproduce the PDG values of the and masses as bound state energies of calculated with the perturbative treatment of the imaginary potential. When we directly calculate the pole position of the scattering amplitude in Lippmann-Schwinger equation with the present potential, we get the value MeV for the parameter set (A). This is obtained above the threshold in the first Riemann sheet as a virtual state. This pole is consistent with the pole position of scattering amplitude obtained by the chiral unitary approach Oller:1997ti ; Oller:1997tiE . In the chiral unitary approach, -wave scattering amplitudes with and were reproduced well by coupled channels of , and . The and meson are obtained as the resonance poles at 993.5 MeV for and 1009.2 MeV for Oller:1997tiE . The is described dominantly by scattering, while for the scattering is also important as well as scattering. These values are slightly higher than the masses reported by PDG, which are given by the peak position of the spectra. We will discuss ambiguity of the interactions in later section.

#### ii.2.3 interaction

We construct the interaction based on observed scattering lengths KNint . The interactions are known to be strong repulsion in the channel and very weak in the channel. The experimental values of the scattering lengths for the and channels are fm and fm KNint . In the present calculation, we assume no interaction in the channel. For the channel, we use phenomenological interaction with the one-range Gaussian form again,

(5) |

where the range parameter is chosen to be the same value as that of the interaction. We adjust the strength to reproduce the experimental scattering strength and obtain MeV and MeV for fm and fm, respectively. We refer the former parametrization as “KN(A)” and the latter one as “KN(B)”.

parameter set of interactions | ||

(A) | (B) | |

HW-HNJH | AY | |

(fm) | 0.47 | 0.66 |

(MeV) | ||

(MeV) | ||

state | ||

Re (MeV) | -11 | -31 |

Im (MeV) | -22 | -20 |

- distance (fm) | 1.9 | 1.4 |

KK(A) | KK(B) | |

(fm) | 0.47 | 0.66 |

(MeV) | ||

state | ||

Re (MeV) | -11 | -11 |

Im (MeV) | -30 | -30 |

- distance (fm) | 2.1 | 2.2 |

KN(A) | KN(B) | |

(fm) | 0.47 | 0.66 |

(MeV) | 0 | 0 |

(MeV) | 820 | 231 |

(fm) | 0 | 0 |

(fm) | 0.31 | 0.31 |

### ii.3 Three-body wave function

The three-body wave function is described as a linear combination of amplitudes of three rearrangement channels (Fig. 1). In the present calculation, we take the model space limited to and of the orbital-angular momenta for the Jacobian coordinates and in the channel owing to the fact that the effective local potentials used in the present calculations are derived in consideration of the -wave two-body dynamics. Then the wave function of the system with and is written as

(6) |

where the specifies the isospin configuration of the wave function , meaning that the total isospin for the system is given by combination of total isospin for the subsystem and isospin for the nucleon.

The wave function of the system is obtained by solving the Schrödinger equation,

(7) |

In solving the Schrödinger equation for the system, we adopt the Gaussian expansion method for three-body systems given in Ref. Hiyama03 as same way as done in Ref.KanadaEn'yo:2008wm with the parameters fm and fm, and for all the channels, . We treat the imaginary part of the potentials perturbatively. We first calculate the wave function for the real part of the Hamiltonian () with variational principle in the model space of the Gaussian expansion. After this variational calculation, we take the lowest-energy solution. The binding energy of the three-body system is given as .

Next we estimate the imaginary part of the energy for the total Hamiltonian by calculating the expectation value of the imaginary part of the Hamiltonian with the obtained wave function :

(8) |

The total energy is given as , and the decay width is estimated as . In the present calculation, we have only three-body decays such as , , and decays for the state by the model setting.

The perturbative treatment performed above is justified qualitatively in the case of . In the two-body systems, and , we find that this condition is satisfied reasonably, observing that MeV is much smaller than MeV, and MeV is also much smaller than MeV. Also in the case of the system, it is found that the absolute values of the perturbative energy MeV is much smaller than the real potential energy MeV in the present calculations.

We also calculate quantities characterizing the structure of the three-body system, such as spatial configurations of the constituent particles and probabilities to have specific isospin configurations. These values are calculated as expectation values of the wave functions.

The root-mean-square (r.m.s.) radius of the state is defined as the average of the distribution of , and by

(9) |

which is measured from the center of mass of the three-body system. We also calculate the r.m.s. values of the relative distances between two particles,

(10) | |||||

(11) | |||||

(12) |

Here , and are the distance, distance and the distance, respectively.

We also introduce the probabilities for the three-body system to have the isospin states as

(13) |

where is the projection operator for the isospin configuration , as introduced before. We calculate the probabilities that the three-body system has the isospin configurations of , where the total isospin is given by combination of total isospin for the subsystem and the kaon isospin :

(14) |

where is again the isospin projection operator.

## Iii Results

In this section, we show the results of investigation of the system with and . We consider two parameter sets (A) and (B) for the two-body interactions listed in Table 1. For the interactions, we use (A) the HW-HNJH potential and (B) the AY potential. For the and interactions, we use the phenomenological interactions derived in Sec. II.2.2 and II.2.3: KK(A) and KN(A) for set (A), and KK(B) and KN(B) for set (B). In addition, we study the effect of the repulsion by switching off the interaction in the parameter sets (A) and (B).

#### iii.0.1 Properties of state

First of all, we find that,
in both calculations (A) with the HW-HNJH and
(B) with the AY potentials,
the bound
state is obtained below all threshold energies of
the , and channels, which correspond to
the , and states,
respectively.^{1}^{1}1In the present calculation, because the
interaction is adjusted to
reproduce the and scalar mesons having
the same mass and width, it is independent of the total isospin of the
subsystem and the thresholds of and
are obtained as the same value.
This means that the obtained bound state is stable against breaking up to
the subsystems.
We show
the level structure of the
system measured from the threshold
in Fig. 2.
The values of the real and imaginary parts of the obtained energies are
given in Table 2. The imaginary part of the energy
is equivalent to the half width of the quasi-bound state.
The contribution of each decay mode to the imaginary energy is shown
as an expectation value of the imaginary potential
and the results obtained without the interaction are also given.
For the HW-HNJH potential, since the original HW-HNJH potential
is moderately dependent on the energy of the system,
we have calculated the bound state energy with the potential
for the energy at and
found that the energy ()
dependence of the HW-HNJH potential is small in the result.

parameter set | (A) | (A) | (B) | (B) |
---|---|---|---|---|

HW-HNJH | HW-HNJH | AY | AY | |

on | off | on | off | |

Re (MeV) | ||||

Im (MeV) | ||||

(MeV) | ||||

(MeV) | ||||

(MeV) | ||||

(MeV) |

Let us discuss first the results with the interaction in detail. The binding energy of the state measured from the three-body threshold is larger in the result with (B) than that with (A), as found to be MeV and MeV in the cases of (A) and (B), respectively. This is because the AY potential gives a deeper binding of the state than the HW-HNJH potential due to the stronger attraction. These values have meaning just for the position of the quasi-bound state in spectrum. It is more physically important that the bound state appears about 10 MeV below the lowest two-body threshold, , in both cases (A) and (B). This energy is compatible to nuclear many-body system, and it is considered to be weak binding energy in the energy scale of hadron system. This weak binding system has the following significant feature. The width of the state is estimated to be MeV from the imaginary part of the energy. Comparing the results of the with the properties of the two-body subsystems shown in Table 1, it is found that the real and imaginary energy of the state is almost given by the sum of those of and (or ), respectively. This indicates that two subsystems, and , are as loosely bound in the three-body system as they are in two-body system.

The decay properties of the state can be discussed by the components of the imaginary energy. As shown in Table 2, among the total width MeV, the imaginary potentials of the with and the with give large contributions as about MeV and MeV, respectively. The former corresponds to the decay channel and gives the decay mode with . The latter is given by the decay, which is dominated by . Contrary, the and the interactions provide only small contributions to the imaginary energy. This is because, as we will see later, the subsystem is dominated by the component due to the strong attraction and the subsystem largely consists of the component as a result of the three-body dynamics. The small contributions of the and the interactions to the imaginary energy implies that the decays to and are suppressed. Therefore, we conclude that the dominant decay modes of the state are and . This is one of the important characters of the bound system.

Although the obtained state is located below the thresholds of , and , there could be a chance to access the state energetically by observing the , and channels in the final states, because these resonances have as large widths as the state. Since, as we will show later, the state has the large component, the state could be confirmed in its decay to by taking coincidence of the out of the invariant mass of and the three-body invariant mass of the decay.

Here we comment on theoretical uncertainty of the energy of the state. In the present calculations, the interactions are obtained under the assumption that the attractive potentials provide and as quasi-bound states and are phenomenologically adjusted to reproduce the masses and the widths of and . As discussed above, in the present result, the interaction with gives the dominant contribution to the total width of the state. We estimate theoretical uncertainty of the width of the state by changing the inputs of the width in the range from to as reported in PDG. We obtain the MeV for the state. We also find that the state becomes unbound if the interaction with is less attraction than 70% of the present values, in which with is not bound in the two-body system.

Finally we discuss the role of the repulsion in the system. In Fig. 2 and Table 2, we show the results calculated without the interaction. We find that the binding energy of the state is 20 MeV larger than the case of the calculation with the repulsion in both (A) and (B) cases, and that the absolute value of the imaginary energy also becomes larger as MeV and MeV for (A) and (B), respectively. These values correspond to the MeV width for the state. The reason that the three-body system without the interaction has the more binding and the larger width is as follows. In general, the three-body system has less kinetic energy than the two-body system because of larger reduced mass in the three-body system. With less kinetic energy the system can localize more. As a result of the localization of the system, the system can gain more potential energy and larger imaginary energy in the case of no interaction than the case with the repulsion. In other words, thanks to the repulsion, the state is weakly bound and its width is suppressed to be as small as the sum of the widths of the subsystems.

#### iii.0.2 Structure of state

(A) | (A) | (B) | (B) | |
---|---|---|---|---|

HW-HNJH | HW-HNJH | AY | AY | |

on | off | on | off | |

isospin configuration | ||||

0.93 | 1.00 | 0.99 | 1.00 | |

0.07 | 0.00 | 0.01 | 0.00 | |

0.09 | 0.25 | 0.17 | 0.25 | |

0.91 | 0.75 | 0.83 | 0.75 | |

spatial structure | ||||

(fm) | 1.7 | 1.0 | 1.4 | 1.0 |

(fm) | 2.1 | 1.3 | 1.3 | 1.2 |

(fm) | 2.3 | 1.4 | 2.1 | 1.5 |

(fm) | 2.8 | 1.6 | 2.3 | 1.6 |

We discuss the structure of the system with . For this purpose, we analyze the wave functions obtained in the present few-body calculation in terms of the spatial structure and the isospin configuration of the system.

We first investigate the isospin configuration of the state. We show the isospin components of subsystems and in Table 3. It is found that the subsystem has a dominant component because of the strong interaction in the channel. In the subsystem, the configuration is dominant while the component gives minor contribution. This isospin configuration is caused by the following reason. In both and channels, the attraction is strong enough to provide quasi-bound states of and . In addition, since these scalar mesons have similar masses and widths, the interactions in and adjusted to these masses and widths are similar to each other. In fact, we use the same parametrization for the interactions in the present calculation, which gives isospin-blind potential. Therefore, the interaction plays a major role to determine the isospin configuration of the state. Since the interaction has stronger attraction in the channel than in the channel, the system prefers to have in the subsystem. If the subsystem has pure configuration, which is the case without the repulsion, the subsystem should be composed by and with the ratio of 1:3 to have of . Thus, the with dominates the system, and simultaneously the with is dominant component. The small deviation from the pure configuration in the state originates in the repulsion.

Next we discuss the spatial structure of the bound system. In Table 3, we show the root-mean-square (r.m.s.) radius of , defined in Eq. (9), and r.m.s. values for the -, - and - distances, , , defined in Eqs. (10), (11) and (12), respectively, in the state. The r.m.s. distances of the two-body systems, and , are shown in Table 1. It is interesting that the present result shows that the r.m.s. - and - distances in the three-body state have values close to those in the quasi-bound two-body states, and , respectively. This implies again that the two subsystems of the three-body state have very similar characters with those in the isolated two-particle systems.

The r.m.s. - distance is relatively larger than the r.m.s. - and - distances due to the repulsion. The effect of the repulsive interaction is important in the present system. Without the repulsion, we obtain smaller three-body systems as shown in Table 3. Especially the distances of the two-body subsystems are as small as about 1.5 fm, which is comparable with the sum of the charge radii of proton (0.8 fm) and (0.6 fm). For such a small system, three-body interactions and transitions to two particles could be important. In addition, the present picture that the system is described in non-relativistic three particles might be broken down, and one would need relativistic treatments and two-body potentials with consideration of internal structures of the constituent hadrons.

Combining the discussions of the isospin and spatial structure of the system, we conclude that the structure of the state can be understood simultaneous coexistence of and clusters as shown in Fig. 3. This does not mean that the system is described as superposition of the and states, because these states are not orthogonal to each other. The probabilities for the system to have these states are 90% as seen in Table 3. It means that is sheared by both and at the same time.

It is interesting to compare the obtained state with nuclear systems. As shown in Table 3, the hadron-hadron distances in the state are about 2 fm, which is as large as nucleon-nucleon distances in nuclei. In particular, in the case (A) of the HW-HNJH potential, the hadron-hadron distances are larger than 2 fm and the r.m.s. radius of the three-body system is also as large as 1.7 fm. This is larger than the r.m.s. radius 1.4 fm for He. If we assume a uniform sphere density of the three-hadron system with the r.m.s. radius 1.7 fm, the mean hadron density is evaluated as 0.07 hadrons(fm). Thus the state has large spatial extent and dilute hadron density.

Order of two-body decay widths can be estimated by geometrical argument. Let us suppose that transitions of three-body bound state to two particles are induced by contact interactions. In such cases, the transition probability is proportional to square of density, , where is the radius of the three-body system. In the present calculation, the system of the radius is obtained as 1.7 fm in the parameter set (A). Assuming a typical decay width and radius of baryon resonances as 300 MeV and 0.8 fm, we estimate the two-body decay width as MeV. This is much smaller than the expected three-body decays.

## Iv Summary

We have investigated the system with and in non-relativistic three-body calculation. We have used the effective potentials proposed by Hyodo-Weise and Akaishi-Yamazaki, which reproduce the as a quasi-bound state of . The interactions are determined so as to reproduce and as quasi-bound states in with and channels, respectively. The potentials of are adjusted to provide the observed scattering lengths, having strong repulsion in and no interaction in . The present three-body calculation suggests that a weakly quasi-bound state can be formed below all threshold energies of the +, + and +. The calculated energies of the quasi-bound state are MeV and MeV from the threshold in the results with HW and AY potentials, respectively. The width for three-hadron decays is estimated to be MeV. It has been found that the binding energy and the width of the state is almost the sum of those in and .

Investigating the structure of the system, we have found that, in the state, the subsystems of and dominate the and , respectively, and that these subsystems have very similar properties with those in the two-particle systems. This leads that the quasi-bound system can be interpreted as coexistence state of and clusters and is a constituent of both and at the same time. As a result of this feature, the dominant decay modes are from the decay and from the decay, and the decays to and channels are suppressed.

We also have found that the root-mean-square radius of the state is as large as 1.7 fm and the inter-hadron distances are lager than 2 fm. These values are comparable to, or even larger than, the radius of He and typical nucleon-nucleon distances in nuclei, respectively. Therefore, the system more spatially extends compared with typical hadronic systems. These features are caused by weakly binding of the three hadrons, for which the repulsive interaction plays an important role.

## Acknowledgments

The authors would like to thank Professor Akaishi, Dr. Hyodo and Dr. Doté for valuable discussions. They are also thankful to members of Yukawa Institute for Theoretical Physics (YITP) and Department of Physics in Kyoto University, especially for fruitful discussions. This work is supported in part by the Grant for Scientific Research (No. 18540263 and No. 20028004) from Japan Society for the Promotion of Science (JSPS) and from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan. A part of this work is done in the Yukawa International Project for Quark-Hadron Sciences (YIPQS). The computational calculations of the present work were done by using the supercomputer at YITP.

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