Sophie Hollitt Ross Young, James Zanotti, and QCDSF LATTICE2018, Wednesday 25 th July
Why B decay constants? 2007 New experiments and the CKM matrix: Need to reduce error in theoretical calculations to reduce error on CKM matrix elements ahead of new experimental results from Belle II Decay constant f B could be used alongside measurement of B →τν to pinpoint |V ub | 2014 f B , f Bs also important to |V td |,|V ts | through B 0 B 0 oscillations We want to learn more about the way SU(3) breaking in the lightest quarks affects heavy B mesons Need a strategy for studying SU(3) breaking effects in u,d,s quarks on the lattice 2 Sophie Hollitt SU(3) breaking of B meson decay constant
Choosing light and strange quarks We choose to study SU(3) breaking in a controlled way, by keeping the average mass of these three lightest quarks constant. Lattice configurations for this method are produced by the QCDSF Collaboration. These configurations are simplified with m u = m d , (called m light ) Choose constant average mass m = ( 2 m l + m s ) ⅓ matching the physical average mass Produces controlled breaking of SU(3) symmetry Flavour singlet quantities remain approx. constant ( O ( δ m) removed) 3 Sophie Hollitt SU(3) breaking of B meson decay constant
Choosing light and strange quarks We choose to study SU(3) breaking in a controlled way, by keeping the average mass of these three lightest quarks constant. Lattice configurations for this method are produced by the QCDSF Collaboration. These configurations are simplified with m u = m d , (called m light ) Choose constant average mass m = ( 2 m l + m s ) ⅓ Light flavour singlets on QCDSF configurations, including: matching the physical average mass Produces controlled breaking of SU(3) symmetry Flavour singlet quantities remain approx. constant ( O ( δ m) removed) 4 Sophie Hollitt SU(3) breaking of B meson decay constant
Choosing light and strange quarks We choose to study SU(3) breaking in a controlled way, by keeping the average mass of these three lightest quarks constant. Lattice configurations for this method are produced by the QCDSF Collaboration. These configurations are simplified with m u = m d , called m light m s = constant The kaon is light + strange, so its mass still changes when m s is m K 2 constant SU(3) breaking effects and effects from simulating a heavier vacuum occur together m = ( 2 m l + m s ) ⅓ m π 2 The average quark mass in the 0 0.5 1 vacuum is constant 2 / X π Breaking ratio m π 2 5 Sophie Hollitt SU(3) breaking of B meson decay constant
Choosing light and strange quarks We choose to study SU(3) breaking in a controlled way, by keeping the average mass of these three lightest quarks constant. Lattice configurations for this method are produced by the QCDSF Collaboration. These configurations are simplified with m u = m d , called m light m s = constant The kaon is light + strange, so its mass still changes when m s is m Bs 2 constant SU(3) breaking effects and effects from simulating a heavier vacuum occur together m BX 2 m = ( 2 m l + m s ) ⅓ m B 2 The average quark mass in the 0 0.5 1 vacuum is constant 2 / X π Breaking ratio m π 2 6 Sophie Hollitt SU(3) breaking of B meson decay constant
Generating b -quarks b - quarks are heavy and “fall through” the lattice if a standard quark action is used. We use an anisotropic, clover-improved action (Relativistic Heavy Quark Action), and then tune the free parameters to physical quantities for the B meson. 1 bare mass anisotropy clover coefficient spin-averaged hyperfine splitting dispersion relation meson mass between B* and B Aoki, Y et al (2012). “ Nonperturbative tuning of an improved relativistic heavy- quark action with application to bottom spectroscopy.” 1 Physical Review D , 86 (11), 116003. doi:10.1103/PhysRevD.86.116003 7 Sophie Hollitt SU(3) breaking of B meson decay constant
Generating b -quarks METHOD : On every set of configurations, generate 1. one “central” b -quark and six other b - quarks in a “parameter star” by changing our three free variables. Make a B light and B strange meson for each 2. b quark Calculate the “singlet” B meson, 3. B X = (2/3) B l + (1/3) B s for each of our seven b -quarks. Compare the calculated B X mesons to 4. the physical B X meson, and find the set of parameters matching the physical B. 8 Sophie Hollitt SU(3) breaking of B meson decay constant
Tuning B mesons 9 Sophie Hollitt SU(3) breaking of B meson decay constant
Tuning B mesons Central b value 10 Sophie Hollitt SU(3) breaking of B meson decay constant
Tuning B mesons 11 Sophie Hollitt SU(3) breaking of B meson decay constant
Tuning B mesons 12 Sophie Hollitt SU(3) breaking of B meson decay constant
Tuning B mesons 13 Sophie Hollitt SU(3) breaking of B meson decay constant
Tuning B mesons 14 Sophie Hollitt SU(3) breaking of B meson decay constant
Calculating the decay constant f Bq Once we have chosen the appropriate quarks, the decay constant is calculated mostly using two point functions Improvement term: Renormalisation Lattice decay constant: 2 point correlators & factor: 2 point functions with coefficient c A Ratio of 2 point and 3 different operators in point functions with the quark propagators, Currently take c A = 0, constant coefficient and mass of B Exact value can be ρ = 1 calculated using perturbative QCD 15 Sophie Hollitt SU(3) breaking of B meson decay constant
Calculating the decay constant f Bq Calculate Φ B and Φ Bs for 1. f B at symmetric point m l = m s each of the b -quarks in the tuning “star” For each set of lattice 2. configurations, collect the “best” tuning parameters matching the physical properties of the B X meson (as seen earlier) Use these parameters to 3. interpolate to a “best” Φ B and thus calculate “best” f B Repeat at other light 4. quark masses and lattice spacings! f B for b in tuning star Interpolated best f B 16 Sophie Hollitt SU(3) breaking of B meson decay constant
Configurations used QCDSF Configurations Part of this analysis (systematic error in average mass) New configurations 510 460 410 360 m π 310 260 210 160 110 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 a 2 17 Sophie Hollitt SU(3) breaking of B meson decay constant
Configurations used QCDSF Configurations Part of this analysis (systematic error in average mass) New configurations 510 460 410 360 m π 310 260 BLUE configurations 210 have a systematic 160 error in the SU(3) symmetric point value 110 compared to the 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 a 2 physical point, so we need a more careful approach 18 Sophie Hollitt SU(3) breaking of B meson decay constant
SU(3) breaking of f Bq a = 0.082 fm a = 0.074 fm f Bs a = 0.068 fm a = 0.059 fm SU(3) symmetric On each configuration, point calculate f Bl and f Bs and the average f Bx to cancel most systematic errors from calculation method Visible errors are almost entirely from f B extrapolation to best B meson Linear fit is not sufficient! 19 Sophie Hollitt SU(3) breaking of B meson decay constant
Toward physical f B and f Bs If we take an SU(3) expansion of f Bq / f BX to NLO, and include quenched light quarks ( q ) and ignore the b quark in the SU(3) breaking, we can write: 2 Difference between valence quark mass and SU(3) quark mass ( δ μ b = 0, not part of SU(3) ) Differences between sea quark masses and SU(3) quark mass with a similar equation governing the mass of the B mesons. By using lattice data to fit the coefficients for both f and M, we can: Extrapolate to a value of f Bq at the physical point for each lattice spacing Perform a continuum extrapolation for each f Bq Based on equation in Bornyakov , V. G. et al (2017). “Flavour breaking effects in the pseudoscalar meson decay constants.” 2 Physics Letters B , 767 (3), 366 – 373. doi:10.1016/j.physletb.2017.02.018 20 Sophie Hollitt SU(3) breaking of B meson decay constant
Toward physical f B and f Bs a = 0.082 fm a = 0.082 fm f Bs (partially-quenched) a = 0.074 fm a = 0.068 fm a = 0.059 fm Fits should be performed for each lattice spacing separately… … but for now we have an f B SU(3) overview of the data symmetric collected so far point Fits to the mass and decay constant for each lattice spacing are waiting for more lattice configurations to be δ m q processed. change in quark mass from Next: extrapolate from finite SU(3) symmetric point(s) lattice spacing to continuum QCD 21 Sophie Hollitt SU(3) breaking of B meson decay constant
Summary and future work f B and f Bs calculated for a large number of lattice spacings and SU(3) splittings Additional configurations to be included soon Adding more partially-quenched light quarks Improvement coefficients Future plans include Measurement of f B* Semileptonic form factors B → D (*) lv Studies of Λ b 22 Sophie Hollitt SU(3) breaking of B meson decay constant
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