Phenomenological Importance Actions and Tuning B − B mixing and fB Allocation Request Conclusion B -meson decay constants and B 0 − B 0 -mixing with domain-wall light and relativistic heavy quarks Norman Christ, Taku Izubuchi, Christoph Lehner, Amarjit Soni, Ruth S. Van de Water, Oliver Witzel (RBC Collaboration) http://rbc.phys.columbia.edu/USQCD/B-physics/ Newport News, VA, May 6, 2011
Phenomenological Importance Actions and Tuning B − B mixing and fB Allocation Request Conclusion Phenomenological Importance ◮ B − ¯ B -mixing allows us to determine CKM matrix elements ◮ Dominant contribution in SM: box diagram with top quarks � | V ∗ td V tb | for B d − mixing ∆ m q = G 2 F m 2 W η B S 0 m B q f 2 tq V tb | 2 B q B B q | V ∗ 6 π 2 | V ∗ ts V tb | for B s − mixing ¯ t ¯ ¯ b q ◮ Non-perturbative contribution: f 2 q B Bq B 0 B 0 ◮ Define the SU (3) breaking ratio W W ξ 2 = f 2 B s B B s / f 2 B d B B d b q t ◮ CKM matrix elements are extracted by ¯ b q ¯ ξ 2 | V ts | 2 ∆ m s = m B s B 0 t t B 0 ∆ m d | V td | 2 W m B d q b ◮ Experimental error of ∆ m q is better than a percent; lattice uncertainty for ξ is about 3%
Phenomenological Importance Actions and Tuning B − B mixing and fB Allocation Request Conclusion Unitarity Fit without Semileptonic Decays [Lunghi and Soni 2009] ◮ Avoids 1-2 σ tension between inclusive and exclusive deter- minations of both V ub and V cb ◮ Requires precise determination of f B (and also of B → τν and ∆ M s ) Possible Deviations from the Standard Model [Lunghi and Soni 2010] ◮ Experimental value for sin(2 β ) is 3 . 3 σ lower than SM expectation ◮ Measured value for BR( B → π l ν ) is 2 . 8 σ lower than predicted ◮ Most likely source of deviation is in B d ( s ) mixing and sin(2 β ); less likely in B → τν
Phenomenological Importance Actions and Tuning B − B mixing and fB Allocation Request Conclusion Lattice Calculations of B -meson mixing Parameters 1.13(12) RBC/UKQCD 2010 1.258(33) 190(13) 231(15) HPQCD 2009 1.205(52) 212(8) 256(8) FNAL-MILC 2008/10 ξ f B d f B s 1.0 1.1 1.2 1.3 1.4 180 195 210 225 240 255 270 ◮ HPQCD and FNAL-MILC result both based on the asqtad-improved staggered ensembles generated by MILC (FNAL-MILC uses new r 1 ) ◮ RBC/UKQCD result only exploratory study computed on 16 3 lattices and using static approximation for the b -quarks ◮ This project aims for an independent cross-check at high precision using domain-wall light-quarks and relativistic heavy quarks performing ◮ Project started 2009/10 and we ask for time to continue in 2011/12
Phenomenological Importance Actions and Tuning B − B mixing and fB Allocation Request Conclusion 2+1 Flavor Domain-Wall Gauge Field Configurations ◮ Domain-wall fermions for the light quarks (u, d, s) [Kaplan 1992, Shamir 1993] ◮ Iwasaki gauge action [Iwasaki 1983] s = 0 s = L s − 1 approx. # time L a (fm) m π (MeV) # configs. sources m l m s 24 ≈ 0.11 0.005 0.040 331 1636 1 24 ≈ 0.11 0.010 0.040 419 1419 1 24 ≈ 0.11 0.020 0.040 558 345 8 32 ≈ 0.08 0.004 0.030 307 628 2 32 ≈ 0.08 0.006 0.030 366 889 2 32 ≈ 0.08 0.008 0.030 418 544 2 [C. Allton et al. 2008, Y. Aoki et al. 2010]
Phenomenological Importance Actions and Tuning B − B mixing and fB Allocation Request Conclusion Relativistic Heavy Quark Action for the b -Quarks ◮ Relativistic Heavy Quark action developed by Christ, Li, and Lin for the b -quarks in 2-point and 3-point correlation functions ◮ Builds upon Fermilab approach [El Khadra, Kronfeld, Mackenzie] by tuning all parameters of the clover action non-perturbatively ◮ Matching of lattice action to continuum through O ( pa ) ◮ Errors will be of O ( a 2 p 2 ) ◮ Heavy quark mass is treated to all orders in m b a ⇒ coefficient of the O ( a 2 p 2 ) error is a function of m b a ◮ This function is bounded to be ≤ O (1) [El Khadra, Kronfeld, Mackenzie] ◮ Heavy-light spectrum quantities can be computed with discretization errors of the same order as in light-light quantities
Phenomenological Importance Actions and Tuning B − B mixing and fB Allocation Request Conclusion Tuning the Parameters for the RHQ Action � 2 � � a D m 0 + γ 0 D 0 − aD 2 ic P � ¯ γ · � � 0 S = Ψ n + ζ � D − − a 4 σ µν F µν Ψ n ′ 2 2 n , n ′ µν n , n ′ ◮ Start from an educated guess for m 0 a , c P , and ζ 4.4 σ ζ 4.1 σ c P m 0 a σ m 0 a 0 0 ζ ± , , c P 0 σ c P 0 3.8 σ m 0 a ζ 0 0 σ ζ 3.5 4.16 4.19 4.22 4.25 7.55 7.45 7.35 c P m 0 a 7.25
Phenomenological Importance Actions and Tuning B − B mixing and fB Allocation Request Conclusion ◮ Compute for all seven parameter sets spin-averaged mass M = ( M B s + 3 M B ∗ s ) / 4 → 5403.1(1.1) MeV hyperfine-splitting ∆ M = ( M B ∗ s − M B s ) → 49.0(1.5) MeV M 1 ratio M 2 = M rest / M kinetic → 1 ◮ Assuming linearity M m 0 a = J (3 × 3) + A (3 × 1) ∆ M Y r = c P ( r = 1 , . . . , 7) M 1 ζ M 2 r r and defining M m 0 a � Y 3 − Y 2 � 2 σ m 0 a , Y 5 − Y 4 , Y 7 − Y 6 J = A = ∆ M − J × c P 2 σ c P 2 σ ζ M 1 ζ M 2 1 1 ◮ We extract the RHQ parameters and iterate until result is inside uncertainties PDG RHQ M m 0 a = J − 1 × ∆ M c P − A M 1 ζ M 2
Phenomenological Importance Actions and Tuning B − B mixing and fB Allocation Request Conclusion Improvement of Tuning ◮ Tuning method pioneered on 24 3 (a ≈ 0.11fm) by Min Li [M. Li 2009] Further studies by Hao Peng on 32 3 (a ≈ 0.08fm) [H. Peng 2010] Exploratory studies; results not suitable for production ◮ Improvements and new setup ◮ Use of point-source strange quark operators and Gaussian-smeared heavy quarks ◮ Performed optimization study of smearing parameters ◮ Significantly increased statistics ◮ Only use of heavy-light quantities ◮ Check on linearity assumption
Phenomenological Importance Actions and Tuning B − B mixing and fB Allocation Request Conclusion Improving the Signal by Smearing of Source 3.15 3.12 Sm−Pt: r rms ≈ 0.855 fm 3.09 Sm−Pt: r rms ≈ 0.634 fm cc ≈ 0.423 fm eff Sm−Pt: r rms l m B bb ≈ 0.224 fm Sm−Pt: r rms 3.06 Pt−Pt 3.03 3.00 5 10 15 20 25 time slice ◮ Reduction of excited state contamination ◮ 818 measurements, m l sea = m l val = 0 . 005, m 0 a = 7 . 38 , c P = 3 . 89 , ζ = 4 . 19
Phenomenological Importance Actions and Tuning B − B mixing and fB Allocation Request Conclusion Tuned Parameters 24 3 m l m 0 a c P ζ sea 0.005 8.4(1) 5.7(2) 3.1(1) 0.010 8.5(1) 5.8(3) 3.1(2) Tuned Parameters 32 3 m l ζ m 0 a c P sea 0.004 4.00(8) 3.6(2) 2.0(1) 0.006 in progress 0.008 3.97(9) 3.6(2) 2.0(1)
Phenomenological Importance Actions and Tuning B − B mixing and fB Allocation Request Conclusion Predictions for the Heavy-Heavy Masses ◮ RHQ action describes heavy-light as well as heavy-heavy mesons ◮ Tuning the parameters in the B s system we can predict bottomonium states and mass splittings 9.95 m l m l exp. sea = 0.004 sea = 0.005 9.50 m l m l h b sea = 0.008 sea = 0.010 9.90 χ b1 9.45 ϒ M [GeV] M [GeV] 9.85 χ b0 9.40 η b 9.80 9.35 m l m l sea = 0.004 sea = 0.005 9.75 32 3 m l m l 9.30 exp. sea = 0.008 sea = 0.010 24 3 9.70 0.000 0.005 0.010 0.015 0.000 0.005 0.010 0.015 a 2 [fm 2 ] a 2 [fm 2 ]
Phenomenological Importance Actions and Tuning B − B mixing and fB Allocation Request Conclusion Predictions for the Heavy-Heavy Mass-Splittings 80 ∆ ( η b , ϒ ) 32 3 60 24 3 ∆ [MeV] 40 ∆ ( χ b0 , χ b1 ) 20 m l m l sea = 0.004 sea = 0.005 m l m l exp. sea = 0.008 sea = 0.010 0 0.000 0.005 0.010 0.015 a 2 [fm 2 ] ◮ Publication on tuning and bottomonium spectroscopy is in preparation
Phenomenological Importance Actions and Tuning B − B mixing and fB Allocation Request Conclusion B 0 − B 0 mixing matrix element calculation b b t Op. t 1 t 2 d d ◮ Location of four-quark operator is fixed ◮ Location of B -mesons is varied over all possible time slices ◮ Need: one point-source light quark and one point-source heavy quark originating from operator location ◮ Propagators can be used for B - and B -meson ◮ Project out zero-momentum component using a Gaussian sink
Phenomenological Importance Actions and Tuning B − B mixing and fB Allocation Request Conclusion Operator Improvement and Matching ◮ Rotate b -quark at the source to reduce discretization errors in the heavy-light current and the four-fermion operator ◮ Compute rotation parameter d 1 at tree-level in tadpole-improved lattice PT (improving operator to O ( α s ap )) ◮ Renormalization factors for matching of lattice operators to continuum operator are computed using 1-loop tadpole-improved lattice PT (truncation errors O ( α s ap )) ◮ Only one other operator at O (1 / m b ) mixes with desired operator (at this order) ◮ For ratio ξ much of the perturbative truncation error should cancel Phenomenologically most important quantity should be most reliable
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