Chapter 6 Field Theory Research Team 6.1 Members Yoshinobu Kuramashi (Team Leader) Yoshifumi Nakamura (Research Scientist) Hiroya Suno (Research Scientist, Joint Position with the Nishina Center for Accelerator-based Research) Eigo Shintani (Research Scientist) Yuya Shimizu (Postdoctoral Researcher) Ken-Ichi Ishikawa (Visiting Scientist, Hiroshima University) Shoichi Sasaki (Visiting Scientist, Tohoku University) Takeshi Yamazaki (Visiting Scientist, University of Tsukuba) Shinji Takeda (Visiting Scientist, Kanazawa University) 6.2 Research Activities Our research field is physics of elementary particles and nuclei, which tries to answer questions in history of mankind: what is the smallest component of matter and what is the most fundamental interactions? This research subject is related to the early universe and the nucleosynthesis through Big Bang cosmology. Another important aspect is quantum properties, which play an essential role in the world of elementary particles and nuclei as well as in the material physics at the atomic or molecular level. We investigate nonperturbative prop- erties of elementary particles and nuclei through numerical simulations with the use of lattice QCD (Quantum ChromoDynamics). The research is performed in collaboration with applied mathematicians, who are experts in developing and improving algorithms, and computer scientists responsible for research and development of software and hardware systems. Lattice QCD is one of the most advanced case in quantum sciences: interactions between quarks, which are elementary particles known to date, are described by QCD formulated with the quantum field theory. We currently focus on two research subjects: (1) QCD at finite temperature and finite density. We try to understand the early universe and the inside of neutron star by investigating the phase structure and the equation of state. (2) First principle calculation of nucleon form factors. Proton and neutron, which are called nucleon, consist of three quarks. We investigate their internal structure and low energy properties by the measurement of various form factors. Successful numerical simulations heavily depend on an increase of computer performance by improving algorithms and computational techniques. However, we now face a tough problem that the trend of computer architecture becomes large-scale hierarchical parallel structures consisting of tens of thousands of nodes which individually have increasing number of cores in CPU and arithmetic accelerators with even higher degree of parallelism: we need to develop a new type of algorithms and computational techniques, which should be different from the conventional ones, to achieve better computer performance. For optimized use of K computer 63 0.17 0.16 0.15 0.14 E T 0.13 0 t √ 0.12 Nf = 4 0.11 Nf = 3 2 0.1 c0+c1/Nt c +c /N2+c /N3 0.09 0 1 t 2 t 0.08 0 0.01 0.02 0.03 0.04 0.05 0.06 2 1/Nt 5 4 E /T 3 PS,E m N = 4 2 f Nf = 3 2 c0+c1/Nt 1 2 3 c0+c1/Nt+c2/Nt 0 0 0.01 0.02 0.03 0.04 0.05 0.06 64 2 CHAPTER 6. FIELD THEORY RESEARCH TEAM 1/Nt 0.9 0.17 0.8 0.16 0.7 0.15 0.6 0.14 E PS,E 0.5 T 0.13 0 m t 0 √ t √ 0.4 0.12 Nf = 4 Nf = 4 0.3 0.11 Nf = 3 Nf = 3 2 2 0.2 c0+c1/Nt 0.1 c0+c1/Nt 2 3 c +c /N +c /N c +c /N2+c /N3 0.1 0 1 t 2 t 0.09 0 1 t 2 t 0 0.08 0 0.01 0.02 0.03 0.04 0.05 0.06 0 0.01 0.02 0.03 0.04 0.05 0.06 2 2 1/Nt 1/Nt √ √ √ Figure 15: CutoFigureff dependence 6.1: Continuum of √t T (top) extrapolationm /T of(middle) the critical and √ pointt m t0(bottom).mPS,E (left) and t0TE (right) with t0 the Wilson flow scale. Red0 E and bluePS symbols,E E denote 4 and 30 flavorPS,E cases, respectively. Results for N = 3 [2] are also shown. f 5 our research team aims at (1)13 developing a Monte Carlo algorithm 4 to simulate physical system with negative E weight effectively and (2) improving iterative methods to solve/T large system of linear equations. These technical 3 development and improvement are carried out in the researchPS,E of physics of elementary particles and nuclei based m on lattice QCD. N = 4 2 f Nf = 3 2 c0+c1/Nt 6.2.1 QCD at finite temperature and finite 1 density 2 3 c0+c1/Nt+c2/Nt Establishing the QCD phase diagram spanned by the temperature 0 T and the quark chemical potential µ in 0 0.01 0.02 0.03 0.04 0.05 0.06 a quantitative way is an important task of lattice QCD. We have been working on2 tracing the critical end line in the parameter space of temperature, chemical potential and quark masses in1/N 4,t 3 and 2+1 flavor QCD using the O(a)-improved Wilson quark action and the Iwasaki gauge action. We have determined the critical end point at zero chemical potential µ = 0 in 3 flavor case. 0.9 Our strategy is to identify at which temperature the Kurtosis of physical observable at the transition point on 0.8 several different spatial volumes intersects. This method is based on the property of opposite spatial volume 0.7 dependence of the Kurtosis at the transition point between the first order phase transition side and the crossover one. We have carried out a systematic study of 0.6 the critical end point changing the temporal lattice size from Nt = 4 to 10 in 3 flavor case, which corresponds PS,E 0.5 m to change the lattice spacing. In Fig. 6.1 (left and right panels)0 we show the continuum extrapolation of the t √ 0.4 critical pseudoscalar meson mass mPS,E and the critical temperature TE normalized by √t0, where √t0 denotes Nf = 4 the Wilson flow scale. We also make the same study in 4 0.3 flavor case for comparison. We observe that the Nf = 3 2 2 critical temperature seems to follow the O(a ) scaling property 0.2 both in 3 mad 4 flavor cases.c0+c1/N Ont the other c +c /N2+c /N3 hand, √t0mPS,E shows significantly large scaling violation and 0.1 its continuum extrapolation0 gives1 t different2 t values for 3 and 4 flavor cases: rather close to zero for the former and sufficiently deviated from zero for the latter. 0 The origin of the difference between 3 and 4 flavor cases and ,especially, 0 0.01 whether 0.02 0.03mPS,E 0.04in 3 flavor 0.05 case 0.06 may 2 vanish in the continuum limit are intriguing theoretical issues. Currently we are performing1/Nt a simulation at finer lattice spacing to investigate the possibility that mPS,E may vanish in the continuum limit. Figure 15: Cutoff dependence of √t0TE (top) mPS,E/TE (middle) and √t0mPS,E (bottom). 6.2.2 Nucleon form factorsResults for Nf = 3 [2] are also shown. Nucleon form factors are good probes to investigate the internal structure of the nucleon which is a bound state of quarks. Study of their properties requires nonperturbative method and much13 effort has been devoted to calculate them with lattice QCD since 1980’s. Unfortunately, the current situation is that we are still struggling for reproducing the well-established experimental results, e.g., the electric charge radius and the axial vector coupling. This means that we have not yet achieved proper treatment of a single hadron in lattice QCD calculation. The left panel of Fig. 6.2 shows a summary plot of the electric charge radius calculated with lattice QCD as of 2014. We focus on two major systematic uncertainties in the current lattice QCD simulations: one is heavier quark masses than the physical values and the other is finite spatial volume effects. In order to get rid of them we have carried out calculation of the nucleon form factors on a (10.8 fm)4 lattice at the physical point in 2+1 flavor QCD. Thanks to the large spatial volume we can get access to small momentum transfer region up to q2 =0.013 GeV2. The right panel of Fig. 6.2 plots our results for the electric charge radius, whose 6.2.Hadron RESEARCH Structure ACTIVITIES Martha Constantinou 65 1.0 0.6 0.9 Experiment (ep scatt.) 1 Experiment (µH atom) 0.8 Isovector, t /a={14,16} 0.5 sep 0.7 0.95 ) 0.4 ) 2 2 0.6 fm (fm (fm 0.5 0.3 1/2 > u-d u-d > > E 2 2 2 0.40.9 1 2 <r <r 0.2 <r 0.3 RBC/UKQCD ’09 (DWF, Nf =2+1) 0.1 RBC/UKQCD ’09 (DWF, Nf =2+1) 0.2 RBC/UKQCD ’13 (DWF, Nf =2+1) RBC/UKQCD ’13 (DWF, Nf =2+1) QCDSF/UKQCD ’11 (Clover, Nf =2) 0.85 ETMC ’10 (TMF, Nf =2) QCDSF/UKQCD ’11 (Clover, Nf =2) ETMC ’10 (TMF, Nf =2) QCDSF ’11 (Clover, Nf =2) 0.1 ETMC ’13 (TMF, Nf =2+1+1) QCDSF ’11 (Clover, Nf =2) ETMC ’13 (TMF, Nf =2+1+1) ETMC ’14 (TMF&clover, Nf =2) PoS(LATTICE2014)001 0.0 CSL/MAINZ ’12 (Clover, Nf =2) CSL/MAINZ ’12 (Clover, Nf =2) ETMC ’14 (TMF&clover, Nf =2) LHPC ’09 (DWF, Nf=2+1) PNDME ’13 (HISQ, Nf=2+1=1) 0.0 PNDME ’13 (HISQ, Nf=2+1=1) LHPC ’10 (DWF/asqtad, Nf=2+1) PDG ’12 LHPC ’10 (DWF/asqtad, Nf=2+1) PDG ’12 µp LHPC ’14 (Clover, Nf=2+1) 0.8 LHPC ’14 (Clover, Nf=2+1) 0 0.1 0.2 0.3 0.4 0.5 0 Linear0.1 0.2Dipole 0.3 0.4 Quadrature0.5 z-exp, kmax=3 m (GeV) 2 2 m (GeV)2 2 2 2 2 2 π q cut=0.013 GeV π q cut=0.102 GeV q cut=0.102 GeV q cut=0.102 GeV Figure 10: Dirac (left) and Pauli (right) radii as a function of mπ .