1. 1.Introduction 2.Formalism 3.Results 4.Summary I=2 pi-pi scattering length with dynamical overlap fermion I=2 pi-pi scattering length with dynamical overlap fermion Takuya Yagi(Univ.Tokyo,KEK) Munehisa Ohtani(Univ.Regensburg) Osamu Morimatsu (KEK) Shoji Hashimoto (KEK) July 31 @Lattice07

2. Introduction Study of hadron interactions from QCD –nuclear force –interesting physics when strangeness is involved I=2 Pion-Pion: simplest among other hadron interactions –well controlled by ChPT –do not have annihilation or rectangular topology Overlap fermion –exact chiral symmetry application of ChPT straightforward (If not overlap, mixed action ChPT with domain-wall was worked out by NPLQCD(2006))

3. Overlap fermion The overlap action with quark mass “m” is defined as where This operator respects the chiral symmetry on the lattice. (GW relation) On the lattice Disadvantages - Numerically costly - So, limited volume

4. Lüscher’s formula Two particles with momentum “k” confined in a large volume box L : Length of the box : Phase Shift M. Lüscher(1986,1991) where For s-wave, this formula can be expanded as a function of scattering length divided by L. a 0 : scattering length

5. Setup Machine –BlueGene/L @ KEK Lattice Size Number of flavors Topological Charge 16 3 32 2 0 Gauge Action Fermion Action Iwasaki Action Overlap Fermion Source Type Gauge Fixing Wall Source Coulomb Gauge We used gauge configurations generated by JLQCD (Matsufuru) – –Lattice spacing = 0.1184(12) [fm], from r 0 ~ 0.49 [fm] – – – –pick configs every 100 traj to avoid autocorrelations. – – with Low mode averaging maNo.Conf 0.015 0.025 0.035 0.050 0.070 0.100 99 96 93 92

6. Periodic boundary (1) Our lattice is periodic in temporal direction Contaminations to the correlation functions exist 1.Two point correlation (1-Pion) 2.Four point correlation (2-Pion) 1-Pion Line (not quark line) T : T : size of the temporal direction

7. Periodic boundary (2) Ex) m=0.050 excited state : independent of A & B Good plateau is identified from this function.

8. Finite size effects (FSE) Lattice volume is not sufficiently large. Still, FSE can be estimated using ChPT a) Corrections to the pion mass and Decay constants (Recent study: Colangelo et al) b) Subleading terms to the Lüscher’s formula (Badeque and Sato (2006),(2007)) ref) R. Brower, S.Chandrasekharan, J.W.Negele, and U.-J. Wiese (2003) S. Aoki, H. Fukaya, S. Hashimoto, and T. Onogi (2007) Additional effect due to fixed topology - -Correction term to the n-point Green function - -I=2 Pi-Pi scattering length has no correction at LO, but pion mass has correction at LO - -In our analysis, only pion mass is corrected From Noaki’s talk

9. Result (1): energy spectrum We performed chiral extrapolation using NLO ChPT where F and F 0 are pion decay constants in the chiral limit. Because this quantity can be described only by decay constant in the chiral limit, we used F 0 from 2pt function in the chiral limit (Noaki’ talk). energy spectra for two- and four- point correlators Lüscher’s formula + Scattering Length

10. Result (2): correction exp after FSE correction (blue) before FSE correction (yellow) From the decay constant Energy difference is converted to the scattering length. Massless limit is from the decay constant data. FSE correction is made: relevant only near the chiral limit. Energy difference is converted to the scattering length. Massless limit is from the decay constant data. FSE correction is made: relevant only near the chiral limit.

11. Result (3): chiral fit exp Data show curvature toward chiral limit. One- loop ChPT has too strong curvature to fit the data. Data at smallest mass (ma=0.015) is away from the fit curve. Possibly FSE? Data show curvature toward chiral limit. One- loop ChPT has too strong curvature to fit the data. Data at smallest mass (ma=0.015) is away from the fit curve. Possibly FSE?

12. Summary We calculated I=2  scattering length with two- flavors of dynamical overlap fermions. –Exact chiral symmetry allows us to use the standard ChPT formulas. Lattice volume is not large enough –Finite size effect is corrected using known analytic results. Fit with one-loop ChPT is attempted. –Not well fitted over the whole mass region. –Two-loop analysis to be done.