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Hadron Spectrum from Lattice QCD


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International Journal of Modern Physics A c World Scienti?c Publishing Company

arXiv:hep-lat/0510009v2 6 Oct 2005

Hadron Spectrum from Lattice QCD

AKIRA UKAWA Center for Computational Sciences, University of Tsukuba Tsukuba, Ibaraki 305-8577, Japan

Received (Day Month Year) Revised (Day Month Year) A brief review is given of the lattice QCD calculation of the hadron spectrum. The status of current attempts toward inclusion of dynamical up, down and strange quarks is summarized focusing on our own work. Recent work on the possible existence of pentaquark states are assessed. We touch upon the PACS-CS Project for building our next machine for lattice QCD, and conclude with a near-term physics and machine prospects. Keywords: lattice QCD; hadron spectrum; pentaquark

1. Introduction The spectrum of hadrons is a fundamental entity for lattice QCD elucidation of the dynamics of the strong interactions. Calculating the spectrum of standard hadrons such as pion, nucleon, J/ψ, Υ, and D’s and B’s provides a basic mean for con?rmation of the theory and a benchmark for the lattice calculational methods. Finding the spectrum of non-standard hadrons, e.g., glueballs, hybrids and multi-quark states, o?ers chances for predictions and thereby further tests of QCD. For these reason the hadron spectrum has been repeatedly and routinely calculated over the years with increasing accuracy and sophistication. At present, the focus of spectrum calculations and many other subjects with phenomenological impact centers around dynamical simulations including the sea quark e?ects of all three light quarks, up, down and strange. There are two major attempts, one using the staggered quark action2 and the other using the Wilsonclover quark action1 . In addition, a third attempt with the domain-wall quark action, made possible with the commissioning of QCDOC with 10T?ops-class capability, are beginning3 . In Sec. 2 we discuss results from our own attempt on the light hadron spectrum and quark masses, and recent preliminary results on heavy quark systems. Search for possible pentaquark states has occupied a lot of experimental and theoretical e?ort. A number of lattice QCD simulations have also been made to see if there is signal indicating their theoretical presence. A crucial and special
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issue with the pentaquark concerns distinguishing bound or resonance states from scattering states. Recent lattice studies focused on applying theoretical criteria on this point, and we shall review them in Sec. 3. We conclude with a brief summary in Sec. 4. 2. Nf = 2 + 1 lattice QCD simulations with the Wilson-clover quark action There are two major lattice QCD research collaborations in the Tsukuba area in Japan, the CP-PACS Collaboration based at University of Tsukuba and the JLQCD Collaboration based at KEK. Since 2001 the two collaborations have jointly pursued a project to carry out simulations including dynamical up, down and strange quarks. The strategy adopted, and the necessary preparations carried out prior to actual runs, is as follows: (i) use Iwasaki RG-improved gluon action to span a range of lattice spacing toward coarse lattices and avoid the arti?cial phase transition observed for the plaquette gluon action4 , (ii) use Wilson-clover quark action with a fully O(a)-improved clover coe?cient calculated with the Shcr¨dinger functional o methods for three dynamical ?avors5, and (iii)apply the polynomial HMC to handle the strange quark6 in addition to the standard HMC for the up and down quarks which are treated as degenerate. In the table below we list the parameters of our simulations. Runs are made at three values of lattice spacing equally spaced in a2 . At each lattice spacing, ?ve values are taken for the degenerate up and down quark mass in the range mP S /mV ≈ 0.6 ? 0.8, and two values for the strange quark mass at mP S /mV ≈ 0.7. Hadron masses calculated for these quark masses are ?tted with a general quadratic ?1 polynomial of VWI quark masses mV W I = (K ?1 ?Kc )/2, and are extrapolated to the physical point de?ned either by the experimental π, ρ and K meson masses (Kinput) or π, ρ and φ meson masses (φ-input). The agreement of the lattice spacing determined by the two types of input as seen in Table 2 provides a simple but important check on the internal consistency of the Nf = 2 + 1 calculation, which is not realized in quenched and Nf = 2 simulations. 2.1. Light quark sector In Fig. 1 we plot the continuum extrapolation of meson masses in quenched(triangles), Nf = 2(squares) and the present Nf = 2 + 1(circles) Parameters of our simulations β size a [fm] (K-input) 1.83 163 × 32 0.1222(17) 1.90 203 × 40 0.0993(19) 3 2.05 28 × 56 0.0758(48)

a [fm] (φ-input) 0.1233(30) 0.0995(19) 0.0755(48)

trajectory 7000 ? 8600 5000 ? 9200 3000 ? 4000

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Hadron spectrum from Lattice QCD
1.050 meson masses [GeV]

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Quenched(RG+PT clover) Nf=2(RG+PT clover) Nf=2+1(RG+NPT clover) experiment

0.895 0.890 K* (φ-input) 0.550

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0.900 0.890 0.880 0.870 0 0.01

K* (K-input)

0.500 K (φ-input) 0.02 0.03 2 2 a [fm ] 0.04 0.05 0 0.01 0.02 0.03 2 2 a [fm ] 0.04 0.05

Fig. 1. Continuum extrapolation of meson masses, compared with those for quenched and Nf = 2 QCD 7 . Note that the quenched and Nf = 2 simulations are made with the one-loop perturbatively O(a)-improved clover action. Thus extrapolations are made linearly in a.

simulations8 . The solid lines for the Nf = 2 + 1 data are pure quadratic ?ts to the results for the two coarse lattice spacings for which runs and measurements have been completed. The agreement with experiment in the continuum limit is encouraging, but we need further data at the ?nest lattice spacing for a solid conclusion. Figure 2 shows the continuum extrapolation for the light quark masses8 . Values sizably smaller than the quenched estimate as has been strongly suggested in the previous Nf = 2 simulations are con?rmed. Again, we wait completion of the analyses at the ?nest lattice spacing to quote the ?nal result for light quark masses in the continuum limit. An important issue with the analyses in the light quark sector concerns chiral extrapolation. Since Wilson-clover quark action involves explicit chiral symmetry breaking, the chiral behavior of physical quantities deviate from that in the continuum. The Wilson chiral perturbation theory9 provides the procedure to work out the chiral behavior for ?nite lattice spacings for Wilson-type actions, and work is under way to calculate the consequence of the approach for the spectroscopic quantities including pseudo scalar meson masses, decay constants, and quark masses on the one hand, and to analyze data based on those results. 2.2. Heavy quark sector Calculating the physical quantities of hadrons containing heavy quarks is important to constrain the parameters of the Standard Model and to help explore physics at ?ner scales. A serious obstacle is large systematic errors of form O(mH a) where mH denotes the heavy quark mass, which are large at currently accessible lattice spacings. A standard lore for overcoming this problem is to resort to heavy quark e?ective theory approach such as NRQCD. We wish to pursue a di?erent approach

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5.0 (?=2GeV) [MeV]
(?=2GeV) [MeV] 120

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WI

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WI
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0.01

0.02 0.03 2 2 a [fm ]

0.02 0.03 2 2 a [fm ]

Fig. 2. Continuum extrapolations of the up, down and strange quark masses obtained with the K-input. The data at the ?nest lattice is not included in the continuum extrapolations. For comparison, results for quenched and Nf = 2 QCD are overlaid.

0.35

0.3

0.3

RHQ(PT) RHQ(NP) clover-iso[9]

(a) fDs [GeV]
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RHQ(PT) RHQ(NP) NRQCD[12] NRQCD[13]

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0.05

0.1

0.15

0.2

0.15

0

0.05

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a [fm]

a [fm]

Fig. 3. Decay constant for (a) Ds and (b) Bs in quenched QCD from relativistic heavy quark approach11 as compared to those of e?ective theories12,13,14 .

in which parameters of the quark action are tuned demanding Lorentz invariance of on-shell quantities for arbitrary large mH a to O(a)10 . This approach allows the continuum extrapolation to be taken in contrast to e?ective theory approaches. We have been testing this approach for charm and bottom systems on Nf = 2 and quenched con?gurations prior to applying it to Nf = 2 + 1 QCD. In Fig. 3 we show results for the decay constant of Ds and Bs mesons in quenched QCD11 and compare them with those of e?ective theory approaches12,13,14 . Mild cuto? dependence of the present results and consistency at ?nite lattice spacings indicate success of our approach. 2.3. The next step At present the largest limitation in our data is a rather large value of the up and down quark mass mud . In terms of the physical strange quark mass mphys , it only s goes down to mud ≈ mphys /2 while experimentally mud ≈ mphys /25. We wish to s s reduce the value to at least mud ≈ mphys /5 or below in order to control the chiral s behavior. This requires both an enhancement of the computing power and an improvement

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of the algorithm. The situations in both respects are very promising at present. For the latter, the domain-decomposition acceleration of HMC15 o?ers a very promising resolution. On the latter, in December 2004, Japanese Government formally approved our proposal to develop a massively parallel cluster at the Center for Computational Physics, University of Tsukuba. We list in table below the speci?cation of the cluster, which is named PACS-CS (Parallel Array Computer System for Computational Sciences). The development is well under way, and the installation and start of operation are scheduled in June-July of 2006. We plan to combine these developments to break through the current limitations in our Wilson-clover program. 3. Lattice pentaquark search A standard lattice QCD calculation of the mass of a hadron starts with a preparation of the operators having the desired quantum numbers and an examination of the large-time behavior of the two-point function to extract the ground state signal. With lattice pentaquark searches, we have to distinguish a possible pentaquark bound state or resonance from the nucleon-kaon scattering state. Hence, we need to disentangle at least two states in the correlation function, and we also have to distinguish a bound state/resonance from scattering states. In addition, since the spin-parity of the pentaquark state is yet unknown, we have to explore over a large operator space. The initial lattice studies16 did not explore these points in detail. Subsequent calculations17,18 addressed them using two techniques. For multi-state analyses, a variational method using a set of operators and diagonalizing the normalized correlator matrix has been known for a long time. This method has been employed in a number of recent pentaquark searches, and has been reasonably successful. For distinguishing a scattering state, a basic strategy is to examine the spatial size dependence of the energy eigenvalues obtained in the multi-state analysis. A scattering state, if the interactions between the scattering hadrons are weak as in the case of the nucleon and kaon for the pentaquark case, would show a size dependence Design speci?cation Number of nodes Peak performance Total memory Total disk space Interconnect OS Programming System size of PACS-CS 2560 14.3 T?ops 5 TByte 0.41 PByte 16 × 16 × 10 3-dim hyper-crossbar Linux and SCore Fortran90, C, C++, MPI 59 racks

Node con?guration single LV Xeon 2.8GHz 5.6G?ops 2 GByte memory 6.4GByte/s 160 GByte×2(RAID1) local disk Interconnect bandwidth 250MByte/s×2/link 750MByte/s×2/node Estimated power 545 kW

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Fig. 4.

Spatial size dependence of energies in the pentaquark channel in quenched QCD18 .

expected from the sum of two energies m2 + (2π?/L)2 with ? = 1, 2, 3, · · ·. Another criteria is to look at the residue of the state in question in the two point function : →∞ < O(t)O(0) > → Zexp(?mt). For a scattering state we expect the overlap of the two particles decreasing as Z ∝ 1/V . In Fig. 4 we reproduce the spatial size dependence of the energies in the 1/2? and 1/2+ channel calculated with ?ve operator basis in quenched QCD18 . The dashed lines indicate the size dependence expected for scattering states, and the horizontal dotted lines show the experimental value quoted for the Θ+ state. As the ?gure indicates, the authors conclude that there is no evidence for the existence of pentaquark states in their data. While conclusions are not unanimous among the studies17,18 , it is our view that the data published to data are more consistent with the absence of pentaquark states at present. Caution should be stated that all data so far have been generated in quenched QCD, mostly at a ?xed lattice spacing and at relatively large quark masses. Truly realistic tests with dynamical light quarks with large volume and small lattice spacing require further work in the future.

4. Conclusions Currently lattice QCD simulations are in a state of transition. Realistic simulations including dynamical up, down and strange quarks are becoming routine, and chirally invariant quark actions are beginning to be used increasingly frequently. This trend will accelerate as computers with 10 T?ops-class capability, starting with QCDOC and followed by ApeNEXT, PACS-CS and also BlueGene/L at several institutions, becomes available. We hope that single hadron properties will be fully understood, inviting us to venture into the World beyond, in the near future.

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Acknowledgments This work is supported in part by the Grants-in-Aid of the Ministry of Education (No. 15204015). References
1. T. Kaneko et al. [CP-PACS and JLQCD Collaborations], Nucl. Phys. B (Proc. Suppl.) 129 (2004) 188; T. Ishikawa et al. [CP-PACS and JLQCD Collaborations], Nucl. Phys. B (Proc. Suppl.) 140 (2005) 225. 2. C. Aubin et al. [HPQCD Collaboration, MILC Collaboration and UKQCD Collaboration], Phys. Rev. D70 (2004) 031504(R); Phys. Rev. D70 (2004) 114501; A. Kronfeld, these proceedings. 3. R. Mawhinney, parallel talk at Lattice 2005 (to appear in the proceedings). 4. S. Aoki et al. [JLQCD Collaboration], To be published in Phys. Rev. D. 5. S. Aoki et al. [CP-PACS and JLQCD Collaborations], hep-lat/0508031. 6. S. Aoki et al. [JLQCD Collaboration], Phys. Rev. D65 (2002) 094507. 7. A. Ali Khan et al. [CP-PACS Collaboration], Phys. Rev. D65 (2002) 054505; Erratum, D67 (2003) 059901. 8. T. Ishikawa et al., [CP-PACS and JLQCD Collaboration], hep-lat/0509142 (2005). 9. S. Aoki, Phys. Rev. D68 (2003) 054508; S. Aoki et al., hep-lat/0509049. 10. S. Aoki, Y. Kuramashi and S. Tominaga, Prog. Theor. Phys. 109 (2003) 383. 11. Y, Kuramashi, parallel talk at Lattice 2005 (to appear). 12. A. J¨ttner and J. Rolf, Phys. Lett. B560 (2003) 59. u 13. S. Collins et al., Phys. Rev. D63 (2001) 034505. 14. K-I. Ishikawa et al., [JLQCD Collaboration], Phys. Rev. D61 (2000) 074501. 15. M. L¨scher, hep-lat/0409106; these proceedings. u 16. F. Csikor, Z. Fodor, S. D. Katz and T. G. Kovacs, JHEP 0311, 070 (2003); S. Sasaki, Phys. Rev. Lett. 93, 152001 (2004). 17. N. Mathur et al., Phys. Rev. D70, 074508 (2004); N. Ishii et al., Phys. Rev. D71, 0340001 (2005); T. W. Chiu and T. H. Hsieh, Phys. Rev. D72, 034505 (2005); hep-ph/0501227; B. G. Lasscock et al., hep-lat/0504015; Phys. Rev. D72, 104502 (2005); C. Alexandrou and A. Tsapalis, hep-lat/0503013; T. T. Takahashi, T. Umeda, T. Onogi and T. Kunihiro, Phys. Rev. D71, 114509 (2005); K. Holland and K. Juge, hep-lat/0504007. 18. F. Csikor, Z. Fodor, S.D. Katz, T.G. Kov?cs, and B.C. T?th, hep-lat/0503012. a o


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