Light pseudoscalar mesons in 2 + 1 flavor QCD Laurent Lellouch with S. Dürr, Z. Fodor, C. Hoelbling, S. Katz, S. Krieg, T. Kurth, T. Lippert, K. Szabo, G. Vulvert arXiv:0710.4769 [hep-lat], arXiv:0710.4866 [hep-lat] CPT, Marseille All results are preliminary EuroFlavor ’07 Orsay, November 14-16, 2007 Laurent Lellouch EuroFlavor ’07, Orsay, November 14-16, 2007
Motivation Goal: calculate hadronic observables on the lattice, relevant for fundamental quark property determination with controlled extrapolations to the physical limit of QCD: M π → 135 MeV , a → 0 , L → ∞ Pseudo-Goldstone boson (PGB) masses and decay constants give access to: Fundamental parameters: m ud and m s Flavor mixing parameters: π, K → µ ¯ ν allows precise determination of | V us / V ud | given a precise calculation of F K / F π ⇒ important check of | V ud | 2 + | V us | 2 + | V ub | 2 = 1 and universality qq � and F Properties of QCD vacuum: � ¯ Higher order couplings of chiral Lagrangian: ( 2 L 6 − L 4 ) , ( 2 L 8 − L 5 ) , L 4 , L 5 . . . Laurent Lellouch EuroFlavor ’07, Orsay, November 14-16, 2007
Our two approaches In both cases: N f = 2 + 1 tree-level, O ( a ) -improved Wilson seas (break SU ( 3 ) A ) 1. “Unitary” simulations: valence quarks are discretized in the same way as the sea quarks 2. “Mixed-action” simulations: valence quarks are chirally symmetric overlap (Ginsparg-Wilson) fermions Laurent Lellouch EuroFlavor ’07, Orsay, November 14-16, 2007
Why use a mixed action approach? + Recent algorithmic (multiple time-scale integration, Hasenbusch acceleration, RHMC, DDHMC . . . ) (Sexton & Weingarten ’92, Hasenbusch ’01, Clark et al ’06, Lüscher ’05, Urbach et al ’06, . . . ) and hardware advances ⇒ N f = 2 + 1 QCD with e.g. M lat π ∼ 190 MeV , a ∼ 0 . 09 fm and L ∼ 4 . 2 fm becoming accessible to Wilson fermions ⇒ near-continuum chiral p -regime w/out conceptual pbs of staggered fermions + Overlap inversions are numerically feasible on these backgrounds ⇒ full χ S (in valence sector) w/out cost of dynamical overlap fermions ⇒ simplified renormalization ⇒ full O ( a ) improvement w/ only NP O ( a ) -improved Wilson sea action + To extrapolate to physical and chiral limits in a model independent-way − → finite-volume (FV) mixed action (MA) PQ χ PT (Sharpe ’90 ’92, Bernard & Golterman ’92 ’94, Sharpe & Shoresh ’00 ’01, Sharpe & Singleton ’98, Aoki ’03, Bär et al ’03 ’04, Sharpe ’06, . . . ) − Discretization induced unitarity violations, but should be able to describe low energy manifestations with MA PQ χ PT (Golterman et al ’05) Laurent Lellouch EuroFlavor ’07, Orsay, November 14-16, 2007
Finite-volume mixed action PQ χ PT An effective theory in finite volume for the PGBs of χ SB which includes discretization errors (Sharpe & Singleton ’98) . Expansion in: ( M PGB / 4 π F π ) 2 ∼ 0 . 03 ÷ 0 . 2 ( p / 4 π F π ) 2 ∼ ( 1 / 2 LF π ) 2 ∼ 0 . 06 α s a Λ QCD ∼ 0 . 06 ← we use tree-level O ( a ) -improved Wilson seas Take here ( M PGB / 4 π F π ) 2 ∼ ( p / 4 π F π ) 2 ∼ α s a Λ QCD → p -regime and above phase transitions (Aoki or 1st order) Allow for O ( a 2 ) unitarity violations Allow sea and valence quarks to have different masses (Sharpe ’90 ’92, Bernard & Golterman ’92 ’94, Sharpe & Shoresh ’00 ’01) ⇒ in continuum (or w/ GW quarks), can consider G c ≡ [ SU ( N f + N v | N v ) L ⊗ SU ( N f + N v | N v ) R ] ⊗ U ( 1 ) L + R → SU ( N f + N v | N v ) L + R ⊗ U ( 1 ) L + R − Laurent Lellouch EuroFlavor ’07, Orsay, November 14-16, 2007
Inclusion of discretization errors at NLO (Sharpe & Singleton ’98, Aoki ’03, Bär et al ’03 ’04, Sharpe ’06, Chen et al ’07) Executive summary: construct Symanzik effective action of Wilson fermions at O ( a 2 ) (Symanzik ’75 ’83, Sheikholeslami & Wohlert ’85) for discretization operators which break G c → additional spurions ∼ a , a 2 construct χ -Lagrangian using spurions in all possible ways consistent with G c and power counting operators which preserve G c contribute to LO and NLO continuum LECs at NNLO and NNNLO and O ( 4 ) -breaking operators at NNNLO Upshot of analysis: W-on-W: → 8 + 1 coupling constants of O ( ap 2 , a 2 ) − GW-on-W: → 1 extra LEC of O ( a 2 ) − Laurent Lellouch EuroFlavor ’07, Orsay, November 14-16, 2007
Unitarity violations: the a 0 propagator (Golterman et al ’05, Chen et al ’07) Assume light sea ( ℓ ) and valence ( v ) are tuned such that M vv = M ℓℓ ≡ M π · Then, MA PQ χ PT at LO gives ( m 1 = m 2 ≡ m v ) · C a 0 ( t ) a 3 X q 2 q 1 ( � x , t )¯ q 1 q 2 ( 0 ) � � ¯ ≡ x � B 2 3 C πη ( t ) − 2 a 2 ∆ ff K ( t ) + 2 t → + ∞ C K ¯ ( M π t + 1 ) C ππ ( t ) − → L 3 M 2 π ⇒ in a 0 channel O ( a 2 ) unitarity violations are LO, only vanish in continuum limit and are exponentially and polynomially enhanced in t PQ result also has m val − m sea unitarity violations Laurent Lellouch EuroFlavor ’07, Orsay, November 14-16, 2007
Charged PGB masses at NLO in finite volume Ω 12 ) NLO n h ( M 2 ( m 1 + m 2 ) B PQ-logs ( µ, M 11 , M 22 , M ℓℓ , M ss ) 1 = 1 + Ω ( 4 π F ) 2 +( 2 α 6 − α 4 )( µ )( 2 M 2 ℓℓ + M 2 ss ) + ( 2 α 8 − α 5 )( µ ) M 2 12 io + a β M + a 2 ∆ × UV-logs ( µ, M 11 , M 22 ) + a 2 γ M ( µ ) + FV with α i ( µ ) ≡ 8 ( 4 π ) 2 L i ( µ ) Continuum or GW-on-GW m 1 , m 2 : Lagrangian masses ∆ = γ M = 0 = β M W-on-W m 1 , m 2 : NLO, AWI masses β M = O (Λ 3 QCD ) for W, O ( α s Λ 3 QCD ) for TL O ( a ) –W, 0 for NP O ( a ) –W ∆ = γ M = 0 Laurent Lellouch EuroFlavor ’07, Orsay, November 14-16, 2007
. . . and their decay constants GW-on-W m 1 , m 2 : GW Lagrangian masses β M = O (Λ 3 QCD ) for W, O ( α s Λ 3 QCD ) for TL O ( a ) –W, 0 for NP O ( a ) –W ∆ , γ M = O (Λ 4 QCD ) ⇒ MA unitarity violations for a � = 0 Charged PGB decay constants at NLO in Ω n h ( F 12 ) NLO F PQ-logs ( µ, M 11 , M 22 , M ℓℓ , M ss ) 1 = 1 + 2 ( 4 π F ) 2 Ω + α 4 ( µ )( 2 M 2 ℓℓ + M 2 ss ) + α 5 ( µ ) M 2 12 io + a β F + a 2 ∆ × UV-logs ( M 11 , M 22 ) + a 2 γ F + FV Same three cases here as for masses, but in GW-on-W case, MA unitarity violations ∝ a 2 ∆ are SU ( 3 ) val -breaking and do not depend on µ Laurent Lellouch EuroFlavor ’07, Orsay, November 14-16, 2007
Simulation ingredients Gauge action: tree-level Symanzik improved Sea quarks: smeared-link, tree-level O ( a ) -improved Wilson fermions Valence quarks: same as sea (“unitary”) or smeared-link overlap fermions (“mixed-action”) Algorithm: Rational HMC with even-odd preconditioning, multiple time-scale Omelyan integration and Hasenbusch acceleration (Clark et al ’06, Sexton & Weingarten ’92, Omelyan et al ’03, Hasenbusch ’01, Urbach et al ’06) Renormalization: non-perturbative à la Rome-Southampton Parameters: a ∼ 0 . 09 fm M lat π ∼ 190 , 300 , 410 , 490 , 570 MeV with M lat π L > ∼ 4 Overlap roughly matched with Wilson m lat s such that M lat K ≃ 1 . 07 M K and 2 valence m lat s at 190 , 300 MeV 34 configs at 190 MeV , 68 at 300 MeV and O ( 100 ) at other points Calculations performed on BG/L ’s at FZ Jülich and on clusters at the University of Wuppertal and CPT Marseille Laurent Lellouch EuroFlavor ’07, Orsay, November 14-16, 2007
No metastabilities and stable algorithm e.g. a ∼ 0 . 15 fm , Ω / a 4 = 16 3 × 32 and M lat π ≃ 300 MeV (difficult simulation according √ q λ min ) 2 � ≃ a / � ( λ min − ¯ to Ω criterion (Del Debbio et al ’05) ) Laurent Lellouch EuroFlavor ’07, Orsay, November 14-16, 2007
Unitarity violations in the a 0 propagator (preliminary) 1 parameter ( a 4 ∆ ) fit of scalar-isovector propagators to chiral expression for C a 0 ( t ) at M lat π ∼ 190 MeV and 300 MeV 0.004 GW on W M π lat ~ 190 MeV 0.002 M π lat ~ 300 MeV bare (t) 3 C a 0 0 a -0.002 5 10 15 20 25 30 t/a Find a 4 ∆ = 0 . 015 ( 6 ) and 0 . 024 ( 10 ) , i.e. compatible √ ⇒ a ∆ ∼ 0 . 27 GeV and 0 . 35 GeV , which compete with meson masses in chiral expressions Laurent Lellouch EuroFlavor ’07, Orsay, November 14-16, 2007
Preliminary fit to the PGB decay constants aF 12 obtained using AWI → no renormalization needed thanks to valence χ S Fit 8 points with M lat π ≤ 500 MeV and M lat K ≤ 590 MeV to NLO expression with FV corrections and unitarity violations constrained with a 0 prior, a 4 ∆ = 0 . 024 ( 10 ) GW on W 0.06 0.05 aF 12 Lattice data (a~0.09 fm) Finite-V, a>0 fits Physical pion curve 0.04 Physical kaon curve SU(3) curve 0.03 0 0.02 0.04 0.06 0.08 2 (aM ll ) Good χ 2 / dof and find a 4 ∆ = 0 . 025 ( 8 ) Get a from self-consistent extrapolation to physical point Laurent Lellouch EuroFlavor ’07, Orsay, November 14-16, 2007
Preliminary fit to the W-on-W PGB masses Unitary theory Fit 6 points with M lat π ≤ 500 MeV and M lat K ≤ 590 MeV are fitted to NLO expression with FV corrections for aB bare 12 ≡ ( aM 12 ) 2 / ( am 1 + am 2 ) bare AWI 1.3 W on W 1.2 1.1 bare aB 12 a~0.09 fm 1 Finite-V, a>0 fits Physical pion curve Physical kaon curve 0.9 SU(3) curve 0.8 0 0.02 0.04 0.06 0.08 2 (aM ll ) Good χ 2 / dof Laurent Lellouch EuroFlavor ’07, Orsay, November 14-16, 2007
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