Nucleon sigma term and strange quark content from

Nucleon sigma term and strange quark content from dynamical overlap simulations ar. Xiv: 0806. 4744 [hep-lat] Lattice 2008　@ College of William and Mary Hiroshi Ohki (YITP and Kyoto University) for JLQCD Collaboration 1

Outline o o o Introduction and motivation Methods Simulation details Results Summary 2

JLQCD collaboration KEK 　 　　　　　 S. Hashimoto, H. Ikeda, T. Kaneko, H. Matsufuru, J. Noaki, E. Shintani, N. Yamada Niels Bohr H. Fukaya Tsukuba 　　　 S. Aoki, T. Kanaya, N. Ishizuka, K. Takeda, Y. Taniguchi, 　A. Ukawa, T. Yoshie Hiroshima 　 K. -I. Ishikawa, M. Okawa YITP H. Ohki, T. Onogi, T. Yamazaki 3 KEK Blue. Gene (10 racks, 57. 3 TFlops)

Introduction o What is sigma term? scalar form factor of the nucleon at zero recoil and Low energy parameter of Ch. PT o Strange quark content of Nucelon 4

Motivation o Y and f parameters are quite important phenomenologically. Neutralino Dark matter search The interaction with nucleon is mediated by the higgs boson exchange in the t-channel. K. Griest, Phys. Rev. Lett. 62, 666(1988) Phys, Rev, D 38, 2375(1988) Baltz, Battaglia, Peskin, Wizanksy Phys. Rev. D 74, 103521 (2006). heavy quark loop strange quark contribution is dominant and important. H, h 5

Introduction o y (and f) parameters are calculated by connected and disconnected contributions. It is necessary for the determination of y parameter to calculate each contribution separately. 6

How well are the parameters known? c. f. Recent work of nucleon mass for plenary talk of Walker-Loud The strange quark content has an almost 100% uncertainty. 7

Uncertainties in y parameter o Ch. PT: Low Energy Constants (higher order). o Previous lattice calculations (Wilson type fermion). Mixing of connected and disconnected contributions (Matrix methods and spectrum methods) due to lattice artifact. The most crucial uncertainty is the additive mass shift. c. f. C. Michael et. al. Nucl. Phys. Proc. Suppl. 106, 293 (2002) Sea quark mass derivative with fixed bare valence quark mass is contaminated by physical valence quark mass derivative Which is unwanted lattice artifact ( red arrow). Spectrum methods with Wilson type fermions 8

Our strategy Determine the nucleon sigma term in unquenched QCD using the dynamical quark (overlap fermion), which has an exact chiral symmetry on the lattice. o The advantage of the exact chiral symmetry n o No mixing of connected and disconnected contributions In this study, we work in nf=2 unquenched QCD. Result for nf=2+1 QCD will also soon appear. 9

Our method the sigma term from the nucleon mass spectrum. Feynman - Hellman theorem partial derivatives with respect to the valence and sea quark masses give contributions from ‘connected’ and ‘disconnected’ diagrams. Exact chiral symmetry no additive mass shift which causes dangerous lattice artifact 10

Numerical simulation Measurement of the nucleon 2 pt function o 6 pts(sea) and 9 pts(valence) for quark masses o Low mode averaging is employed (#eigenmodes=100) Nf=2 overlap fermion configurations o 16^3 x 32, a=0. 12 fm, L=1. 9 fm o 6 values of sea quark mass o fixed topology o At Q=0 accumulated 10, 000 trajectories 11

Results o Nucleon masses from 2 -pt functions Effective mass plot for amq=0. 035 Solid lines are the mass from the fit 12

Chiral extrap. (unitary point) extraction of nucleon sigma term o Fit without lightest quark mass data(5 pts) n several fit forms to study chiral extrapolation errors Fit with finite volume correction (5 and 6 pts) o n fits including finite volume effects estimated by Ch. PT. 13

Ch. PT Fit of nucleon mass spectrum Fit formula with Heavy Baryon chiral perturbation theory c. f. E. E. Jenkins et. al. , PLB 255, 558 (1991) M. Procura et. al. PRD 69, 034505(2004) I : II : with input 　　 　　　　　 , 　 III : with input , (0 : simplified version of Fit I) 14

Fit results with and without finite volume corrections raw data Solid …fit 0 dot…fit I Dashed…Fit II dot-dashed…Fit III Nicely fit to the Ch. PT formula without lightest point. 　 Finite volume corrected(Fit 0) Raw data Finite volume corrected data Successful with all data point Fit uncertainty is O(10)%. 15

Results of sigma term 1. The systematic error is mainly the chiral extrap. error. 2. Finite volume effect (FVE) is sub-leading (~ 9%). 3. We quate final results from Fit 0(FVE uncorrected). 16

PQCh. PT fit (partially quenched data points) extraction of y parameter Fit with partially quenched Ch. PT (5 X 8 data points) o n n consistency check of the unitary point fit interpolation to the strange quark mass. Separate extraction of connected and disconnected contributions o 17

PQCh. PT fit function J. W. Chen et al. , PRD 65, 094001(2002) S. R. Beane et al. NPA 709, 319 (2002) • Fit a: 6 parameters • Fit b: 7 parameters • Fit b: 8 parameters 18

Fit results (PQCh. PT) PQCh. PT fit works very well. It gives consistent results with the unitary point fit. 19

Connected and disconnected contributions at valence Sea The disconnected contribution (sea quark content) is always smaller than the connected contribution (valence quark content). 20

Connected and disconnected contributions at Strictly speaking, it is not possible to extract the strange quark content within two-flavor QCD. For the final result, we should wait for 2+1 -flavor QCD result (coming soon). o o We present semi-quenched estimate of the y parameter Semi quenched estimate of y 21

Comparison with other results n n Our results of is consistent with Ch. PT. Finite Volume correction is controllable. Previous lattice result of sea/valence is larger than 1. Our result is 0~0. 3. n Ch. PT predicts n Previous lattice results due to large sea quark contribution without removing lattice artifact. n After removing lattice artifact previous y is -0. 3(3) n Our result gives n 22

6. summary o o We studied the nucleon mass spectrum for nf=2 unquenched QCD using exactly chiral symmetric fermions. Our calculation is free from the dangerous lattice artifacts (mixing of connected and disconnected contributions) Our result of sigma term is consistent with the Ch. PT prediction. We found that the disconnected (strange quark content) part is tiny. 23

Thank you 24

o Backup slide 25

Fit of the quark mass dependence (5 pt) a means fit with input g. A B means fit with g. A free The solid, dot, dashed, dot-dashed curves represent the Fit 0 a, II, and III, respectively. 26

Fit of the quark mass dependence (5 and 6 pt) Box size of L=1. 9 fm is rather small for baryon with light quarks. We can estimate the Finite Size Effect(FSE) using Ch. PT. ＦＶＣ　Ｒｅｆする After correcting the lattice data including FSE, we can redo the Ch. PT fit. Raw data Finite volume corrected data 27

Results of sigma term 5ｐｔと６ｐｔ 1. Although there are finite size effects (FSE), sigma term gets only about 5% change. 2. Ch. PT fit (g. A fixed) without considering FSE gives reasonable result. 3. We take Ch. PT fit (g. A fixed, FSE uncorrected) as our best fit, assuming possible 10% finite size error. 28

JLQCD’s Simulations o Overlap fermion ( explicit construction by Neuberger) Exact chiral symmetry on the lattice　(index theorem) 　　　　　 Hasenfratz, Laliena and Niedermayer, Phys. Lett. B 427(1998) 125 Luscher, Phys. Lett. B 428(1998)342. 29

Our analysis Fit 5 pt data without finite volume correction • Fit 0 a and 0 b: 3 and 4 parameters fit • Fit Ia and Ib: 3 and 4 parameters fit • Fit II : 3 parameters with c_2=3. 2[Ge. V-1] and c_3=-3. 4[Ge. V-1] • Fit III : 3 parameters with c_2=3. 2[Ge. V-1] and c_3=-4. 7[Ge. V-1] a means fit with input g. A=1. 267, B means fit with g. A free Fit 5 and 6 pt data with finite volume correction • Fit 0 a: 3 parameters fit • Fit Ia: 3 parameters fit • Fit II : 3 parameters with c_2=3. 2[Ge. V-1] and c_3=-3. 4[Ge. V-1] • Fit III : 3 parameters with c_2=3. 2[Ge. V-1] and c_3=-4. 7[Ge. V-1] 30

Fit of the quark mass dependence (5 pt) The solid, dot, dashed, dot-dashed curves represent the Fit 0 a, II, and III, respectively. 31

Fit the finite volume corrected data (5 and 6 pt) Box size of L=1. 9 fm is rather small for baryon with light quarks. We can estimate the Finite Size Effect(FSE) using Ch. PT. c. f. A. Ali Khan et al. , Nucl, Phys. B 689, 175(2004) For the input parameter, we use the nominal values of g. A, c 1, c 2 and c 3, after correcting the lattice data including FSE, we carry out the Ch. PT fit. Raw data Finite volume corrected data 32