Light Meson spectrum with Nf=2+1 dynamical overlap fermions

Transcription

1 Light Meson spectrum with Nf=2+1 dynamical overlap fermions Jun Noaki (KEK) for JLQCD-TWQCD Collaborations: S.Aoki, T-W.Chiu, H.Fukaya, S.Hashimoto, T-H.Hsieh, T.Kaneko, H.Matsufuru, T.Onogi, E.Shintani and N.Yamada

2 QCD vs ChPT? (mk OK?) Introduction ,000 M ev K N Lattice QCD (1 st principle) ChPT Chiral properties of meson masses and decay consts. Numerical simulation with dynamical overlap fermions Neuberger, 1998 Exact chiral symmetry, No O(a) errors Direct application of ChPT: no need of WchPT, SchPT, tmchpt... Many technical challenges see S.Hashimoto's plenary, H.Matsufuru's poster 1/ 15

3 Plan Numerical simulation (Nf=2 and 2+1) Setup parameters Correlation functions with eidenmodes Analysis (Nf=2+1) Finite size effects Non-perturbative renormalization Chiral extrapolation Results on the Nf=2 calculation Preliminary results on the Nf=2+1 Summary 2/ 15

4 Simulation setup Parameters action: S o v S Iwasaki S ~~~~~ ex-wilson Q t o p = 0 N f = 2 N f = 2 1 (two ms's) volume: #config: Cutoff: (r0=0.49 fm) mass range: ,000 trajs./20 2,500 trajs./ GeV GeV quark mass: 6 5 x 2 fm fm MeV MeV MeV 3/ 15

5 With low-lying eigenmodes (Nf=2+1) Lowest 80 eigenpairs stored on disk D o v u i = i u i Quark propagator Giusti et al, 2003 S q x, y = 80 i=1 u i x u i y i m q S q H i g h x, y Meson correlator DeGrand & Schaefer, 2004; Giusti et al., 2004 C t = C H H t C L H t C H L t C L L t ~~~~ Average over source locations (time slices) 4/ 15

6 Finite size effects m L 2.7 for the lightest mass Standard FSE Fixed topology effect (Q=0) Brower et al, 2003 Correction with ChPT exact chiral symm. Standard FSE: resummed Luscher's formulae Colangelo et al, 2005 Fixed topology effect (NLO ChPT): eg. Nf=2 ChPT m,q=0 m =0 = V top [1 1 ln m / 3 2 ] f,q=0 f =0 = V top 1 ln m / 4 2 Aoki et al, 2007; JLQCD-TWQCD, 2007; see also Chiu's talk 5/ 15

7 Actual Corrections (Nf=2+1) m 2 / m q f - Two corrections almost cancel. +2% at most. - Fixed Q correction is tiny. +8% at most. 6/ 15

8 Quark mass renormalization RI/MOM condition In Landau gauge: Martinelli et al., 1995 Z q 1 Z p = 1 Vertex functions (WTI+OPE) P p = A m q Z q Z m B P m q 2 S p = A m q Z q Z m B S m q 2 ms=0.100,mud=0.035, simul fit for same color P Control of m_q dependence with real data: 80 = 1 V i=1 2 m q m 2 2 q i S 3-loop matching at chiral limit Z m M S 2GeV = / 15

9 Plan Numerical simulation (Nf=2 and 2+1) Setup parameters Correlation functions with eidenmodes Analysis (Nf=2+1) Finite size effects Non-perturbative renormalization Chiral extrapolation Results on the Nf=2 calculation Preliminary results on the Nf=2+1 Summary

10 Nf=2 NLO ChPT Chiral expansion m 2 /m q = 2B 1 x ln x c 3 x O x 2 f = f 1 2 x ln x c 4 x O x 2 where x = 2B m q 4 f 2 Equivalent expansion parameters in the valid region of NLO, x x = m 4 f 2 or = m 4 f 2 Effectively resum higher order effects Three fit curves on one figure rescale of horizontal axis m 2 = m 2 / m q curve m q, m 2 = 4 f curve 2 -Fits as a func of m_q, m_pi^2 and xi -Simultaneous fit and independent fits -Comparison through chi^2 is inadequate. Direct comparison is possible. 8/ 15

11 Fit curves (Nf=2) Using the lightest 3 data points, m 2 / m q f NLO is OK for the lightest three data. Xi-fit describes the data beyond the fitted region. 9/ 15

12 Validity of NLO? Fit parameters for different mass ranges Convergence at the 3 rd pt Threshold is between 3,4 th points ~450MeV Two options: Analyse data below 450 MeV instead of full info from lattice. Statistical error is larger. Use heavier mass points by including NNLO terms. Only the xi-fit is useful. 10/ 15

13 Extension to NNLO (Nf=2) NLO m 2 / m q = 2B 1 ln c 3 f = f 1 2 ln c 4 NNLO m 2 m q = 2B[ 1 ln 7 2 ln 2 c 4 2 f 4 3 L 16 2 ln ] c ln K 1 2 [ f = f 1 2 ln 5 ln 2 3 L 2 53 ] 2 2 ln c ln K 2 2 input: L = 7 ln 1 4 f 2 8ln 2 4 f 2 from phenomenology Large shift of c3 and c4. Simultaneous fit is necessary for NNLO. 11/ 15

14 Convergence of ChPT (Nf=2) Using all data points, chi^2/dof = 1.42 m 2 / m q f 500MeV 500MeV = 0.3 and 0.7 NLO-LO 12/ 15

15 Low energy parameters (Nf=2) NLO vs NNLO NLO for >500 MeV is indeed problematic. LECs f = M e V M S, 2 G e V =[ (stat.)(6-5pts)(latt. scale) p h l ys 3 = p h l ys 4 = (stat.)(6-5pts)(renorm. points) 12.7 M e V] 3 Physical quantities M S, 2 G e m V ud = M e V f = M e V 13/ 15

16 Preliminary result for Nf=2+1 Simultaneous fit to the full NNLO expressions Amoros et al m 2 / m q, m K 2 / m av f, f K 16 params for 20 data points with input L_1,2,3,7. Using chi^2/dof = 9.7 Preliminary result:, s M S, 2 G e m V M S, 2 G e ud = M e V, m V s = M e V, f K / f = study of LECs is ongoing. 14/ 15

17 Summary Nf=2 and 2+1 dynamical overlap fermions Exact chiral symm. No need of XChPT. Improvements of the data with eigenmodes. FSE corrections using ChPT calculations. (shevere with current resource.) Non-perturbative renormalization for quark mass. Nf=2 ChPT is tested Xi-expansion shows better convergence behavior. ~450 MeV is the upper limit of NLO ChPT (Kaon is out). NNLO analysis needed beyond this scale. 1/a is a source of large systematic error Extension to Nf=2+1 ChPT test is to be completed on a 16^3x48 lattice. Generation on a 24^3x48 lattice has started. 15/ 15