Lattice QCD with Dynamical up, down and Strange Quarks with the Earth Simulator

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1 Lattice with Dynamical up, down and Strange Quarks with the Earth Simulator Project Representative Akira Ukawa Center or Computational Sciences, University o Tsukuba, Proessor Authors Tomomi Ishikawa, Sinya Aoki 2, Sadataka Furui 3, Koichi Hashimoto 4, Shoji Hashimoto 5, Ken-Ichi Ishikawa 6, Naruhito Ishizuka, 2, Yoichi Iwasaki 2, Kazuyuki Kanaya 2, Takashi Kaneko 5, Yoshinobu Kuramashi, 2, Masanori Okawa 6, Tetsuya Onogi 7, Takayuki Matsuki 8, Hideo Nakajima 9, Akira Ukawa, 2 and Tomoteru Yoshie, 2 Center or Computational Sciences, University o Tsukuba 2 Graduate School o Pure and Applied Sciences, University o Tsukuba 3 School o Science and Engineering, Teikyo University 4 Graduate School o Science and Engineering, Kanazawa University 5 Institute o Particle and Nuclear Physics, KEK 6 Graduate School o Sciences, Hiroshima University 7 Institute o Fundamental Physics, Kyoto University 8 Tokyo Kasei University 9 Institute or Inormation Engineering, Utsunomiya University The Standard Model is a uniied theory o elementary particles which includes Weinberg-Salam theory or electro-weak interactions and or strong interactions. Lattice and its numerical simulations oer a undamental tool or veriying and extracting predictions o the model. Our project aims to carry out a large scale lattice simulation in which the three light quarks, up, down and strange, are treated dynamically, thereby ully exonerating quenching eects that have plagued past attempts. This year we have completed the runs on a and lattices, and have started the run on a lattice. The results or the meson spectrum show that the dynamical eects o three quarks are just right to ill in the disagreement o the quenched results rom. The light quark masses are signiicantly lighter than expected phenomenologically, conirming the trend rom our earlier results or two dynamical lavors. Keywords: elementary particle physics, Standard Model, quark, hadron, lattice, Monte Carlo simulation. The physics goal o our project The Standard Model o elementary particles describes all known particle contents and orces in Nature except or gravity. This model includes quantum chromodynamics () describing strong interactions and Weinberg-Salam theory describing electroweak interactions. Veriication o the Standard Model is a crucial step, both to establish the current gauge-theoretic understanding o Nature and to help explore the hierarchy o even iner microscopic scales. At low energy scales non-perturbative analysis is needed or since the coupling becomes strong toward the inrared due to the remarkable property o asymptotic reedom. The only method versatile enough at present or this purpose is lattice and its numerical simulation. In lattice simulations in the past, the so-called "quenched approximation", in which vacuum polarization eects o quarks are ignored, have been used. The reason was technical; ull simulations treating quarks dynamically requires computational resources two to three orders o magnitude larger than the quenched one. More undamentally, simulation algorithms or odd number o quarks had not been developed. Fortunately, both these limitations are now overcome. In our project, we aim to carry out a large-scale three lavor ull simulation, in which up, down and strange quarks are treated dynamically, i.e., there is no quenching 77

2 Annual Report o the Earth Simulator Center April 24 - March 25 eect. Among the many important issues which should be addressed in such simulations, we initially concentrate on the ollowings: Veriication o the hadron mass spectrum. Determination o quark masses and the coupling which are the undamental parameters o. Determination o hadronic weak matrix elements or constraining the Cabibbo-Konayashi-Maskawa quark mixing matrix and or understanding o CP violation. Elucidation o the U() problem related to the eta' meson mass and gluon ield topology, and other long-standing issue o strong interactions. As a irst step we work on an analysis o the light hadron spectrum and quark masses using the unquenched gauge conigurations. 2. Gluon coniguration generation In usual lattice ull simulations, gluon coniguration generation and measurement o physical quantities are perormed separately. The Earth Simulator is a powerul tool or the ormer part which takes large cost computationally. The conigurations generated and accumulated are used or o-line measurements o various physical quantities such as hadrons masses, topology and so on. 2.. Optimization o the PHMC code In our simulation we assume that up and down quarks are degenerate but that strange quark has a distinct mass. All three quarks are treated dynamically. Thereore, we call this simulation as a 2 + lavor ull simulation. For up and down quarks we employ the standard Hybrid Monte Carlo algorithm; or strange quark an odd-lavor algorithm is needed or which we apply the Polynomial Hybrid Monte Carlo (PHMC) algorithm. It is based on a polynomial approximate inversion o the Dirac matrix. A noisy Metropolis test or a correction o the approximation makes this algorithm exact []. Our PHMC code was originally developed or Hitachi SR8 at KEK or which it achieved 4% o peak speed. We have ported this code to the Earth Simulator, and made extensive rewriting and optimizations. In detail, our code uses the strategy in which sites on the z-t plane are numbered by a one-dimensional array and divided by our to realize a large vector length without list vectors [2]. In addition, we use a library supplied by the Earth Simulator support team upon our request which enables an overlap o computations and communications. The use o the library enhances the sustained perormance by 2-3%. The total eiciency o the code or Earth Simulator has now reached 46% or our largest lattice size o used in this year's runs Simulation parameters or the coniguration generation In our simulation we use a Renormalization Group (RG) improved Wilson gauge action and a non-perturbatively O(a) improved Wilson quark action [3]. In order to take the continuum limit our simulation is perormed at β = 2.5,.9 and.83 corresponding to the three lattice spacings a ~ /2.,. and 3/2. m, which are at even intervals in a 2. The lattice sizes are L 3 T = , and respectively so that the physical volume is ixed at 2. m. Since this volume is not suiciently large to avoid inite-size eects or baryon masses, we ocus on the analysis o the meson sector in this report. We take ive values or the degenerate up and down quark mass in the range m PS /m V ~.6.78 or chiral extrapolation, and two values or strange quark mass around m PS /m V ~.7 or interpolation. Production o gluon conigurations at β =.9 was in good progress last year, which we completed early this year. We also carried out some runs or β =.83, but soon decided that this run, being small in lattice size and so demands much less computing resources, can be handled by conventional supercomputers elsewhere. The bulk o our eort this year has been concentrated on the largest lattice at β = 2.5. The simulation parameters and current statistics are listed in Table. Table Simulation parameters. At β = 2.5, the number o trajectory is 2 at present. The lattice spacing described in this table represents the current estimate. lattice spacing (m) # o trajectory 7-86 (inished) lattice size The light hadron spectrum.223(7).(9).797(92) 5-9 (inished) 4-5(target) 2(current) 3.. Analysis method Meson masses and decay constants are calculated rom a correlated it o meson correlator using single hyperbolic cosine or hyperbolic sine it orms. For the chiral extrapolation to obtain the quantities at the physical point, we assume a polynomial it orm in quark masses up to quadratic order. This it is made to all combinations o valence quarks simultaneously ignoring correlations among these masses. We ix the physical point using the al values o m π, m ρ, m Κ, (Κ-input) or m π, m ρ, m φ (φ-input). We have completed the analyses at β =.9 and.83, which are presented below. A preliminary continuum extrapolation using these two points are also made, and compared with provisional values at β =

3 3.2. Light meson spectrum In Fig. we present results or meson masses. Since we use a non-perturbatively O(a) improved Wilson quark action, the scaling o meson masses as a unction o lattice spacing should be a 2. With this assumption our results are consistent with. This is an encouraging result showing the importance o including the vacuum polarization eects or precision agreement with at a percent level. meson mass [GeV] φ φ-input Fig. Continuum extrapolation o light meson masses in 2 + lavor ull and its comparison with al value. Horizontal axis is a 2 and star symbols represent the al value. The results at a ~ /2. m in this igure are preliminary and not used or the continuum extrapolation a 2 [m 2 ] a 2 [m 2 ] um limit in our current analysis. The preliminary estimate or the continuum value or quark masses in 2 + lavor ull are given by m up,down = 3.4(24)MeV, m strange = 88.6(7.)MeV The continuum extrapolation leading to these values and a comparison with earlier results in two-lavor ull and quenched cases are shown in Fig. 2. The 2 + lavor ull results are somewhat smaller than those o two-lavor ull at inite lattice spacings, but the dierence are not visible ater the continuum extrapolation. Precision results at our inest lattice at β = 2.5 are needed to settle the continuum value. m ** MS ( = 2GeV) [MeV] N = 2, VWI N =, VWI N =, AWI N = 2, AWI N = 2 +, AWI a [m] m 5 MS ( = 2GeV) [MeV] N =, AWI N = 2, AWI N = 2 +, AWI N =, VWI N = 2, VWI 7..2 a [m] 3.3. Light quark masses The quark masses are undamental parameters in. These masses cannot be determined by because o the quark coninement property o. At present the only way to obtain the quark masses rom the irst principle is lattice. We calculate the physical quark mass m q using the axialvector Ward-Takahashi identity (AWI) deinition: lim An alternative deinition employs the vector Ward- Takahashi identity (VWI). In our analysis we apply the AWI deinition since the scaling violation o the quark mass is empirically smaller than that in the VWI deinition. The renormalization actors which match the quark mass on the lattice to that with MS scheme in the continuum theory are calculated perturbatively. Hence errors remain in our result where represents the strong coupling constant. The renormalized quark masses are evolved to µ = 2 GeV using the 4-loop β -unction. We assume that the remaining eects are small, and take the a 2 continu- Fig. 2 Continuum extrapolation o up and down quark masses (let) and strange quark mass (right) in 2 + lavor ull (N = 2 + ) compared with 2 lavor ull (N = 2) and quenched (N = ) [4]. Horizontal axis denotes lattice spacing a. The quark masses deined by VWI are also shown or N = 2 and cases or comparison Meson decay constants The decay constants o pseudo-scalar meson and vector meson are deined through where ε i is a polarization vector. Historically, previous simulations using the Wilson type quark action have not reproduced well the al values o the decay constants. In this analysis we use a local current or A 4 and a conserved current or V i. The advantage o using the conserved vector current is that a renormalization actor is not needed. Our calculation o the decay constants has O(a) errors; we assume that they are small and employ a 2 scaling in taking continuum limit as in the quark masses. Fig. 3 shows our current values, exhibiting a reasonable 79

4 Annual Report o the Earth Simulator Center April 24 - March build a data grid o lattice gauge coniguration archive so that they can be shared by the lattice community. We plan to put the 2 + conigurations to our Lattice Archve [3] which is the Japanese site o the ILDG. PS [MeV] V [MeV] a 2 [m 2 ]..2 a 2 [m 2 ] Fig. 3 Continuum extrapolation o pseudo-scalar decay constant (let) and vector decay constant (right). Horizontal axis is a 2. Star symbols represent the al value. agreement with within the limitation o the twopoint continuum extrapolation and large statistical errors. The inal results at β = 2.5 are needed or urther analyses. 4. Plan or Fiscal Runs and analyses This year, we have completed the gluon coniguration generations at β =.83 and.9. The results so ar obtained rom these conigurations are quite encouraging, as we reported here, and also at a number o international workshops and conerences this year [5-9]. We think that this calls or accelerated coniguration generations and analyses o the β = 2.5 run. So ar, we have accumulated 2 trajectories at quark parameter sets at β = 2.5. In iscal 25 we hope to increase the number up to 4-5 trajectories. Using the ull set o conigurations we hope to report the inal result o the light hadron spectrum by the end o iscal 25. Achieving this goal requires a control o chiral and continuum extrapolations. For the chiral extrapolation, we intend to pursue Wilson chiral perturbation theory which will provide the chiral it unction including inrared chiral logarithms and inite lattice spacing corrections to them []. For the continuum extrapolation, we are calculating renormalization actors and O(a) improved coeicients non-perturbatively. This year we have separately carried out extensive tests o a relativistic heavy quark action using quenched and two-lavor ull []. Application o this ramework to 2 + lavor ull to calculate heavy meson quantities will also be pursued 4.2. Sharing o the gauge conigurations International Lattice Data Grid (ILDG) [2] is a project to Acknowledgements We express our gratitude toward the ES Center or strong support in running our simulations on the Earth Simulator. Part o the present calculations has been made on CP-PACS and SR8/G at Center or Computational Sciences o University o Tsukuba, VPP5 at Academic Computing and Communications Center o University o Tsukuba, and SR8/F o KEK. This research greatly owes its success to the SuperSINET provided by National Institute o Inormatics, or executing eicient and timely data analyses by sharing the load with computing resources at collaboration institutes. Reerences [ ] S. Aoki et al. (JL collaboration), Phys. Rev. D65 (22) [ 2 ] S. Aoki et al., Nucl. Phys. B (Proc. Suppl.) 29 (24) 859. [ 3 ] K-I. Ishikawa et al. (JL and CP-PACS collaborations), Nucl. Phys. B (Proc. Suppl.) 29 (24) 444. [ 4 ] A. Ali Khan et al. (CP-PACS collaboration), Phys. Rev. D65(22)5455. [ 5 ] T. Ishikawa et al., Parallel talk at Lattice 24 (FNAL, USA, June 25), Nucl. Phys. B (Proc. Suppl.) 4 (25) 225. [ 6 ] T. Ishikawa, talk at the st ILFT Network Workshop "Lattice simulations via international research network" (Shuzenji, Japan, September 24). [ 7 ] T. Ishikawa, talk at German-Japanese Workshop "Towards Precision Physics rom Lattice Simulated on Tera-Flops Computers" (Berlin, Germany, November 24). [ 8 ] K. Kanaya, talk at ZiF Workshop "Quantum Fiels in the era o Teralop-computing" (Bieleeld, Germany, November 24). [ 9 ] T. Ishikawa, talk at the 2nd ILFT Network Workshop "From actions to " (Edinburgh, United Kingdom, March 25). [] S. Aoki, Phys. Rev. D68 (23) [] Y. Kayaba et al., Parallel talk at Lattice 25 (FNAL, USA, June 24), Nucl. Phys. B (Proc. Suppl.) 4 (25) 479. [2] [3] 8

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