Progress on B-physics lattice calculations by ETMC Petros - PowerPoint PPT Presentation

Progress on B-physics lattice calculations by ETMC Petros Dimopoulos University of Rome Tor Vergata on behalf of ETM Collaboration September 19, 2012 “New Frontiers in Lattice Gauge Theory” Galileo Galilei Institute for Theoretical Physics

Outline • ETMC computation • Method based on Ratios of heavy-light ( h , ℓ/ s ) observables using relativistic quarks and exact knowledge of static limit for the appropriate ratios • Interpolation of ( h , ℓ/ s ) observables to the b-region from the charm region and the static limit • Application of Ratio method to b -quark mass, decay constants and Bag parameters. • Summary ETMC , JHEP 1201 (2012) 046; JHEP 1004 (2010) 049; N. Carrasco and A. Shindler @ LAT2012 ETMC-b : B. Blossier, N. Carrasco, P. Dimopoulos, R. Frezzotti, V. Gimenez, G. Herdoiza, K. Jansen, V. Lubicz, G. Martinelli, C. Michael, D. Palao, G. C. Rossi, F. Sanfilippo, A. Shindler, S. Simula, C. Tarantino, M. Wagner

ETMC – N f = 2 twisted-mass formulation • Mtm lattice regularization of N f = 2 QCD action is [Frezzotti, Grassi, Sint, Weisz, JHEP 2001; Frezzotti, Rossi, JHEP 2004] � ∇ − i γ 5 τ 3 � � � + a 4 � − a S ph ¯ γ · � N f =2 = S YM ψ ( x ) 2 r ∇ ∗ ∇ + M cr ( r ) + µ q ψ ( x ) L x • ψ is a flavour doublet, M cr ( r ) is the critical mass and τ 3 acts on flavour indices • From the “physical” basis (where the quark mass is real), the non-anomalous ψ = exp( i πγ 5 τ 3 / 4) χ , ¯ χ exp( i πγ 5 τ 3 / 4) ψ = ¯ transformation brings the lattice action in the so-called “twisted” basis � 2 r ∇ ∗ ∇ + M cr ( r ) + i µ q γ 5 τ 3 � + a 4 � ∇ − a S tw N f =2 = S YM γ · � χ ( x ) ¯ χ ( x ) L x • Unlike the standard Wilson regularization, in Mtm Wilson case the subtracted Wilson operator − a 2 r ∇ ∗ ∇ + M cr ( r ) is “chirally rotated” w.r.t. the quark mass = ⇒ offers important advantages...

ETMC – N f = 2 twisted-mass formulation • Automatic O( a ) improvement for the physical quantities • Dirac-Wilson matrix determinant is positive and (lowest eigenvalue) 2 bounded from below by µ q 2 • Simplified (operator) renormalization ... • Multiplicative quark mass renormalization • No RC for pseudoscalar decay constant (PCAC) • O( a 2 ) breaking of parity and isospin Frezzotti, Rossi, JHEP 2004; ETMC , Phys.Lett.B 2007

ETMC N f = 2 simulations 0.10 1 / 1 . 25 a [fm] 1 /L [fm − 1 ] 0.05 1 / 2 . 5 N f = 2 exp t 1 / 5 0.00 0 100 200 300 400 500 600 100 200 300 400 500 600 m PS [MeV] m PS [MeV] • a = { 0 . 054 , 0 . 067 , 0 . 085 , 0 . 098 } fm • m ps ∈ { 270 , 600 } MeV • L ∈ { 1 . 7 , 2 . 8 } fm , m ps L ≥ 3 . 5

ETMC N f = 2 simulations β a µ ℓ a µ s (valence) a µ h (valence) 3.80 0.0080, 0.0110 0.0175, 0.0194, 0.0213 0.1982, . . . , 0.8536 3.90 0.0030, 0.0040, 0.0159, 0.0177, 0.0195 0.1828, . . . , 0.7873 0.0064, 0.0085, 0.0100 4.05 0.0030, 0.0060, 0.0080 0.0139, 0.0154, 0.0169 0.1572, . . . , 0.6771 4.20 0.0020, 0.0065 0.0116, 0.0129, 0.0142 0.13315, . . . , 0.4876 • µ ℓ ∈ [ ∼ m s / 6 , ∼ m s / 2] • µ h ∈ [ ∼ m c , ∼ 3 m c ]

M ( hl ) eff plateau quality β = 3 . 90 ( a = 0 . 085 fm); µ h ∼ 2 . 6 GeV β = 3 . 80 ( a = 0 . 098 fm); µ h ∼ 2 . 6 GeV 2.20 LL LL 1.70 SL SL 2.00 SS SS WL WL 1.60 1.80 aM eff ( x 0 ) aM eff ( x 0 ) 1.50 1.60 1.40 1.40 1.30 1.20 5 10 15 20 5 10 15 20 x 0 /a x 0 /a • Smearing techniques improve signal; reduce the coupling between the ground and excired states; safe good plateaux at earlier times; absolutely necessary for obtaining safe plateau in the calculation of 3-point correlation functions (when large heavy quark mass ( > 1 GeV) are employed). • Employ ”optimal” source: Φ source ( w ) = w Φ S + (1 − w )Φ L ; W Check vs. GEVP – when two states matter – seems OK (in progress).

Ratio method • we use correlators with relativistic quarks • c -mass region computations are reliable (’small’ discr. errors) • construct HQET-inspired ratios of the observable of interest at successive (nearby) values of the heavy quark mass ( µ ( n ) = λµ ( n − 1) ) h h • ratios show smooth chiral and continuum limit behaviour • ratios at the ∞ -mass (static) point are exactly known (= 1) • physical values of the observable at the b -mass point is related to its c -like value by a chain of the ratios ending up at the static point: use HQET-inspired interpolation

b-quark mass computation - 1 � � M h ℓ • observing that lim = constant (HQET) µ pole µ pole →∞ h h µ ( n ) ¯ • construct (taking h = λ ): µ ( n − 1) ¯ h µ ( n ) µ ( n − 1) µ ( n − 1) M h ℓ (¯ h ; ¯ µ ℓ , a ) · ¯ · ρ (¯ , µ ∗ ) µ ( n ) h h y (¯ h , λ ; ¯ µ ℓ , a ) ≡ = µ ( n − 1) µ ( n ) µ ( n ) M h ℓ (¯ ; ¯ µ ℓ , a ) ¯ ρ (¯ h , µ ∗ ) h h µ ( n ) µ ( n ) λ − 1 M h ℓ (¯ h ; ¯ µ ℓ , a ) · ρ (¯ h /λ, µ ∗ ) = , n = 2 , . . . , N µ ( n ) µ ( n ) M h ℓ (¯ h /λ ; ¯ µ ℓ , a ) ρ (¯ h , µ ∗ ) µ pole = ρ (¯ µ h , µ ∗ ) ¯ µ h ( µ ∗ ) (with ¯ µ h ← MS scheme ) h µ h , µ ∗ ) known up to N 3 LO ρ (¯ – relevant only for the ’1 / ¯ µ h ’ interpolation → In the static limit (and in CL) obviously: µ h →∞ y (¯ lim µ h , λ ; ¯ µ ℓ , a = 0) = 1 ¯

b-quark mass computation - 2 µ (1) µ (1) • Triggering input mass: M h ℓ (¯ h ) PS meson mass (at ¯ ∼ m c ) affected by h (tolerably) small cutoff effects. µ ( n ) • Ratios y (¯ h , λ ; ¯ µ ℓ , a ) have small discretisation errors 1.14 2.4 β = 3 . 80 β = 3 . 80 2.3 β = 3 . 90 β = 3 . 90 1.13 2.2 β = 4 . 05 β = 4 . 05 h ) β = 4 . 20 β = 4 . 20 µ (3) 2.1 h ) (GeV) h ) /M hℓ (¯ 1.12 CL - phys. point CL - phys. point 2.0 1.9 µ (1) 1.11 µ (4) M hℓ (¯ 1.8 M hℓ (¯ 1.7 1.10 1.6 1.5 1.09 1.4 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 ¯ µ ℓ (GeV) µ ℓ (GeV) ¯

b-quark mass computation - 3 1.01 1.00 0.99 µ h ) y (¯ 0.98 0.97 µ − 1 ¯ b 0.96 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 µ h (GeV − 1 ) 1 / ¯ µ h ) = 1 + η 1 µ h + η 2 • fit ansatz y (¯ (inspired by HQET) ¯ µ 2 ¯ h µ ( K +1) ) ≡ M expt • Determine K (integer) such that M hu / d (¯ : h B µ ( K +1) � ρ (¯ µ (1) ) = λ − K M hu / d (¯ ) h , µ ∗ ) µ (2) µ (3) µ ( K +1) � h y (¯ h ) y (¯ h ) . . . y (¯ · h µ (1) µ ( K +1) M hu / d (¯ h ) ρ (¯ , µ ∗ ) h (strong cancellations of perturbative factors in the ratios) µ (1) • One adjusts ( λ, ¯ h ) such that K integer µ (1) • our calculation: λ = 1 . 1784 and ¯ = 1 . 14 GeV (in MS , 2 GeV) h µ b = λ K ¯ µ (1) → ¯ ( K = 9) h • y deviates from its static value ∼ 1% for ¯ µ h ≤ m b . 2 contribution to ratios y . • curvature denotes a large 1 / ¯ µ h

b-quark mass computation - 4 1.01 1.01 1.00 1.00 0.99 0.99 µ h ) µ h ) y s (¯ y (¯ 0.98 0.98 0.97 0.97 µ − 1 µ − 1 ¯ ¯ b 0.96 b 0.96 ✏✏✏✏ ✶ 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 µ h (GeV − 1 ) µ h (GeV − 1 ) 1 / ¯ 1 / ¯ µ h ) = 1 + η 1 µ h + η 2 • y (¯ (similar results if M hs data is used as input) ¯ µ 2 ¯ h µ ( K +1) ) ≡ M expt • Determine K (integer) such that M hu / d (¯ : h B µ ( K +1) � ρ (¯ µ (1) ) = λ − K M hu / d (¯ ) h , µ ∗ ) µ (2) µ (3) µ ( K +1) � h y (¯ h ) y (¯ h ) . . . y (¯ · h µ (1) µ ( K +1) M hu / d (¯ h ) ρ (¯ , µ ∗ ) h (strong cancellations of perturbative factors in the ratios) • One adjusts ( λ, ¯ µ (1) h ) such that K integer µ (1) • our calculation: λ = 1 . 1784 and ¯ = 1 . 14 GeV (in MS , 2 GeV) h µ b = λ K ¯ µ (1) → ¯ ( K = 9) h • y deviates from its static value ∼ 1% for ¯ µ h ≤ m b . 2 contribution to ratios y . • curvature denotes a large 1 / ¯ µ h

b-quark mass computation - 5 1.01 1.01 ETMC 2012 ETMC 2011 1.00 1.00 0.99 0.99 µ h ) µ h ) y (¯ y (¯ 0.98 0.98 0.97 ✟✟✟✟✟✟✟ 0.97 ✯ µ − 1 µ − 1 ¯ ¯ b 0.96 b 0.96 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 µ h (GeV − 1 ) µ h (GeV − 1 ) 1 / ¯ 1 / ¯ µ h ) = 1 + η 1 µ h + η 2 • y (¯ (Comparison with less precise data from ETMC-2011 paper) ¯ µ 2 ¯ h µ ( K +1) ) ≡ M expt • Determine K (integer) such that M hu / d (¯ : h B µ ( K +1) � ρ (¯ µ (1) ) = λ − K M hu / d (¯ ) h , µ ∗ ) µ (2) µ (3) µ ( K +1) � h y (¯ h ) y (¯ h ) . . . y (¯ · h µ (1) µ ( K +1) M hu / d (¯ h ) ρ (¯ , µ ∗ ) h (strong cancellations of perturbative factors in the ratios) • One adjusts ( λ, ¯ µ (1) h ) such that K integer µ (1) • our calculation: λ = 1 . 1784 and ¯ = 1 . 14 GeV (in MS , 2 GeV) h µ b = λ K ¯ µ (1) → ¯ ( K = 9) h • y deviates from its static value ∼ 1% for ¯ µ h ≤ m b . 2 contribution to ratios y . • curvature denotes a large 1 / ¯ µ h

b-quark mass - results • m b ( m b , MS ) | N f =2 = 4 . 35(12) GeV ( PRELIMINARY! ) • compatible result for m b when ( hs )-data and M expt as input are used Bs PDG 2012 Chetyrkin et al. 2009 • Main source of uncertainty of the HPQCD 2010 (Nf=2+1) ETMC result is due to quark mass ALPHA 2011 (Nf=2) RC and scale setting uncertainties; ETMC 2011 (Nf=2) stat & fit errors very small. ETMC 2012 - Prel. (Nf=2) 4 4.2 4.4 4.6 m b ( m b ) [GeV]