# Spin 3/2 Penta-quarks in anisotropic lattice QCD

###### Abstract

A high-precision mass measurement for the pentaquark (5Q) in channel is performed in anisotropic quenched lattice QCD using a large number of gauge configurations as . We employ the standard Wilson gauge action at and the improved Wilson (clover) quark action with on a lattice with the renormalized anisotropy as . The Rarita-Schwinger formalism is adopted for the interpolating fields. Several types of the interpolating fields with isospin are examined such as (a) the NK-type, (b) the (color-)twisted NK-type, (c) a diquark-type. The chiral extrapolation leads to only massive states, i.e., GeV in channel, and GeV in channel. The analysis with the hybrid boundary condition(HBC) is performed to investigate whether these states are compact 5Q resonances or not. No low-lying compact 5Q resonance states are found below 2.1GeV.

###### pacs:

12.38.Gc, 12.39.Mk, 14.20.-c, 14.20.Jn## I Introduction

The recent discovery of the manifestly exotic baryon by the LEPS group at SPring-8 has made a great impact on the exotic hadron physics nakano . Apart from other pentaquark baryon candidates, NA49 and H1 , several other candidates of exotic hadrons have also been discovered, such as , , , and exotics . They also receive an increasing interest from theoretical side as well. is supposed to have baryon number , charge and strangeness . Since is the simplest quark content to implement this quantum number, is a manifestly exotic penta-quark(5Q) state. The penta-quark had been considered several times even before the experimental discovery diakonov ; jaffe-76 ; strottman ; weigel ; praszalowicz . In particular, LABEL:diakonov provided the direct motivation of the experimental search nakano . The discovered peak in the invariant mass is centered at GeV with a width smaller than 25 MeV. At the present stage, some groups confirmed the LEPS discoverydiana ; clas ; saphir ; experiments , while the others reported null resultsnull . It will still take a while to establish the existence or non-existence of experimentally hicks . LABEL:saphir claims that must be isoscalar, since no is observed in the invariant mass spectrum.

Enormous theoretical efforts have been devoted to 5Q baryons diakonov ; jaffe-76 ; strottman ; weigel ; praszalowicz ; oka ; zhu.review ; cohen ; itzhaki ; kim ; hosaka ; jaffe ; lipkin ; carlson-positive ; stancu ; jennings ; glozman ; enyo ; zhu ; matheus ; sugiyama ; carlson ; shinozaki ; huang ; maezawa ; shlee ; narodetskii ; sugamoto ; suganuma ; okiharu ; bicudo ; oset ; hosaka32 ; nishikawa32 ; takeuchi32 ; sugiyama32 ; zhu32 ; inoue32 ; jaffe32 ; huang32 ; capstic32 ; dudek32 ; hyodo32 ; nam32 . One of the most challenging problems in understanding its structure is its extremely narrow decay width as MeV pdg . Several ideas have been proposed: (1) possibility capstic32 , (2) Jaffe-Wilczek’s diquark picture jaffe , (3) hepta-quark picture bicudo ; kishimoto , (4) string picture suganuma ; sugamoto , (5) possibility hosaka ; enyo . Although each gives a mechanism to explain the narrow decay width, none of them can satisfy all the known properties of simultaneously.

In this paper, we are interested in the possibility, in particular. Note that the spin of has not yet been determined experimentally. In the constituent quark picture, the narrow decay width of penta-quarks can be understood in the following way hosaka ; enyo . We expect that the special configuration is dominant in the 5Q ground-state in channel. Although penta-quarks can decay to KN in the d-wave, the spectroscopic factor to find d-wave KN states in the dominant configuration vanishes. Since the decay is thus allowed only through its sub-dominant d-wave configuration, the decay width is suppressed. Note that it is further suppressed by the d-wave centrifugal barrier, leading to the significantly narrow decay width of penta-quarks. A possible disadvantage of assignment is that such a state tends to be massive due to the color-magnetic interaction in the constituent quark models, which seems to be one of the main reasons why there are only a limited number of effective model studies for spin 3/2 penta-quarks hosaka32 ; nishikawa32 ; takeuchi32 ; sugiyama32 ; zhu32 ; inoue32 ; jaffe32 ; huang32 ; capstic32 ; hyodo32 ; nam32 ; enyo . However, it is not clear whether these conventional framework is applicable to a new exotic 5Q system as without involving any modifications. Indeed, a model was proposed where a part of the role of the color-magnetic interaction can be played by the flavor-spin interaction, which makes the mass-splitting between the and the states smaller takeuchi32 .

There have been several lattice QCD calculations of 5Q states by todayscikor12 ; sasaki ; chiu ; kentacky ; ishii12 ; rabbit ; lasscock12 ; alexandrou12 ; csikor122 ; holland ; lasscock32 . However, these studies are restricted to channels except for a very recent one lasscock32 . Enormous efforts are being devoted to more accurate studies of states, using the variational technique to extract multiple excited states, among which a compact resonance state is sought for. Indeed, quite large scale calculations are planned and being performedfleming attempting to elucidate some of the mysterious natures of such as its diquark structure and/or non-localities desired in interpolating fields. Here, we emphasize again that these studies are aiming at states, not at states.

In this paper, we present anisotropic lattice QCD results on 5Q states in channels using a large number of gauge configurations as as an attempt to search for a low-lying 5Q state in channel. We adopt the standard Wilson gauge action at on the lattice with the renormalized anisotropy . The anisotropic lattice is known to serve as a powerful tool for high-precision measurements of temporal correlators klassen ; matsufuru ; nemoto ; ishii-gb . The large number of gauge configurations plays a key role in our calculation, because 5Q correlators in channels are found to be quite noisy. For quark action, we adopt -improved Wilson (clover) action with four values of the hopping parameter as . One of the purpose of our calculation is to examine how the results depend on the choice of interpolating field operators. We employ several types of interpolating fields as (a) the NK-type, (b) the (color-)twisted NK-type, (c) a diquark-type, and adopt a smeared source to enhance the low-lying spectra.

In channel, we obtain massive states GeV except for the diquark-type interpolating field, which involves a considerable size of the statistical error. In channel, we obtain more massive states GeV. None of these 5Q states appear below the NK threshold. Note that the NK threshold is raised up by about MeV due to the finite extent of the spatial lattice as fm, from which we expect the penta-quark signal to appear below the (raised) NK threshold considering the empirical mass difference between N+K(1440) and . To clarify whether our 5Q states are compact resonance or not, we perform an analysis with the hybrid boundary condition(HBC), which was recently proposed in LABEL:ishii12. HBC analysis indicates that no compact 5Q resonance is contained in our 5Q states both in channels.

The paper is organized as follows. In Sect. II, we discuss the general formalisms. We begin by introducing several types of interpolating fields, determining their parity transformation properties. We next consider the temporal correlator and its spectral decomposition. We finally discuss the two-particle scattering states involved in 5Q spectra, and introduce the hybrid boundary condition (HBC) to examine whether a state of our concern is a compact resonance state or not. Sect. III is devoted to the brief descriptions of our lattice action and parameters. In Sect. IV, we present our numerical results for channels in the standard periodic boundary condition(PBC). We show 5Q correlators of various interpolating fields, i.e., the NK-type, the (color-)twisted NK-type, the diquark-type. In Sect. V, we attempt to determine whether these 5Q states are compact 5Q resonance states or two-particle scattering states by using the HBC. In Sect. VI, we summarize our results.

## Ii General formalisms

### ii.1 Interpolating fields

We consider an iso-scalar interpolating field of NK-type in Rarita-Schwinger form ioffe ; benmerrouche ; hemmert as

where denotes the Lorentz index, refer to the color indices, and denotes the charge conjugation matrix. Unless otherwise indicated, the gamma matrices are represented in the Euclidean form given in LABEL:montvay.

We are also interested in the (color-)twisted NK-type interpolating field as

which is an extension to the one originally proposed in LABEL:scikor12 to study 5Q states. It has a slightly more elaborate color-structure than Eq. (II.1), suggesting somewhat stronger coupling to a genuine 5Q state, if it exists, than simple NK states.

Another interpolating fields of our possible interests are diquark-type interpolating fields such as

(3) |

which is an extension to the one proposed in Refs. sasaki ; sugiyama . The first factor corresponds to the scalar diquark (color , , ), which is expected to play important roles in hadron physics jaffe-exotica . The second factor corresponds to the vector diquark (color , , ). Note that, although the axial-vector diquark (color , , ) is considered to play a more important role than the vector diquark, it cannot replace the vector diquark due to its iso-vector nature. Unless otherwise indicated, we refer to Eq. (3) as the “diquark-type” interpolating field. We can also consider another interpolating field of diquark-type as

(4) |

which consists of the pseudo-scalar diquark (color , , ) and the vector diquark. However, actual lattice QCD calculation shows that its correlator is afflicted with quite a huge statistical error. A possible reason could be attributed to the fact that both of these diquark fields do not survive the non-relativistic limit. Hence, we do not consider this interpolating field in this paper.

Under the spatial reflection of the quark fields as

(5) |

all of these interpolating fields transform as

(6) |

for .

### ii.2 5Q correlators and parity projection

We consider the Euclidean temporal correlator as

(7) |

where projects the total 5Q momentum to zero. Since the spin 3/2 contribution from the temporal component of Rarita-Schwinger spinor vanishes in the rest frame, we can restrict ourselves to the spatial parts, i.e., . Now, Eq. (7) is decomposed in the following way:

(8) |

where denote the spatial part of the Lorentz indices, and denote the projection matrices onto the spin 3/2 and 1/2 subspaces defined as

(9) | |||||

They satisfy the following relations as

(10) | |||||

Here, summations over repeated indices are understood. and in Eq. (7) denote the spin 3/2 and 1/2 contributions to , respectively, which can be derived by operating and on , respectively. (In our practical lattice QCD calculation, we construct the Rarita-Schwinger correlator for and (fixed), and multiply from the left to obtain .)

In the asymptotic region (), contaminations of
excited states are suppressed. Considering the parity transformation
property Eq. (6), and are
expressed in this region as

where denote the projection matrices onto the “upper” and “lower” Dirac subspaces, respectively. and denote the lowest-lying masses in and channels, respectively. and represent the couplings to the interpolating field Eq. (II.1) with and states, respectively. In Eq. (II.2), we adopt the anti-periodic boundary condition along the temporal direction. A brief derivation of Eq. (8) and Eq. (II.2) is presented in Appendix. A. The forward propagation is dominant in the region , while the backward propagation is dominant in the region . To separate the negative (positive) parity contribution, we restrict ourselves to the region , and examine the “upper” (“lower”) Dirac component.

### ii.3 Scattering states involved in 5Q spectrum

We consider the (two-particle) scattering states involved in 5Q spectrum. For iso-scalar penta-quarks, NK and NK scattering states play an important role. (K does not couple to the iso-scalar channel.) These states are expressed as

(13) |

where and denote the spin of the nucleon and K, and denotes the spatial momentum allowed for a particular choice of the spatial boundary condition adopted. For instance, if these hadrons are subject to the spatially periodic boundary condition, their momenta are quantized as

(14) |

where denotes the spatial extent of the lattice. In contrast, if they are subject to the spatially anti-periodic boundary condition, their momenta are quantized as

(15) |

We first perform the parity projections. The positive and the negative parity states are obtained in the following way:

Assuming that the interactions between N and K and between N and K are weak, their energies are approximated as

(18) | |||||

(19) |

respectively. The scattering states which couple to penta-quarks are obtained as spin-3/2 projections of Eq. (II.3) and Eq. (II.3). The d-wave NK states and the s-wave NK states can couple to the channel, while the p-wave NK states and NK states can couple to the channel.

The scattering states with vanishing spatial momentum are exceptional in the following sense. On the one hand, the positive parity states vanish, because the first terms coincides with the second terms in Eq. (II.3) and Eq. (II.3) in the right hand side. On the other hand, the negative parity states are constructed only from the spin degrees of freedom, i.e., the spin degrees of freedom of the nucleon in Eq. (II.3), and the spin degrees of freedoms of the nucleon and K in Eq. (II.3). By counting the degeneracy of the resulting states, it is straightforward to see that no d-wave states are contained, i.e., Eq. (II.3) gives only s-wave NK states in channel, and that Eq. (II.3) gives only s-wave NK states in and channels.

### ii.4 Hybrid boundary condition(HBC)

0.1210 | 0.1220 | 0.1230 | 0.1240 | emp. | ||
---|---|---|---|---|---|---|

NK(s-wave) | PBC | 2.996 | 2.815 | 2.633 | 2.445 | 1.830 |

NK(p-wave) | PBC | 3.222 | 3.052 | 2.883 | 2.710 | 2.163 |

NK(p/d-wave) | PBC | 2.987 | 2.806 | 2.624 | 2.438 | 1.865 |

NK(s/p-wave) | HBC | 3.167 | 2.995 | 2.823 | 2.647 | 2.084 |

NK(p/d-wave) | HBC | 2.924 | 2.739 | 2.553 | 2.363 | 1.770 |

In order to determine whether a state of our concern is a compact 5Q resonance state or a scattering state of two particles, we use two distinct spatial boundary conditions(BC), i.e., the (standard) periodic BC(PBC) and the hybrid BC(HBC), which is recently proposed in LABEL:ishii12. In PBC, one imposes the spatially periodic BC on u,d and s-quarks. As a result, all the hadrons are subject to the periodic BC. In this case, due to Eq. (14), all hadrons can take zero-momentum, and the smallest non-vanishing momentum is of the form as

(20) |

which gives

(21) |

On the other hand, in HBC, we impose the spatially anti-periodic BC on u and d-quarks, whereas the spatially periodic BC is imposed on s-quark. Since N(), K() and K() contain odd numbers of u and d quarks, they are subject to the anti-periodic BC. Therefore, due to Eq. (15), N, K and K cannot have a vanishing momentum in HBC. The smallest possible momentum is of the form as

(22) |

Hence, its norm is expressed as

(23) |

In contrast, () is subject to the spatially periodic BC, since it contains even number of u and d quarks. Therefore, can have the vanishing momentum.

Switching from PBC, HBC affects the low-lying two-particle scattering spectrum. A drastic change is expected in the s-wave NK channel. In PBC, the energy of the lowest NK state is given as

(24) |

In contrast, in HBC, since both N and K are required to have non-vanishing momenta , the energy of the lowest NK state is raised up as

Note that the shift amounts typically to a few hundred MeV for fm.

HBC affects NK(d-wave), NK(p-wave), NK(p-wave) as well. However, these changes are not as drastic as that in NK(s-wave), because they are induced by the minor change in the minimum momentum from to . In PBC, the energies of the lowest two-particles states are expressed as

In HBC, they are shifted down as

Numerical values of NK and NK thresholds for each hopping parameter in spatial lattice of the size fm for both PBC and HBC are summarized in Table 1.

In contrast to the scattering states, HBC is not expected to affect a compact 5Q resonance so much. Since can have vanishing momentum also in HBC, the shift of the penta-quark mass originates only from the change in its intrinsic structure. In this case, the shift is expected to be less significant than the shift induced by the kinematic reason as is the case in N, K, and K. Now our way to find a compact 5Q resonance state is to seek for such a state which is not affected by HBC.

## Iii Lattice actions and parameters

[GeV] | Size | Values of | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

5.75 | 3.2552 | 4 | 1.100(6) | 1000 | 0.7620(2) | 0.9871(0) | 3.909 | 0.12640(5) | 0.1240, 0.1230, 0.1220, 0.1210 |

To generate gauge field configurations, we use the standard plaquette action on the anisotropic lattice of the size as

where denotes the plaquette operator in the --plane. The lattice parameter and the bare anisotropy parameter are fixed as and , respectively. These values are determined to reproduce the renormalized anisotropy as klassen . Adopting the pseudo-heat-bath algorithm, we pick up gauge field configurations every 500 sweeps after skipping 10,000 sweeps for the thermalization. We use totally 1000 gauge field configurations to construct the temporal correlators. Note that the high statistics of is quite essential for our study, because the 5Q correlators for spin 3/2 states are found to be rather noisy. In fact, a preliminary analysis with less statistics leads to a spurious resonance-like state ishii320 . The lattice spacing is determined from the static quark potential adopting the Sommer parameter MeV ( fm) as GeV ( fm). Note that the lattice size amounts to in the physical unit.

We adopt the -improved Wilson (clover) action on the anisotropic lattice for quark fields and as matsufuru

(31) | |||||

where and denote the hopping parameters for the spatial and the temporal directions, respectively. The field strength is defined through the standard clover-leaf-type construction. denotes the Wilson parameter. and denote the clover coefficients. To achieve the tadpole improvement, the link variables are rescaled as and , where and denote the mean-field values of the spatial and temporal link variables, respectively matsufuru ; nemoto . This is equivalent to the redefinition of the hopping parameters as the tadpole-improved ones (with tilde), i.e., and . The anisotropy parameter is defined as , which coincides with the renormalized anisotropy for sufficiently small quark mass at the tadpole-improved level matsufuru . For given , the four parameters , , and should be, in principle, tuned so that “Lorentz symmetry” holds up to discretization errors of . Here, , and are fixed by adopting the tadpole improved tree-level values as

(32) |

Only the value of is tuned nonperturbatively by using the meson dispersion relation matsufuru . It is convenient to define as

(33) |

Then the bare quark mass is expressed as in the spatial lattice unit in the continuum limit. This plays the role of the hopping parameter “” in the isotropic formulation. For detail, see Refs. nemoto ; matsufuru , where we take the lattice parameters. The values of the lattice parameters are summarized in Table 2.

0.1210 | 0.1220 | 0.1230 | 0.1240 | ||
---|---|---|---|---|---|

1.007(2) | 0.897(1) | 0.785(2) | 0.658(2) | 0.140 | |

1.240(4) | 1.157(5) | 1.074(7) | 0.991(11) | 0.823(13) | |

0.846(2) | 0.785(1) | 0.722(2) | 0.658(2) | 0.476(2) | |

1.119(6) | 1.076(7) | 1.033(9) | 0.991(11) | 0.902(15) | |

1.877(4) | 1.739(3) | 1.600(4) | 1.454(5) | 1.164(8) | |

2.325(17) | 2.194(21) | 2.059(28) | 1.918(42) | 1.648(53) |

We adopt four values of the hopping parameter as and , which correspond to and , respectively. These values roughly cover the region . For temporal direction, we impose anti-periodic boundary condition on all the quark fields. For spatial directions, we impose periodic boundary condition on all the quarks, unless otherwise indicated. We refer to this boundary condition as “periodic BC (PBC)”.

By keeping fixed for quark, and by changing for and quarks, we perform the chiral extrapolation to the physical quark mass region. In the following part of the paper, we will use

(34) |

as a typical set of hopping parameters in presenting correlators and effective mass plots. For convenience, we summarize masses of , , K, K, N and N( baryon) for each hopping parameter together with their values at the physical quark mass in Table 3. Here, the chiral extrapolations of these particles are performed by a linear function in . Unless otherwise indicated, we adopt the jackknife prescription to estimate the statistical errors.

We use a smeared source to enhance the low-lying spectra. More precisely, we employ spatially extended interpolating fields of the gaussian size fm, which is obtained by replacing the quark fields in 5Q interpolating fields by the smeared quark fields in the Coulomb gauge as

(35) |

where is an appropriate normalization factor. For a practical use, we extend Eq. (35) appropriately so as to fit a particular choice of the spatial boundary condition. In this paper, we present correlators with a smeared source and a point sink.

## Iv Numerical results on 5Q spectrum

We present our lattice QCD results on 5Q spectrum in the standard periodic boundary condition(PBC) in this section.

### iv.1 5Q spectrum in PBC

We consider 5Q spectrum in channel. In Fig. 1, we show the effective mass plots in channel for three interpolating fields, i.e., (a) the NK-type, (b) the twisted NK-type, (c) a diquark-type. The dotted lines indicate the s-wave NK and the d-wave NK thresholds, which happen to coincide accidentally in Fig. 1 for the spatial lattice size fm.

We define the effective mass as a function of by

(36) |

where denotes the temporal correlator. At sufficiently large , the correlator is dominated by the lowest-lying state with energy as . Then Eq. (36) gives a constant as . Thus a plateau in indicates that the correlator is saturated by a single-state. In such cases, we can perform a single-exponential fit in the plateau region.

Fig. 1 (a) shows the effective mass plot for the NK-type interpolating field. In the region , the contamination of the higher spectral contributions are gradually reduced, which is indicated by the decreases in . There is a plateau in the interval , where a single-state is expected to dominate the 5Q correlator. Beyond , the statistical error becomes large. In addition, the effect of the backward propagation becomes gradually more significant as is approached. Hence, we simply neglect the data for , and perform the single-exponential fit in the region . We obtain GeV, which is denoted by the solid line. One sees that the 5Q states appears above the s-wave NK and the d-wave NK thresholds.

Fig. 1 (b) shows the effective mass plot for the twisted NK-type interpolating field. There is a plateau in the interval , where the single-exponential fit is performed leading to GeV. The 5Q state is again above the s-wave NK and the d-wave NK thresholds.

Fig. 1 (c) shows the effective mass plot for the diquark-type interpolating field. We see that the statistical error is too large to identify the plateau unambiguously. Hence, we do not perform the fit. Note that this plot is obtained by using gauge configurations. A possible reason for such a large noise is that the interpolating field Eq. (3) does not survive the non-relativistic limit due to the vector diquark.

I.F. | 0.1210 | 0.1220 | 0.1230 | 0.1240 | ||
---|---|---|---|---|---|---|

(a) | 3.08(1) | 2.90(2) | 2.72(2) | 2.54(3) | 2.17(4) | |

(b) | 3.08(1) | 2.89(1) | 2.70(2) | 2.49(3) | 2.11(4) | |

(c) | – | – | – | – | – | |

(a) | 3.52(2) | 3.34(3) | 3.17(11) | 3.00(5) | 2.64(7) | |

(b) | 3.27(3) | 3.11(4) | 2.95(5) | 2.83(9) | 2.48(10) | |

(c) | 3.34(2) | 3.16(2) | 2.98(3) | 2.78(5) | 2.42(6) |

Now, we perform the chiral extrapolation. As mentioned before, we keep fixed for -quark, and vary for and quarks. Fig. 2 shows the 5Q masses in channel against . Circles and boxes denote the data obtained from the NK-type and the twisted NK-type 5Q correlators, respectively. Note that they agree with each other within the statistical error. The open symbols refer to the direct lattice QCD data. Since these data behave almost linearly in , we adopt the linear chiral extrapolation in to obtain in the physical quark mass region. Note that the ordinary non-PS mesons and baryons show similar linearity in nemoto . The closed symbols denote the results of the chiral extrapolation. We see that all the 5Q states appear above the s-wave NK and the d-wave NK thresholds. As a result of the chiral extrapolation, we obtain only massive 5Q states as GeV from the NK-type and the twisted NK-type correlators, respectively, which is too heavy to be identified with the experimentally observed . Numerical values of at each hopping parameter together with their chirally extrapolated values are summarized in Table 4. To obtain a low-lying state at MeV, a 5Q state should appear below these thresholds at least in the light quark mass region. In this case, a significantly large chiral effect is required. Of course, this point can be in principle clarified by an explicit lattice QCD calculation with chiral fermions.

### iv.2 5Q spectrum in PBC

We consider 5Q spectrum in channel. is an interesting quantum number from the view point of the diquark picture of Jaffe and Wilczekjaffe . In this picture, the pair of diquarks has angular momentum one, which is combined with the spin 1/2 of quark. Hence, there are two possibilities as and , i.e., the diquark picture can support possibility as well. Its mass splits from the state depending on a particular form of the LS-interaction. If it is massive, it is expected to have a large decay width. If it is light enough, its exotic structure may work to implement the narrow decay width as in case.

In Fig. 3, we show the 5Q effective mass plots in PBC employing three types of interpolating fields, i.e., (a) the NK-type, (b) the twisted NK-type, (c) the diquark-type. The dotted lines indicate the s-wave NK, the p-wave NK and the p-wave NK thresholds in the spatial lattice of the size fm, respectively.

Fig. 3 (a) shows the 5Q effective mass plot employing the NK-type interpolating field. In the region, , the contaminations of higher spectral contributions become gradually reduced. There is a flat region , which is still afflicted with slightly large statistical errors. The single-exponential fit in this region gives GeV. Note that this value agrees with the s-wave NK threshold GeV. (See Table 3 for .)

Fig. 3 (b) shows the 5Q effective mass plot corresponding to the twisted NK-type interpolating field. We have a rather stable plateau in the interval , where the single-exponential fit is performed. We obtain GeV. The result is denoted by the solid line.

Fig. 3 (c) shows the 5Q effective mass plot for the diquark-type interpolating field. We find a plateau in the interval , where the single-exponential fit is performed. We obtain GeV, which is denoted by the solid line.

Now, we perform the chiral extrapolation. In Fig. 4, is plotted against . Circles, boxes and triangles denote the data obtained from the NK-type, the twisted NK-type and the diquark-type 5Q correlators, respectively. Note that the latter two agree with each other within the statistical errors.

As a result of the chiral extrapolation, we obtain GeV from the NK-type correlator, GeV from the twisted NK-type correlator, and GeV from the diquark-type correlator. Numerical values of in channel at each hopping parameter together with their chirally extrapolated values are also summarized in Table 4. The two data from the twisted NK-type and the diquark-type correlators are considered to be almost consistent with the p-wave NK threshold, while the data from the NK-type correlator seems to correspond to a more massive state, which is likely to be consistent with the NK(s-wave) threshold. We see again that all of our data of appear above the NK threshold(p-wave), which is located above the artificially raised NK threshold(p-wave) due to the finiteness of the spatial lattice as fm. As a result, we are left only with such massive 5Q states.

Now, several comments are in order. (1) LABEL:lasscock32 reported the existence of a low-lying 5Q state in channel using NK-type interpolating field. However, we have not observed such a low-lying 5Q state in our calculation. (2) Recall that, except for a single calculationchiu , lattice QCD calculations indicate that state is heavy scikor12 ; sasaki ; kentacky ; ishii12 ; rabbit ; lasscock12 ; alexandrou12 ; csikor122 ; holland , for instance GeV in LABEL:ishii12. From the viewpoint of the diquark picture, it could be natural to obtain such massive 5Q states in channel. If there were a low-lying 5Q state in channel, then the diquark picture could suggest also a low-lying 5Q state in channel nearby.

## V Analysis with HBC

In the previous section, we have only massive 5Q states, which are obtained by using the linear chiral extrapolation in . However, the chiral behavior may deviate from a simple linear one in the light quark mass region, which could lead to somewhat less massive states. Considering this, we think it of worth at this stage to analyze whether our 5Q states are compact 5Q resonances or not. This is done by switching the spatial periodic BC to the hybrid BC(HBC) introduced in Sect. II.

### v.1 5Q spectrum in HBC

Fig. 5 shows the 5Q effective mass plots in HBC employing the three types of interpolating fields, i.e., (a) the NK-type, (b) the twisted NK-type, and (c) the diquark type. These figures should be compared with their PBC counterparts in Fig. 1. The dotted lines denote the s-wave NK and the d-wave NK thresholds. For the typical set of hopping parameters, i.e., Eq. (34), the s-wave NK threshold(the thick dotted line) is raised up by MeV, and the d-wave NK threshold(the thin dotted line) is lowered down by MeV due to HBC in the finite spatial extent as fm. (See Table 1.)

Fig. 5 (a) shows the 5Q effective mass plot for the NK-type interpolating field in HBC. We find a plateau in the interval , where the single-exponential fit is performed leading to GeV, which is denoted by the solid line. We see that is raised up by 80 MeV due to HBC. The value of is consistent with the s-wave NK threshold within the statistical error. Therefore, we regard this state as an NK scattering state.

Fig. 5 (b) shows the 5Q effective mass plot for the twisted NK-type interpolating field. We find a plateau in the interval , where the single exponential fit is performed leading to GeV, which is denoted by the solid line. The situation is similar to the NK-interpolating field case. We see that is raised up by 90 MeV due to HBC. Since the value is consistent with the s-wave NK threshold within the statistical error, we regard it as an NK scattering state.

Fig. 5 (c) shows the 5Q effective mass plot for the diquark-type interpolating field. We see that it is afflicted with considerable size of statistical errors as before, due to which the best-fit is not performed.

In this way, all of our 5Q states in channel turn out to be NK scattering states. More precisely, we do not observe any compact 5Q resonance states in channel below the raised s-wave NK threshold, i.e., in the following region:

(37) |

with MeV.

### v.2 5Q spectrum in HBC

Fig. 6 shows the 5Q effective mass plots in HBC employing the three types of interpolating fields, i.e., (a) the NK-type, (b) the twisted NK-type, and (c) the diquark-type. These figures should be compared with their PBC counterparts in Fig. 3. The meanings of the dotted and the solid lines are the same as in Fig. 3.

HBC may not be useful in channel, since it induces only minor changes in the two-particle spectra. For the typical set of hopping parameters, i.e., Eq. (34), the p-wave NK threshold is lowered down only by MeV, and the p-wave NK threshold is lowered down only by MeV. We see that these shifts are rather small. This is because they are induced by the changes in the minimum non-vanishing momentum, i.e., MeV to MeV as mentioned before. In channel, NK(s-wave) threshold shows the most drastic change, i.e., the upper shift by 170 MeV, which however plays a less significant role, since its location is at rather high energy.

Fig. 6 (a) shows the 5Q effective mass plot employing the NK-type interpolating field. There is a flat region , which is still afflicted with slightly large statistical errors. The single-exponential fit in this region leads to GeV, which is denoted by the solid line. We see that is raised up by 40 MeV. Although the shift of 40 MeV is rather small, is again almost consistent with the s-wave NK threshold. Considering its rather large statistical error, this 5Q state is likely to be an s-wave NK scattering state. To draw a more solid conclusion on this state, it is necessary to improve the statistics further more.

Fig. 6 (b) shows the 5Q effective mass plot employing the twisted NK-type interpolating field. There is a plateau in the interval , where we perform the single-exponential fit. The result GeV is denoted by the solid line. We see that is lowered down by 90 MeV, which is considered to be consistent with the shift of the NK(p-wave) threshold. Therefore, this state is likely to be an NK(p-wave) scattering state.

Fig. 6 (c) shows the 5Q effective mass plot employing the diquark-type interpolating field. Although the data is slightly noisy, there is a plateau in the interval . A single-exponential fit in this plateau region leads to GeV, which is denoted by the solid line. is lowered down by 80 MeV due to HBC. The situation is similar to Fig. 6 (b). This state is likely to be an NK(p-wave) scattering state.

In this way, all of our 5Q states are likely to be either NK(s-wave) or NK(p-wave) states rather than compact 5Q resonance states. Of course, because HBC induces only minor changes in the 5Q spectrum in channel, and also because 5Q correlators still involve considerable size of statistical error, more statistics is desirable to draw a more solid conclusion on the real nature of these 5Q states. Here, we can at least state that these 5Q states are all massive, which locate above NK(p-wave) threshold.

## Vi Summary and conclusion

We have studied penta-quark(5Q) baryons in anisotropic lattice QCD at the quenched level with a large number of gauge field configurations as for high precision measurements. We emphasize that the spin of has not yet determined experimentally, and that the assignment provides us with one of the possible solutions to the puzzle of the narrow decay width of hosaka32 . We have employed the standard Wilson gauge action on the anisotropic lattice of the size with the renormalized anisotropy at , which leads to fm and fm. We have found that correlators of 5Q baryons in channels are rather noisy. Hence, the large statistics as has played a key role to get a solid result in our calculation. For the quark part, we have employed -improved Wilson (clover) action with four values of the hopping parameters as , which roughly cover the quark mass region as . To avoid the contaminations of higher spectral contributions, we have employed the spatially extended source in the 5Q correlators.

We have examined several types of the 5Q interpolating fields as (a) the NK-type, (b) the (color-)twisted NK-type, (c) the diquark-type. In channel, there are plateaus in the effective mass plots for the NK-type and the twisted NK-type interpolating field, whereas no plateau has been identified in that for the diquark-type interpolating field due to the significantly large statistical error. The former two give almost identical results. We have employed the linear chiral extrapolations in , which have lead to and GeV for the NK-type and the twisted NK-type 5Q correlators, respectively. In channel, we have recognized plateaus in all the three effective mass plots. However, the plateau for the NK-type inte