The Lambda parameter and strange quark mass in two-flavor QCD

Transcription

1 The Lambda parameter and strange quark mass in two-flavor QCD Patrick Fritzsch Institut für Physik, Humboldt-Universität zu Berlin, Germany talk based on [arxiv: ] in collaboration with F.Knechtli, B.Leder, M.Marinkovic, S.Schaefer, R.Sommer, F.Virotta LPHA ACollaboration lavi net The 7th International Workshop on Chiral Dynamics August 6-10, 2012, Jefferson Lab, Newport News, VA P. Fritzsch Chiral Dynamics

2 Introductory remarks Toolbox: lattice regularisation of QCD, including N f = 2 dynamical quarks 1 Compute fundamental parameters of QCD, i.e., renormalization group invariant (RGI) quantities Λ (N f ) QCD ; M i, i = u, d, s,... non-perturbatively Λ (2) MS, m s(2gev) 2 Good control over systematic errors, like autocorrelations; finite volume, discretisation,..., chiral extrapol. 3 Scale setting through physical quantities as f π, f K, m Ω,... (Coordinated Lattice Simulations, CLS, effort) to convert dim.less numbers (a f π, a f K,... ) a, L,... in physical units 4 Compute further quantities of interest decay constants, masses, form factors,..., ALPHAs B-physics program P. Fritzsch Chiral Dynamics

3 GENERAL SETUP P. Fritzsch Chiral Dynamics

4 Non-perturbative Renormalization Ingredient: Schrödinger functional as intermediate renormalization scheme massless, finite volume renorm. scheme in the continuum IR regulator on the lattice (Dirichlet b.c. in time) m = 0 on the lattice NP definition of a running coupling g 2 (µ), w/ box size L = 1/µ N f = 2: QCD running coupling [ALPHA 04] and mass [ALPHA 05] known through finite size scaling technique NP running coupling: NP running mass: Λ-parameter, low energy scale µ 1/L 1 RGI quark masses M i P. Fritzsch Chiral Dynamics

5 Dynamical fermion simulations e.g.: O7 ensemble, , m π 270 MeV Lattice framework: plaquette gauge action mass-degenerate doublet of non-perturbatively improved Wilson fermions Our criteria: FV effects small by construction Lm π data for chiral extrapolation uses 400 m π 500 MeV 300 three lattice spacings 200 < 0.08 fm 100 Goal: controlled extrapol. to physical point a/fm mπ/mev P. Fritzsch Chiral Dynamics

6 TWO STRATEGIES FOR CHIRAL EXTRAPOLATIONS P. Fritzsch Chiral Dynamics

7 Setting the scale Standard procedure, still room for improvements though calibrate lattice spacing a through dimensionful reference quantity Q: a 1 [MeV] = Q exp [MeV] [ ] aq latt choose well behaved quantity Q according to, Q { f π, f K, m N, m π,...} experimentally available input reasonable signal-to-noise ratio well-controlled and understood chiral behaviour mild cut-off effects... Our choice: kaon decay constant f K milder chiral extrapolation compared to f π better control over systematic errors (2 strategies) P. Fritzsch Chiral Dynamics

8 Two chiral extrapolations strategy 2 strategy 1 P. Fritzsch Chiral Dynamics

9 Strategy 1 fix ratio: m 2 ( K κ1, h(κ 1 ) ) fk 2 ( κ1, h(κ 1 ) ) = R! K = m2 K,phys fk,phys 2 trajectory with m K m K,phys and thus m s + m light const + O(m 2 ) κ 3 h(κ 1 ) determined by interpolation PQ-SU(3) ChPT: f K ( κ1, h(κ 1 ) ) f K,phys systematic expansion in [Sharpe 97] strategy 1 m 2 π, m 2 K m2 K,phys P. Fritzsch Chiral Dynamics

10 Strategy 2 fix strange quark s PCAC mass: am 34 ( κ1, s(κ 1, µ) )! = µ µ fixed for 2 values (independent of κ 1 ) κ 3 s(κ 1, µ) determined by interpolation SU(2) ChPT: [Roessl:1999; AlltonEtAl:2008] ( a f K κ1, s(κ 1, µ) ) p(µ) [am K ] 2( κ 1, s(κ 1, µ) ) q(µ) strategy 2 solve numerically for µ s : q(µ) p(µ) 2 m2 K,phys µ=µs fk,phys 2 P. Fritzsch Chiral Dynamics

11 Results for chiral extrapolations β = 5.2 β = 5.3 β = 5.5 global fit expansion variable: afk m y 1 2 π(κ 1 ) 8π 2 fk 2 ( ) 0.1 κ1, κ y π y 1 ChPT to 1-loop, i.e., LEC or combinations thereof multiply (y 1 y π ) O(y 2 ) small f K,phys = 155 MeV [FLAG 11] (isospin symmetric limit & QED effects removed) Strategy 1: NLO partially quenched SU(3) ChPT κ 3 = h(κ 1 ) a = [a f K ] / f K,phys Strategy 2: NLO SU(2) ChPT κ 3 = s(κ 1, µ s ) a = p(µ s ) / f K,phys P. Fritzsch Chiral Dynamics

12 Results for chiral extrapolations Systematics at finest lattice spacing afk yπ y1 cut at y 1 = 0.1 y 1 m2 π(κ 1 ) 8π 2 f 2 K (κ 1) global fit: α 4, α f independent of β check systematic from chiral extrapolation: chiral vs. linear fit (see above) Strategy 1 - Strategy 2 LEC s: α 4 = 0.57(12), α f = 1.13(8) taking all systematics into account: β a f K a[fm] (7)(6) (9)(7) (6)(6) (7)(7) (4)(3) (4)(5) P. Fritzsch Chiral Dynamics

13 RESULTS Λ, M s P. Fritzsch Chiral Dynamics

14 The Λ parameter of N f = 2 QCD Master formula: Λ (2) 1 MS = [ ] f K fk L 1 cont [ALPHA 05] [ L 1 Λ (2) SF ]cont Λ(2) MS Λ (2) SF Renorm. scale set through SF coupling: g 2 (L 1 ) = 4.484, µ 1 = 1/L 1 Exact relation between schemes: / Λ (2) SF = (3) Λ (2) MS Non-perturbative running coupling: L 1 Λ (2) SF = 0.264(15) Missing piece: [ fk L 1 ] cont L1fK strategy 1 strategy (afk) 2 x 10 3 P. Fritzsch Chiral Dynamics

15 The Λ parameter of N f = 2 QCD Master formula: Λ (2) 1 MS = [ ] f K fk L 1 cont using result from strategy 1 [ALPHA 05] [ L 1 Λ (2) SF ]cont Λ(2) MS Λ (2) SF L 1 f K = 0.315(8)(2) Λ (2) SF / f K = 0.84(6) Renorm. scale set through SF coupling: g 2 (L 1 ) = 4.484, µ 1 = 1/L 1 Exact relation between schemes: / Λ (2) SF = (3) Λ (2) MS Non-perturbative running coupling: L 1 Λ (2) SF = 0.264(15) Missing piece: [ fk L 1 ] L1fK cont Λ (2) = 310(20) MeV MS strategy 1 strategy (afk) 2 x 10 3 P. Fritzsch Chiral Dynamics

16 The strange quark mass in N f = 2 QCD RGI strange quark mass: M s = Z M m s = M m(µ 1 ) m s(µ 1 ) = M m(µ 1 ) Z A Z P (µ 1 ) m s Renorm. scale set through SF coupling: g 2 (L 1 ) = 4.484, µ 1 = 1/L 1 Non-perturbative running mass: [ALPHA 05] M / m(µ 1 ) = 1.308(16) P. Fritzsch Chiral Dynamics

17 The strange quark mass in N f = 2 QCD RGI strange quark mass: M s = Z M m s = M m(µ 1 ) m s(µ 1 ) = M m(µ 1 ) Z A Z P (µ 1 ) m s Renorm. scale set through SF coupling: g 2 (L 1 ) = 4.484, µ 1 = 1/L 1 Non-perturbative running mass: [ALPHA 05] M / m(µ 1 ) = 1.308(16) Strategy 1: only combination m s + m directly accessible remove average light quark mass m additional systematic uncertainty combination of LEC α 4 & α 6 from constrained global fit Strategy 2: (conceptually preferred) no additional LEC s involved m s / f K = 0.678(12)(5) M s = 138(3)(1) MeV, ( ) m MS s µ =2 GeV = 102(3)(1) MeV P. Fritzsch Chiral Dynamics

18 Summary complete analysis of CLS based N f = 2 ensembles; a = ( )fm Conservative error estimates through autocorrelation analysis Scale setting with f K Two strategies for chiral extrapolation in agreement simple linear extrapolation also agrees within errors small cut-off effects control over systematic effects in scale setting Main results Λ (2) = 310(20) MeV, M MS m MS s s = 138(3)(1) MeV ( ) µ =2 GeV = 102(3)(1) MeV controlled reduction of statistical & systematic error achieved (since 2005) consistent with scale setting through r 0 (not coverred) P. Fritzsch Chiral Dynamics

19 Outlook unclear situation N f Λ MS experiment theory 0 238(19) MeV m K, K µν µ, K πµν µ lattice gauge theory [ALPHA 93] 2 310(20) MeV m K, K µν µ, K πµν µ this work 5 212(12) MeV world average perturb. theory [Bethke 11] remove remaining systematic uncertainty due to quenching N f = 3, 4? TODO? other low energy constants m / m s... Thank you for your attention!... and many thanks to my colleagues F.Knechtli, B.Leder, M.Marinkovic, S.Schaefer, R.Sommer, F.Virotta P. Fritzsch Chiral Dynamics

20 BACKUP SLIDES P. Fritzsch Chiral Dynamics

21 Autocorrelations in HMC generated data... grow towards the continuum limit autocorrel. depends on observable F = 1 N N i=1 F i autocorrelation function Γ F (t) = (F t F)(F 0 F) integrated autocorrelation time τ int = 1 2 t=1 ρ F(t), ρ F (t) = Γ F (t) / Γ F (0) ρfk (t) W u W l t τ exp (β) = 200 c τ R act e 7[β 5.3] τ exp estimated from β = 5.3 and quenched scaling [Schaefer,Sommer,Virotta 10] R act fraction of active links (DD-HMC: 30%; MP-HMC: 100%) c τ=2 = 1; c τ=0.5 = 2 Exponential tail accounts for 50% of the total uncertainty! P. Fritzsch Chiral Dynamics

22 Masses and decay constants A handle on excited states ( lattice, a = 0.045fm, m π = 268MeV ) x ameff x min x0 Taking τ exp into account: ampcac t 1 st step: double exponential fit to {c 1, c 2, m} with m = m + 2m π ( ) ( ) F(x 0 ) = c 1 e mx 0 + e m(t x 0) + c 2 e m x 0 + e m (T x 0 ) + criterion: stat.error(am eff ) 4 1 st excited state contribution x min 0 2 nd step: single exponential fit (c 2 0) c 2 0 well justified theoretically; see talk by M.Golterman P. Fritzsch Chiral Dynamics

23 Results for chiral extrapolations Strategy 1 afk β = 5.2 β = 5.3 β = 5.5 strat.1: strat.2: cut at y 1 = 0.1 m y 1 2 π(κ 1 ) 8π 2 fk 2 ( κ1, h(κ 1 ) ) yπ y1 constrained global fit: α 4 independent of a partially quenched ChPT: ( f K κ1, h(κ 1 ) ) ] = f K,phys [1 + L K (y 1, y K ) + (α ) (y 1 y π ) + O(y 2 ), L K (y 1, y K ) = L K (y 1, y K ) L K (y π, y K ), L K (y 1, y K ) = 2 1 y 1 log(y 1 ) 1 8 y 1 log(2y K /y 1 1). y 3 y 1 (π K) does not appear, since y 3 = 2y K y 1 + O(y 2 ) combination of chiral logs overall much smaller with less curvature compared to L π P. Fritzsch Chiral Dynamics

24 Results for chiral extrapolations Strategy 2 afk β = 5.2 β = 5.3 β = 5.5 strat.1: strat.2: cut at y 1 = 0.1 m y 1 2 π(κ 1 ) 8π 2 fk 2 ( κ1, s(κ 1, µ) ) yπ y1 constrained global fit: α f, α m independent of a SU(2) ChPT: ( a f K κ1, s(κ 1, µ) ) ] = p(µ) [1 3 8 [y 1 log(y 1 ) y π log(y π )] + α f (µ) (y 1 y π ) + O(y 2 1 ), a 2 m 2 K( κ1, s(κ 1, µ) ) ] = q(µ) [1 + α m (µ) (y 1 y π ) + O(y 2 1 ) for three different fixed µ q(µ s ) p(µ s ) 2 = m2 K,phys f 2 K,phys a = p(µ s) f K,phys P. Fritzsch Chiral Dynamics

25 The strange quark mass in N f = 2 QCD RGI strange quark mass: M s = Z M m s = M m(µ 1 ) m s(µ 1 ) = M m(µ 1 ) Z A Z P (µ 1 ) m s Renorm. scale set through SF coupling: g 2 (L 1 ) = 4.484, µ 1 = 1/L 1 Non-perturbative running mass: [ALPHA 05] M / m(µ 1 ) = 1.308(16) Strategy 1: ( 2m 13 κ1, h(κ 1 ) ) ( Z P f K κ1, h(κ 1 ) ) = m s + m [ ] 1 + L m (y f 1, y K ) + (α 4,6 4 1 )(y 1 y π ) + O(y 2 1 ) K,phys Strategy 2: L m (y 1, y K ) = L m (y 1, y K ) L m (y π, y K ), α 4,6 = 3α 4 4α 6, L m (y 1, y K ) = (y K 3 8 y 1) log(2y K /y 1 1) y K log(y 1 ) M s f K,phys = M m(l 1 ) 1 (conceptually preferred) Z P (L 1 ) [1 b µ s P µ s ] FK,phys bare, P. Fritzsch Chiral Dynamics

26 The strange quark mass in N f = 2 QCD RGI strange quark mass: M s = Z M m s = M m(µ 1 ) m s(µ 1 ) = M m(µ 1 ) Z A Z P (µ 1 ) m s m s / f K = 0.678(12)(5) Strategy 1: use y 1 = m 2 π(κ 1 ) / 8π 2 fk 2 (κ 1) < 0.1 Renorm. scale set through SF coupling: remove avg. light quark contribution m by correction g 2 (L 1 ) = factor 4.484, ρ: M s = (M s + µ 1 M)(1 = 1/L ρ) 1 Non-perturbative LOχPT : running ρ = M /( mass: M s + M ) [ALPHA 05] m2 π M / LECm(µ α 4,6 from 1 ) = constrained 1.308(16) global fit Strategy 2: 2m 2 K neglect tiny O(a) impr. term, ( b A b P )am sea Strategy 1: ( 2m 13 κ1, h(κ 1 ) ) ( Z P f K κ1, h(κ 1 ) ) = m s + m [ ] 1 + L m (y f 1, y K ) + (α 4,6 4 1 )(y 1 y π ) + O(y 2 1 ) K,phys Strategy 2: L m (y 1, y K ) = L m (y 1, y K ) L m (y π, y( K ), ) α 4,6 = 3α 4 4α 6, M s = 138(3)(1) MeV L m (y 1, y K ) = (y K 8 3 y m MS s µ = 2GeV = 102(3)(1) MeV 1) log(2y K /y 1 1) y K log(y 1 ) M s f K,phys = M m(l 1 ) 1 (conceptually preferred) Z P (L 1 ) [1 b µ s P µ s ] FK,phys bare, P. Fritzsch Chiral Dynamics

27 Generic strategy... to connect low- & high-energy regime NP ly one more important ingredient: How to connect hadronic observables from low-energies to the widely used MS-scheme? µ lattice PT (MS) 100 GeV m Z 10 GeV m τ m D 1 GeV m p 100 MeV m π 1/L µ, E, p 1/a µ pert Λ P. Fritzsch Chiral Dynamics

28 Generic strategy... to connect low- & high-energy regime NP ly one more important ingredient: How to connect hadronic observables from low-energies to the widely used MS-scheme? µ lattice intermediate (SF) µ PT (MS) 100 GeV m Z intermediate renorm. scheme Schrödinger functional (finite volume, continuum scheme) 10 GeV m τ m D RG scale evolution solved NP ly 1 GeV m p 1/L max thus continuum limit needs to be well controled (small cutoff effects) 100 MeV m π low-energy scale fixed by imposing g 2 (L max ) u max 1/L µ, E, p 1/a µ pert Λ P. Fritzsch Chiral Dynamics

29 Scale dependence of QCD parameters Running coupling and mass, g 2 (µ) g 2 = 5.5 g 2 = 4.61 Renormalization group (RG) equations 1 coupling µ g g 0 = β(g) g 3 (b 0 + b µ 1 g ) 2 m(µ) M mass µ m g 0 = τ(g) g 2 (d 0 + d m µ 1 g ) µ/λ in a massless scheme, b 0, b 1, d 0 universal Solution leads to exact equations in mass-independent scheme P. Fritzsch Chiral Dynamics

30 Scale dependence of QCD parameters Running coupling and mass, Renormalization Group Invariants (RGI) g 2 (µ) m(µ) M g 2 = 5.5 g 2 = µ/λ Renormalization group (RG) equations 1 coupling µ g µ Λ µ [ b 0 g 2] b 1 /(2b 2 0 ) e 1/(2b 0g 2) exp 2 mass µ m m µ M m(µ)[2b 0 g 2 ] d 0/(2b 0 ) exp = β(g) g 0 g 3 (b 0 + b 1 g ) { g [ 1 dg 0 β(g) + 1 b 0 g 3 b ]} 1 b0 2g = τ(g) g 0 g 2 (d 0 + d 1 g ) { g [ τ(g) dg 0 β(g) d ]} 0 b 0 g in a massless scheme, b 0, b 1, d 0 universal Solution leads to exact equations in mass-independent scheme P. Fritzsch Chiral Dynamics