Light pseudoscalar masses and decay constants with a mixed action

Transcription

1 Light pseudoscalar masses and decay constants with a mixed action Jack Laiho Washington University Christopher Aubin and Ruth Van de Water Lattice 2008 July 15, 2008 William + Mary, July 15, 2008 p.1/21

2 Mixed action simulations Work done with Christopher Aubin and Ruth Van de Water These simulations use MILC lattices (with 2+1 Asqtad staggered quarks in the sea sector) and domain wall quarks in the valence sector. Advantages A large number of ensembles with different volumes, sea quark masses and lattice spacings exist and are publicly available. The existing ensembles have 2+1 flavors of light sea quarks (m strange /10 for the lightest quarks) The good chiral properties of the valence sector make things much simpler than the staggered case. There are only two additional parameters (over pure domain wall) that appear at one loop in the mixed action ChPT for m π, f π, and B K. They can both be obtained from spectrum calculations. NPR can be carried through in the same way as in domain wall. William + Mary, July 15, 2008 p.2/21

3 Mixed action calculations In 1-loop Mixed Action χpt only two parameters beyond those of domain-wall: m 2 dw = 2µ dw (m v + m res ), (1) m 2 I = 2µ stag m s + a 2 I, (2) m 2 mix = µ dw (m v + m res ) + µ stag m s + a 2 mix, (3) William + Mary, July 15, 2008 p.3/21

4 Run parameters Table 1: Lattice parameters a(fm) a bm /am s L(fm) m π L 10/g 2 Lat Dim # Confs / / / / / / / / William + Mary, July 15, 2008 p.4/21

5 The residual mass circles: a=0.12fm; squares: a=0.09fm am l /am s = 0.005/0.05 am l /am s = 0.007/0.05 am l /am s = 0.01/0.05 am l /am s = 0.02/0.05 am l /am s = /0.031 am l /am s = /0.031 m res m val William + Mary, July 15, 2008 p.5/21

6 Determining the splitting mix 0.8 circles: a=0.12fm; squares: a=0.09fm (m π,mix ) 2-1/2(m π,dw ) mixed action simulation data MILC staggered taste splittings m stag William + Mary, July 15, 2008 p.6/21

7 Scalar bubble prediction π du a 0 du du du η B(t) = " µ 2 X 2 3L 3 9 k e 2ω vvt ω 2 vv e 2ω vvt 2ω 4 vv e (ω vv+ω ηi )t (m2 SI m2 )2 UI ω vv ω ηi (m 2 vv m 2 η I ) 2 " 3m 2 vv (m 2 vv 2m2 η I ) + 2m 4 S + m 4 # I U I 3(m 2 η I m 2 vv )2 (ω vv t + 1) (m2 U I m 2 vv )(m2 S I m 2 vv ) m 2 η I m 2 vv e 2ω vut ω 2 vu e 2ω vst ω 2 vs # Prelovsek, PR D73, (2006), hep-lat/ William + Mary, July 15, 2008 p.7/21

8 Scalar bubble prediction vs. data am l /am s = 0.007/ am val = 0.01 am val = 0.02 am val = C(t) t/a William + Mary, July 15, 2008 p.8/21

9 Scalar bubble prediction vs. data am l /am s = 0.007/0.05, am val = 0.01 am l /am s = /0.031, am val = C(t) (t/a)( coarse /r1 ) William + Mary, July 15, 2008 p.9/21

10 Approach to chiral fits We have generated data with relatively high statistics so that we can resolve a correlation matrix and obtain reliable confidence levels in fits. Using SU(3) chiral perturbation theory in order to interpolate about the strange quark mass and extrapolate in the light quark mass. We are using one-loop SU(3) mixed action χpt and higher order analytic terms. Plans to investigate SU(2) χpt and 2-loop corrections. Separate fits to m 2 π/m q and f π, where leading order µ is taken from linear fits to m 2 π data, evaluated in region of data, rather than chiral limit. f π evaluated at physical pion point. William + Mary, July 15, 2008 p.10/21

11 m 2 π/m q chiral fit m 2 π /(m x + m y + 2m res ) full QCD 01/05 007/05 005/ / / /031 01/ (m x + m y + 2m res ) /2 χ 2 /dof=0.58, C.L.=0.97 William + Mary, July 15, 2008 p.11/21

12 f π chiral fit (f π )/2 1/ full QCD 01/05 007/05 005/05 01/ / / /031 MILC (m x + m y + 2m res ) /2 χ 2 /dof=1.09, C.L.=0.33 William + Mary, July 15, 2008 p.12/21

13 f π chiral fit (compared w/ MILC) (f π )/2 1/ full QCD 01/05 007/05 005/05 01/ / / /031 MILC exp (m x + m y + 2m res ) /2 William + Mary, July 15, 2008 p.13/21

14 Fit results type of f π fit χ 2 /d.o.f. C.L. NNLO analytic % No NNLO No NLO logs No FV % No splittings % type of m 2 π/m q fit χ 2 /d.o.f. C.L. NNLO analytic % No NNLO No NLO logs No FV % No splittings % William + Mary, July 15, 2008 p.14/21

15 Convergence of SU(3) χpt (f SU(3) )/2 1/ LO NLO All orders (m x + m y + 2m res ) /2 William + Mary, July 15, 2008 p.15/21

16 m 2 π/m q with a higher cut on masses m 2 π /(m x + m y + 2m res ) full QCD 01/05 007/05 005/ / / /031 01/03 02/ (m x + m y + 2m res ) /2 χ 2 /dof=1.14, C.L.=0.24 William + Mary, July 15, 2008 p.16/21

17 f π with a higher cut on masses 0.2 (f π )/2 1/ full QCD 02/05 01/05 007/05 005/05 01/ / / /031 MILC (m x + m y + 2m res ) /2 χ 2 /dof=1.25, C.L.=0.12 William + Mary, July 15, 2008 p.17/21

18 f K determination 0.2 (f π )/2 1/ full QCD K + full QCD π 02/05 01/05 007/05 005/05 01/ / / /031 MILC f π MILC f K (m x + m y + 2m res ) /2 William + Mary, July 15, 2008 p.18/21

19 Preliminary decay constant error budget source f K f π f K /f π statistics input chiral/continuum extrap finite volume total error Preliminary error budget for the decay constants and their ratio. Uncertainties are quoted as a percentage. The total combines systematic errors with statistical errors in quadrature. William + Mary, July 15, 2008 p.19/21

20 Preliminary result for V us f K /f π = 1.185(18)(20) where the first error is statistical and the second is systematic. The MILC value is f K /f π = 1.197(3)( ). Following Marciano [PRL 93 (2004) ] and MILC [PR D71 (2005) , arxiv: ] we can obtain V us from the kaon leptonic branching fraction [KLOE Collaboration, Phys. Lett. B 632, 76 (2006)], V us = (51) This is consistent with other recent determinations of V us and with unitarity constraints, given the latest V ud. William + Mary, July 15, 2008 p.20/21

21 For the future Investigate alternative fits: SU(2) χpt, SU(3) 2-loop logarithms in order to study convergence of the chiral expansion, and potentially decrease systematic errors. Move on to B K. We have B K data on the same lattices, and the analysis is in progress. William + Mary, July 15, 2008 p.21/21