Five-quark picture of Lambda(1405) in anisotropic lattice QCD
aa r X i v : . [ h e p - l a t ] J un Five-quark picture of Λ(1405) in anisotropic lattice QCD
Noriyoshi
Ishii ∗ ) , Takumi Doi , , Makoto Oka , and Hideo Suganuma Center for Computational Sciences, Univ. of Tsukuba, Tsukuba 305–8577, Japan Dept. of Physics and Astronomy, Univ. of Kentucky, Lexington KY 40506, USA RIKEN BNL Research Center, Brookhaven National Lab., New York 11973, USA Department of Physics, Tokyo Institute of Technology, Tokyo 152–8551, Japan Department of Phyics, Kyoto University, Kyoto 606–8502, Japan
Five-quark (5Q) picture of Λ(1405) is studied using quenched lattice QCD with an exotic5Q operator of N ¯K type. To discreminate mere N ¯K and Σ π scattering states, Hybrid Bound-ary Condition (HBC), a flavor-dependent boundary condition, is imposed on the quark fieldsalong spatial direction. 5Q mass m ≃ .
89 GeV is obtained after the chiral extrapolationto the physical quark mass region, which is too heavy to be identified with Λ(1405). Then,Λ(1405) seems neither a pure 3Q state nor a pure 5Q state, and therefore we present aninteresting possibility that Λ(1405) is a mixed state of 3Q and 5Q states.
Λ(1405) is an I = 0 , S = − , Q = 0 negative-parity baryon. As is obvious fromits quantum number, it contains a (heavy) strange quark as a valence quark. Nev-ertheless, Λ(1405) is the lightest negative-parity baryon. (The lightest non-strangebaryon is N ∗ (1520).) In a quark-model interpretation, Λ(1405) is identified as aflavor SU(3) singlet baryon. For this assignment, however, the anomalous value ofthe LS force is to be introduced. Then, one may wonder if there may be somethingbehind, i.e., it may involve some exotic structure. Indeed, it has been conjectured,for a long period, that Λ(1405) may be a bound state of N and ¯K via the stronginteraction, i.e., a 5Q object rather than a 3Q one. If this is the case, its bindingenergy is m N + m K − ≃
30 MeV, which seems to be natural magnitude of thebinding energy for the hadronic molecule.Baryon spectra in flavor SU(3) sector was studied by quenched lattice QCD.
It was suggested that Λ(1405) may have an exceptional feature. For instance, inRef.[1], it was reported that baryon masses in flavor SU(3) sector can be reproducedby quenched lattice QCD within about 10 % deviation except for Λ(1405), for whicha significant overestimate of more than 300 MeV is reported. One of the most attrac-tive explanations for this is provided by the 5Q picture of Λ(1405), i.e., if Λ(1405)is dominated by 5Q components, it should be difficult for quenched QCD with aconventional 3Q interpolating field to reproduce it.In this paper, we will consider the 5Q picture of Λ(1405) by making a con-structive use of quenched QCD. We attempt to use the suppressed contributionsfrom q ¯ q loops in quenched QCD in order to obtain an additional information ona possible exotic structure of Λ(1405) in the following way. An ordinary 3Q in-terpolating field fully couples to 3Q intermediate states, whereas it couples to 5Qintermediate states only in an imperfect manner. On the other hand, a 5Q in- ∗ ) e-mail address: [email protected] typeset using PTP
TEX.cls h Ver.0.9 i N. Ishii, T. Doi, M. Oka, and H. Suganuma terpolating field can fully couple to 5Q intermediate states, whereas its couplingsto 3Q intermediate states vanish at all, once we neglect the contribution from theannihilation diagram. We will use these quenched QCD features to study the na-ture of Λ(1405) by comparing the mass spectrum obtained with an ordinary 3Qinterpolating field and that with a 5Q interpolating field neglecting the annihilationdiagram. For 3Q interpolating field, we employ a flavor-singlet interpolating field asΛ (3Q) ≡ ǫ abc (cid:2) ( u Ta Cγ d b ) s c + ( d Ta Cγ s b ) u c +( s Ta Cγ u b ) d c (cid:3) , where a, b, c denote colorindices, and C ≡ γ γ denotes the charge conjugation matrix. For 5Q interpolatingfield, we employ an iso-scalar 5Q interpolating field of N ¯K type as Λ (5Q) ≡ p K − − n ¯K , where p ≡ ǫ abc ( u Ta Cγ d b ) u c , n ≡ ǫ abc ( u Ta Cγ d b ) d c , K − ≡ ¯ uiγ s , and ¯K ≡ − ¯ diγ s .5Q calculation involves an obstacle. Empirically, the mass gap between Λ(1405)and the N ¯K threshold is only about 30 MeV, which is too small for a practical latticeQCD calculation to identify these two states separately. To avoid this, we adopta flavor-dependent boundary condition (BC) along the spatial directions (“ HybridBoundary Condition(HBC) ”), which was proposed in the studies of Θ + (1540).HBC consists of anti-periodic BC on u and d quark fields and periodic BC on s quark field. Since Λ (5Q) field contains even number of u and d fields, it is subjectto the periodic BC, which allows us to consider the rest frame of the 5Q systemΛ(1405). In contrast, since p , n , K − , ¯K fields contain odd number of u and d fields,they are subject to the anti-periodic BC. Their spatial momenta are discretized as ~p = ((2 n x + 1) π/L, (2 n y + 1) π/L, (2 n z + 1) π/L ), where n i ∈ Z , L is the spatialextension of the lattice. Note that | ~p | cannot vanish with HBC. Its minimum valueis p min ≡ √ π/L , owing to which N ¯K threshold is raised from E PBC , th ≃ m N + m K for PBC to E HBC , th ≃ q m N + 3 π /L + q m K + 3 π /L for HBC. For L ≃ . | ~p | amounts to about 499 MeV leading to the shift in thethreshold of more than 200 MeV, which make it possible to distinguish a possibleΛ(1405) state from N ¯K threshold.The 5Q calculation still involves a difficulty. Λ(1405) is expected to be embeddedin π Σ “continuum” even with HBC. Note that Σ π threshold is not raised by HBC.We have to distinguish Λ(1405) as a compact 5Q state from π Σ scattering states.If we were to impose such a spatial BC that the periodic BC is imposed on ¯ u , ¯ d ,and ¯ s fields, while the anti-periodic BC is imposed on u , d and s fields, then theboth the N ¯K and Σ π threshold could be raised more than 200 MeV. However, this isproblematic, since it does not respect the charge conjugation symmetry. Instead, wevirtually introduce additional flavors u ′ and d ′ , and regard ¯ u and ¯ d in 5Q Λ(1405)as anti-quarks for these two additional flavors, i.e., ¯ u ′ and ¯ d ′ , respectively. Weemphasize that this will not change anything, because we neglect the annihilationdiagram as is mentioned before. Now, we consider a modified HBC, which will bereferred to as “ HBC2 ”. HBC2 consists of the periodic BC on u ′ and d ′ fields, and theanti-periodic BC on u , d , s fields. Note that Λ (5Q) field consists of four quarks withoriginal flavor and one anti-quark with additional flavor. N, Σ fields consist of threequarks with original flavor. ¯K and π fields consist of one quark with original flavorand one anti-quark with additional flavor. By repeating a similar consideration, weconvince ourselves that Λ (5Q) is subject to the periodic BC, whereas N, Σ, ¯K and π ive-quark picture of Λ(1405) in anisotropic lattice QCD π while keeping a possible compact 5Q Λ(1405) state unaffected.For precision measurements, we use anisotropic lattice QCD, which has 4 timesfiner mesh along the temporal direction than the spatial directions as a s /a t = 4. We employ the standard Wilson gauge action at β = 5 .
75 and O ( a ) improved (clover)quark action with κ = 0 . . . m s ≤ m u,d ≤ m s . κ s = 0 . s quark, while 0 . ≤ κ ≤ . u and d quark masses. The latticespacing is determined with the Sommer parameter r − = 395 MeV, which leads tothe spatial lattice spacing of a − s = 1 .
10 GeV ( a s ≃ .
18 fm). We use the latticesize 12 × L ≃ . t = 0 plane. We employ a Gaussian smeared source with the Gaussiansize ρ ≃ . t = t ≡
64 plane. We utilise the time-reversaland charge conjugation symmetries to effectively double the statistics. m e ff; Q [ G e V ] t-t [a t ] HBC2HBCPBCthresholds 1.52.02.53.00.0 0.5 1.0 m Λ [ G e V ] m π [GeV ] 5Q(HBC2)5Q(HBC)5Q(PBC)3Qthresholds Fig. 1. The effective mass plot and the chiral extrapolation of the 5Q states which have samequantum number as the Λ(1405). Two types of the hybrid boundary condition (HBC, HBC2)are applied to raise the threshold.
Fig. 1(left) shows the 5Q effective mass plot for HBC2(circle), HBC(triangle)and PBC(cross) for κ = κ s = 0 . π thresholds for PBC. HBC raises N ¯K threshold, which is denoted by the upper dottedline. HBC2 raises also Σ π threshold, which is denoted also by the upper dotted line.Note that, due to the flavor SU(3) symmetry, the (raised) N ¯K threshold coincideswith the (raised) Σ π threshold. We see that a plateau is located at t − t ∈ [29 , π . For HBCand HBC2, the plateaux are raised above by about 200 MeV, which are located at t ∈ [26 , ∼
60 MeV, this ismost probably not an indication of the existence of a compact 5Q states, consideringthe energy shift caused by remaining interaction in the finite volume. To clarifywhich is the case, further investigation is necessary. Since HBC does not affect Σ π threshold, the fact that the plateau for HBC data is raised by about 200 MeV impliesthat our N ¯K-type interpolating field has only a small overlap with Σ π state. The N. Ishii, T. Doi, M. Oka, and H. Suganuma results of single exponential fits in these plateau regions are shown by solid lines.Fig. 1(right) shows the results of linear chiral extrapolations. 5Q data for HBC2,HBC, and PBC are denoted with circle, triangle, and cross, respectively. The resultof the 3Q data is also shown by square. Dotted curves denote the raised and un-raised thresholds for N ¯K and Σ π . The solid lines denote the results of the chiralextrapolations for 5Q data with HBC2 and 3Q data. Note that the 5Q result m =1 . m = 1 . Since
Λ(1405) is neither a pure 3Q statenor a pure 5Q state, we present an interesting possibility that
Λ(1405) is a mixedstate of 3Q and 5Q states. (Note that, the energy is generally reduced, if one seeksfor a solution in a larger space.) The chiral quark effect may be also interesting. Although quenched QCD with 3Q interpolating field leads to an imperfect overlapwith intermediate 5Q states, 5Q contribution increases as the smaller quark massregion is approached. Of course, the annihilation diagram may be important, whichwe have neglected in our calculation to save the computational time. Since Λ(1405)is such an interesting hadron, which may provide us with a possible exotic structure,we will keep studying on this interesting target from every aspect.
Acknowledgements
The authors acknowledge the Yukawa Institute for Theoretical Physics at KyotoUniversity for useful discussions during the YKIS2006 on “New Frontiers in QCD”.T. D. is supported by Special Postdoctoral Research Program of RIKEN and U.S.DOE grant DE-FG05-84ER40154. Lattice QCD Monte Carlo calculations have beendone with NEC SX-5 at Osaka University.
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