Probing the Chiral Limit in 2+1 flavor Domain Wall Fermion QCD

Transcription

1 Probing the Chiral Limit in 2+1 flavor Domain Wall Fermion QCD Meifeng Lin for the RBC and UKQCD Collaborations Department of Physics Columbia University July 29 - August 4, 2007 / Lattice Regensburg Meifeng Lin (Columbia University) Chiral Limit in 2+1-flavor DWF QCD August Lattice / 23

2 Outline 1 Introduction Chiral Extrapolations in General Chiral Extrapolations for Domain Wall Fermions 2 Ensemble Details m P, f P and the Chiral Fits Low Energy Constants and Physical f π, f K and f K /f π 3 Summary and Outlook Meifeng Lin (Columbia University) Chiral Limit in 2+1-flavor DWF QCD August Lattice / 23

3 Chiral Extrapolations Introduction Chiral Extrapolations in General Finished 2+1-flavor DWF simulations: m l /m s 1/5 (dynamical) m l /m s 1/10 (partially quenched) Use SU(3) L SU(3) R chiral perturbation theory to guide the extrapolations. Expand around m u = m d = m s = 0 limit. Example: M 2 π from SU(3) ChPT, NLO M 2 π { = 2B 0 m B 0 + 2B 0 16π 2 f 2 f 2 [(2L 8 L 5 )m + (2L 6 L 4 )(2m + m s )] µ 2 m + 2m s log 2B ] 0(m + 2m s ) } 9 3µ 2. [ mlog 2B 0m Non-analyticity (chiral logarithms) anticipated at small quark masses Meifeng Lin (Columbia University) Chiral Limit in 2+1-flavor DWF QCD August Lattice / 23

4 Introduction DWF AWI and the Residual Mass Chiral Extrapolations for Domain Wall Fermions How does the chiral extrapolation change for DWF? Continuum QCD axial Ward-Takahashi identity is µ A a µ(x)o(y) = 2m f J5(x)O(y) a + i δ a O(y). For DWF Furman & Shamir, Nucl.Phys.B439:54-78,1995. µ A a µ(x)o(y) = 2m f J5(x)O(y) a + 2 J5q(x)O(y) a + i δ a O(y). The additional term J5q a is related to the residual mass m res J5q a = m resj5 a + O(a) All the lattice quark masses get an additive renormalization: m q = m f + m res Instead of using the input quark mass m f in the chiral fits, we need to use the total quark mass m q. Meifeng Lin (Columbia University) Chiral Limit in 2+1-flavor DWF QCD August Lattice / 23

5 Introduction DWF AWI and the Residual Mass Chiral Extrapolations for Domain Wall Fermions How does the chiral extrapolation change for DWF? Continuum QCD axial Ward-Takahashi identity is µ A a µ(x)o(y) = 2m f J5(x)O(y) a + i δ a O(y). For DWF Furman & Shamir, Nucl.Phys.B439:54-78,1995. µ A a µ(x)o(y) = 2m f J5(x)O(y) a + 2 J5q(x)O(y) a + i δ a O(y). The additional term J5q a is related to the residual mass m res J5q a = m resj5 a + O(a) All the lattice quark masses get an additive renormalization: m q = m f + m res Instead of using the input quark mass m f in the chiral fits, we need to use the total quark mass m q. Meifeng Lin (Columbia University) Chiral Limit in 2+1-flavor DWF QCD August Lattice / 23

6 Symanzik Expansions Introduction Chiral Extrapolations for Domain Wall Fermions How about corrections from higher-dimension operators with O(a)? To O(a), the Symanzik action for DWF can be written as Z S = d 4 x[ ψ(x)(i /D m q)ψ(x)] + ae αls c dwf ψ(x)σ µν F µνψ(x) where m q = m f + m res. Similar to Wilson fermions, (Rupak & Shoresh, PRD 66:054503,2002), but the O(a) lattice artefact is exponentially small for domain wall fermions ae αls c dwf m res(aλ QCD) 2 0.1m res, with the assumption that a 0.1 fm, and Λ QCD 500 MeV = O(a) term is NLO correction Our definition for m res guarantees M 2 π vanishes at m f + m res = 0 PCAC quark mass S.Sharpe, arxiv: [hep-lat] m res = P x Ja 5q( x, t)π a (0) P x Ja 5 ( x, t)πa (0) Contributions of all orders in a have been taken into account in m res Meifeng Lin (Columbia University) Chiral Limit in 2+1-flavor DWF QCD August Lattice / 23

7 Symanzik Expansions Introduction Chiral Extrapolations for Domain Wall Fermions How about corrections from higher-dimension operators with O(a)? To O(a), the Symanzik action for DWF can be written as Z S = d 4 x[ ψ(x)(i /D m q)ψ(x)] + ae αls c dwf ψ(x)σ µν F µνψ(x) where m q = m f + m res. Similar to Wilson fermions, (Rupak & Shoresh, PRD 66:054503,2002), but the O(a) lattice artefact is exponentially small for domain wall fermions ae αls c dwf m res(aλ QCD) 2 0.1m res, with the assumption that a 0.1 fm, and Λ QCD 500 MeV = O(a) term is NLO correction Our definition for m res guarantees M 2 π vanishes at m f + m res = 0 PCAC quark mass S.Sharpe, arxiv: [hep-lat] m res = P x Ja 5q( x, t)π a (0) P x Ja 5 ( x, t)πa (0) Contributions of all orders in a have been taken into account in m res Meifeng Lin (Columbia University) Chiral Limit in 2+1-flavor DWF QCD August Lattice / 23

8 Simulation Overview Ensemble Details Used the Iwasaki gauge action S g [U] = β (1 8 c 1 ) 3 x;µ<ν P[U] x,µν + c 1 x;µ ν R[U] x,µν with c 1 = 0.331, chosen to balance between a small m res and fast topology tunneling RBC and UKQCD, PRD75 (2007) RBC and UKQCD 2+1 flavor DWF ensembles β L 3 T L s m input l /m input s m tot l /m phys s # MD Time Units / / / / / / / Will focus on the two lightest ensembles in this talk. Meifeng Lin (Columbia University) Chiral Limit in 2+1-flavor DWF QCD August Lattice / 23

9 Ensemble Details Partially Quenched Measurements Use a set of different valence quark masses at a fixed m sea O = 1 [du]det[d (m sea )D(m sea )]O(m 1, m 2 )e Sg[U] Z More data points to constrain the chiral fits Partially quenched measurement parameters # measurements [sources] m l/m s am 1, /0.04 ( ) [2] (ND) 0.01/0.04 ( ) [2](ND) 90 [2] (ND) 0.02/0.04 ( ) [2](ND) 45 [2] (D) 0.03/0.04 ( ) [2](ND) 45 [2] (D) ND: all the non-degenerate combinations D : degenerate mesons only Meifeng Lin (Columbia University) Chiral Limit in 2+1-flavor DWF QCD August Lattice / 23

10 The Residual Mass Ensemble Details Recall: J a 5q = m resj a 5 + O(a) We compute the ratio R(t) on the lattice R(t) = P x Ja 5q( x, t)π a (0) P x Ja 5 ( x, t)πa (0) At small t s, the correlation function may overlap with unphysical states. Only when t is sufficiently large does R(t) represent the residual mass Fit R(t) where plateaux are reached to a t constant to get m res R(t) R(t) t X X 32 Meifeng Lin (Columbia University) Chiral Limit in 2+1-flavor DWF QCD August Lattice / 23

11 The Residual Mass Ensemble Details am res / / / /0.04 dynamical points chiral limit J 5q and J 5 may depend on the quark mass differently, so m res displays some quark mass dependence. Define the residual mass in the limit of m f 0 : am res am avg All the quark masses in the simulation get an additional mass shift: m q = m f + m res Meifeng Lin (Columbia University) Chiral Limit in 2+1-flavor DWF QCD August Lattice / 23

12 Ensemble Details Determine the Lattice Scale from Ω Mass Advantages of Ω (J P = 3 2+, sss) input also in Toussaint and Davies, NPB (Proc. Suppl.) 140 (2005) 234 Statistically well-determined on the lattice better statistical accuracy for 1/a Representative effective masses m Ω, eff m Ω, eff m l sea = m l sea = 0.02 Free of light quark mass chiral logs simple chiral extrapolations m Ω, eff m l sea = Stable particle in the QCD sector no complications with decays m Ω, eff m l sea = t Meifeng Lin (Columbia University) Chiral Limit in 2+1-flavor DWF QCD August Lattice / 23

13 Ensemble Details Determine the Lattice Scale from Ω Mass One complication: need to interpolate to the physical strange quark mass = get m phys s from the physical ratio of m 2 K/m 2 Ω / m Ω 2 m K m l sea =-mres expt ratio a m s phys Ω - mass m s val = 0.04 m s val = val am s sea m l 1/a = 1.73(2) GeV Other determinations of 1/a: From m ρ: 1.62(5) GeV From r 0 = 0.5 fm: 1.63(2) GeV Both consistent with simulations (hep-lat/ ), but systematically low Meifeng Lin (Columbia University) Chiral Limit in 2+1-flavor DWF QCD August Lattice / 23

14 m P, f P and the Chiral Fits Pseudoscalar Meson Mass m P Simultaneous fit to multiple correlators for reduced systematic and statistical errors One common ground-state mass, and one amplitude for each correlator included CAA LW (t) = A L (t)a W (0) A LW AA [exp( m Pt) + exp( m P(T t))], CPP LW (t) = P L (t)p W (0) A LW PP [exp( m Pt) + exp( m P(T t))], CAP LW (t) = A L (t)p W (0) A LW AP [exp( m Pt) exp( m P(T t))], CPP WW (t) = P W (t)p W (0) A WW PP [exp( m Pt) + exp( m P(T t))], CAP WW (t) = A W (t)p W (0) A WW AP [exp( m Pt) exp( m P(T t))]. Superscripts: W - Coulomb gauge fixed wall source; L - local source Subscripts: A - axialvector operator, P -pseudoscalar operator Matrix elements needed in the f P calculation can be obtained from appropriate ratios Meifeng Lin (Columbia University) Chiral Limit in 2+1-flavor DWF QCD August Lattice / 23

15 m P, f P and the Chiral Fits Pseudoscalar Meson Decay Constant f P f P = Z A 0 A π (def.) or f P = 2(m f + m res ) 0 P π (AWI) m P m P Five ways to determine f P based on different combinations of 0 P π and 0 A π Not all independent, choose the one with best systematic and statistical error I : f 2 P = 2Z2 A m PV II : f 2 P = A LW AA A LW AP A WW AP 8(mf + mres)2 m 3 P V III : fp 2 = 2Z2 A A LW 2 AP m PV A WW PP IV : f 2 P = 4Z A m f + m res m 2 P V V : f 2 P = 2Z2 A m PV A LW 2 AA A WW PP A WW AP 2 A LW PP 2 A WW PP A LW PP A LW AP A WW PP Meifeng Lin (Columbia University) Chiral Limit in 2+1-flavor DWF QCD August Lattice / 23

16 m P, f P and the Chiral Fits Pseudoscalar Meson Decay Constant f P af P I II III IV V Methods with m res involved differ O(a) error? Choose III in the final analysis (use only the definition, systematically clean and has the smallest error bar) am avg am sea l /am sea s = 0.005/0.04 Meifeng Lin (Columbia University) Chiral Limit in 2+1-flavor DWF QCD August Lattice / 23

17 Chiral Fits Revisited m P, f P and the Chiral Fits Small volume data: to check if NLO ChPT is a valid description in the range of quark masses we are working with am PS 2 /(mval +m res ) / / /0.04 af PS / / / am val am val Pseudoscalar meson masses : MeV Simultaneous NLO fit fails! Masses too heavy; Higher-order corrections non-negligible Meifeng Lin (Columbia University) Chiral Limit in 2+1-flavor DWF QCD August Lattice / 23

18 m P, f P and the Chiral Fits Fits Using NLO SU(3) PQChPT Sharpe and Shoresh, hep-lat/ Large volume data allows for simulations with lighter quark masses without introducing large finite volume effects Decreasing the masses included improves the fit quality m avg = (m 1 + m 2)/2 [0.001, 0.02], m sea l [0.005, 0.01] m P < 550 MeV / / / / af P (am P ) 2 /(am avg +am res ) am avg am avg Meifeng Lin (Columbia University) Chiral Limit in 2+1-flavor DWF QCD August Lattice / 23

19 Fits Using NLO SU(3) PQChPT m P, f P and the Chiral Fits Large volume data allows for simulations with lighter quark masses without introducing large finite volume effects Decreasing the masses included improves the fit quality m avg = (m 1 + m 2)/2 [0.001, 0.015], m sea l [0.005, 0.01] m P < 490 MeV / / / / af P (am P ) 2 /(am avg +am res ) am avg am avg Meifeng Lin (Columbia University) Chiral Limit in 2+1-flavor DWF QCD August Lattice / 23

20 Fits Using NLO SU(3) PQChPT m P, f P and the Chiral Fits Large volume data allows for simulations with lighter quark masses without introducing large finite volume effects Decreasing the masses included improves the fit quality m avg = (m 1 + m 2)/2 [0.001, 0.01], m sea l [0.005, 0.01] m P < 420 MeV / / / / af P (am P ) 2 /(am avg +am res ) am avg am avg Meifeng Lin (Columbia University) Chiral Limit in 2+1-flavor DWF QCD August Lattice / 23

21 Low Energy Constants Low Energy Constants and Physical f π, f K and f K /f π With m P < 420 MeV, lattice data can be described by NLO SU(3) PQChPT reasonably well From which we obtained the following LECs Λ χ ab 0 Z m af (2L 8 L 5 ) 10 4 (2L 6 L 4 ) 10 4 L L MeV 2.35(16) (40) 2.47(45) -0.02(42) 1.36(80) 8.62(99) 1 GeV 5.23(45) -0.48(42) -0.71(80) 2.42(99) Also determined a m = (7) from m 2 P, and f π = 124.7(30) MeV Problem: Strange quark mass outside of the range of NLO ChPT How are we going to determine f K? Adding analytic NNLO terms does make the fits go through the data points up to the strange quark mass, but light quark limit is also changed. Alternatives? Meifeng Lin (Columbia University) Chiral Limit in 2+1-flavor DWF QCD August Lattice / 23

22 Low Energy Constants and Physical f π, f K and f K /f π using SU(2) L SU(2) R NLO PQChPT Expand around m u = m d = 0 limit, strange quark mass irrelevant for light quark limit For quantities involving kaons, use SU(2) L SU(2) R ChPT in the presence of a heavy m s heavy meson ChPT, SU(2) chiral limit unaffected Fit several valence m s and then interpolate to the physical strange quark mass af K af K fit, am l =0.005 fit, am l =0.01 am l =0.005 am l =0.01 unitary, am l =am x am l =am x =am ud fit, am l =0.005 fit, am l =0.01 am l =0.005 am l =0.01 unitary, am l =am x am l =am x =am ud am x am val s = 0.04 am x am val s = 0.03 Meifeng Lin (Columbia University) Chiral Limit in 2+1-flavor DWF QCD August Lattice / 23

23 Final Results Low Energy Constants and Physical f π, f K and f K /f π Preliminary final results from SU(2) NLO chiral fits f π = 124.2(35) MeV [130.7(41) MeV] f K = 150.0(36) MeV [159.8(15) MeV] f K /f π = 1.208(14) [1.223(12)] It s also interesting to compare the LECs from SU(3) and SU(2) fits LEC SU(3) SU(2) ab 0 Z m 2.453(75) 2.414(61) af (17) 0.665(21) l (28) 3.13(33) l (5) 4.42(14) For more details of the SU(2) chiral fits and quark masses, see talk by Enno Scholz in H10 15:00 Today. Meifeng Lin (Columbia University) Chiral Limit in 2+1-flavor DWF QCD August Lattice / 23

24 Low Energy Constants and Physical f π, f K and f K /f π Finite Volume Effects? NLO ChPT predictions am 1 = am 2 = am sea l = 0.01, m πl 3.9: 1% FSE on f P Colangelo et al (resummed Luscher + NNLO) predicts 1.5% FSE af P L=24, m l sea = 0.01, ms sea = 0.04 L=16, m l sea = 0.01, ms sea = 0.04 Our data shows a difference of 3.5(9)% between L = 16 and L = 24 results Light valence point with am 1,2 = on the L = 24 lattices may also suffer from visible finite volume effects Further investigations needed am avg Meifeng Lin (Columbia University) Chiral Limit in 2+1-flavor DWF QCD August Lattice / 23

25 Summary Summary and Outlook NLO SU(3) ChPT is possibly accurate up to m P 400 MeV; Within the chiral regime, SU(3) and SU(2) give similar level of accuracy in probing the light quark chiral limit; SU(2) heavy meson ChPT provides a useful tool for extrapolations/interpolations up to the strange quark mass Outlook Heavy meson ChPT works with the assumption: m2 π 1. Lighter m 2 K would be better. To get a more accurate determination of f K, another dynamical m s will be needed. Finer lattice spacing ( 2.1 GeV) and lighter quark masses ( m s /7) under way: moving inside chiral regime, and continuum extrapolation possible Meifeng Lin (Columbia University) Chiral Limit in 2+1-flavor DWF QCD August Lattice / 23