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Calculation of Semileptonic Form Factor for $\bar B\to D^\ast \ell \bar \nu$ Decay Using the Oktay-Kronfeld Action

DC Field Value Language
dc.contributor.advisor이원종-
dc.contributor.author박성우-
dc.date.accessioned2018-11-12T00:54:33Z-
dc.date.available2018-11-12T00:54:33Z-
dc.date.issued2018-08-
dc.identifier.other000000152995-
dc.identifier.urihttp://hdl.handle.net/10371/143030-
dc.description학위논문 (박사)-- 서울대학교 대학원 : 자연과학대학 물리·천문학부(물리학전공), 2018. 8. 이원종.-
dc.description.abstractWe study the calculation of $\bar{B}\to D^\ast \ell \bar{\nu}$ semileptonic form factor using the lattice QCD technique. Our first target is the zero recoil process which is gold-plated channel to determine the flavor mixing between bottom and charm: $V_{cb}$ of the Cabbibo-Kobayashi-Maskawa (CKM) matrix.



The simulation is done on the $N_{f}=2+1+1$ MILC Highly-improved staggered-quark (HISQ) lattices where the lattice spacing $a\approx 0.12~\fm$, and the sea pion mass $M_\pi\approx 300~\MeV$. For valence quarks, we use the HISQ action for the light and strange quarks and the Oktay-Kronfeld (OK) action for the bottom and charm quarks. Here the OK action is improved version of the Fermilab action such that the bilinear operators are tree-level matched to QCD through $\CO(\lambda^3)$ in HQET power counting where $\lambda\sim a\Lambda\sim \Lambda/(2m_Q)$ and $m_Q$ is the heavy quark mass.



We present preliminary results of the $\BtoDst$ semileptonic form factor at zero recoil. For the OK action inputs, we determine the critical hopping parameter $\kcrit$ nonperturbatively, and tune the hopping parameters $\kappa_b$ and $\kappa_c$ for physical bottom and charm.
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dc.description.tableofcontents1 Introduction 1

1.1 CKM Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Complex phase in CKM matrix . . . . . . . . . . . . . . . . 2

1.1.2 CP violation phase . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.3 Wolfenstein parametrization . . . . . . . . . . . . . . . . . . 3

1.2 CKM Matrix Determination . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Leptonic decay of charged pseudoscalar meson . . . . . . . . 5

1.2.2 Semi-leptonic decay . . . . . . . . . . . . . . . . . . . . . . 6

1.2.3 $V_{tq}$ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.4 More on exclusive $\bar{B}\to D^\ast \ell \bar{\nu}$ decays . . . . . . . . . . . . . . 7

1.3 Lattice Gauge Theory . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3.1 QCD on the lattice . . . . . . . . . . . . . . . . . . . . . . . 8

1.3.2 $V_{cb}$ on the lattice . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Lattice Fermion Actions 13

2.1 Naive Fermion Action . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.1 Doubling symmetry . . . . . . . . . . . . . . . . . . . . . . 14

2.1.2 Naive staggered action . . . . . . . . . . . . . . . . . . . . . 15

2.2 Highly Improved Staggered Quarks . . . . . . . . . . . . . . . . . . 15

2.2.1 FN-type action . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.2 Fat7 smearing . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.3 Lepage term . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.4 Naik term . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.5 HISQ action . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.6 Phase factor . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2.6.1 Phase-absorbed staples . . . . . . . . . . . . . . . 22

2.2.6.2 Phase-absorbed longlink . . . . . . . . . . . . . . . 23

2.2.7 HISQ coefficients for the MILC code . . . . . . . . . . . . . 24

2.2.7.1 Tadpole improvement . . . . . . . . . . . . . . . . 25

2.2.7.2 Boundary condition . . . . . . . . . . . . . . . . . 25

2.3 Fermilab Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3.1 Wilson fermion action . . . . . . . . . . . . . . . . . . . . . 26

2.3.2 Clover action . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3.3 $\gamma_5$-hermiticity . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3.4 Fermilab action . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.3.5 Tadpole improved action . . . . . . . . . . . . . . . . . . . . 30

2.4 Oktay-Kronfeld Action . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.4.1 $\gamma_5$-hermiticity . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.4.2 Tadpole improvement . . . . . . . . . . . . . . . . . . . . . 33

3 Correlation Functions 39

3.1 Quark Propagators . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.1.1 Naive fermion propagator . . . . . . . . . . . . . . . . . . . 39

3.1.2 Naive fermion propagator for staggered quark . . . . . . . . 40

3.2 Meson Propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2.1 Wilson-Wilson meson . . . . . . . . . . . . . . . . . . . . . 41

3.2.2 Wilson-Staggered meson . . . . . . . . . . . . . . . . . . . . 42

3.2.3 Signal-to-noise ratio . . . . . . . . . . . . . . . . . . . . . . 44

3.3 3-point Correlation Function . . . . . . . . . . . . . . . . . . . . . . 45

3.3.1 Naive spectator with sequential inversion . . . . . . . . . . 45

3.3.2 Coherent sequential source . . . . . . . . . . . . . . . . . . . 47

3.3.3 3-point function with a Staggered spectator . . . . . . . . . 49

3.3.4 3-point function: heavy-quark spectator . . . . . . . . . . . 49

3.3.5 Current improvement . . . . . . . . . . . . . . . . . . . . . 50

3.4 Discrete Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.4.1 Reflection parity for 3-point functions . . . . . . . . . . . . 52

3.5 Fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.5.1 Constant fit of the ratio of the correlation function . . . . . 55

3.5.2 Covariance matrix estimation . . . . . . . . . . . . . . . . . 56

4 Tuning of the Hopping Parameters in Oktay-Kronfeld Action 59

4.1 Nonperturbative Determination of $\kappa_{crit}$ . . . . . . . . . . . . . . . . 60

4.1.1 Simulation details: Iteration . . . . . . . . . . . . . . . . . . 61

4.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.1.3 Correlator fit results . . . . . . . . . . . . . . . . . . . . . . 65

4.2 Nonperturbative Tuning of $\kappa_b$ and _x0014_$\kappa_c$ . . . . . . . . . . . . . . . . 67

4.2.1 Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.2.2 Fermilab interpretation: Kinetic mass . . . . . . . . . . . . 68

4.2.3 Tuning strategy . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.2.4 More on HQET inspired fitting function . . . . . . . . . . . 70

4.2.4.1 Error propagation . . . . . . . . . . . . . . . . . . 72

4.2.5 Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.2.6 Correlator fit results . . . . . . . . . . . . . . . . . . . . . . 78

4.3 Inconsistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.4 Hyperfine Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.4.1 HQET expansion . . . . . . . . . . . . . . . . . . . . . . . . 93

4.4.2 Generalized masses . . . . . . . . . . . . . . . . . . . . . . . 95

4.4.3 Fit and results . . . . . . . . . . . . . . . . . . . . . . . . . 96

5 _x0016_$\bar{B}\to D^\ast \ell \bar{\nu}$ Semileptonic Form Factor at Zero Recoil 99

5.1 Simulation Details . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.1.1 Valence light quark: HISQ . . . . . . . . . . . . . . . . . . . 100

5.1.2 Valence heavy quark: OK action . . . . . . . . . . . . . . . 100

5.1.3 Improved currents . . . . . . . . . . . . . . . . . . . . . . . 101

5.2 Spectral Decomposition . . . . . . . . . . . . . . . . . . . . . . . . 102

5.3 Ground State Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.3.1 Analysis on $C^{B\to D^\ast}_{A_1}$ . . . . . . . . . . . . . . . . . . . . . . 104

5.3.2 Analysis of R . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.3.3 Summary on ground state analysis . . . . . . . . . . . . . . 107

5.4 Excited State Analysis . . . . . . . . . . . . . . . . . . . . . . . . 109

5.4.1 $B$ and $D^\ast$ meson 2 + 2 excited states . . . . . . . . . . . . 109

5.4.2 $C^{X\to Y}_J$ Simultaneous fit . . . . . . . . . . . . . . . . . . . . 110

5.4.3 Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

6 Conclusion and Future Works 117

A Gauge Ensemble 119

B MILC Library Convention 121

B.1 Standard MILC Phases . . . . . . . . . . . . . . . . . . . . . . . . 121

B.2 Gamma Matrices in MILC Code . . . . . . . . . . . . . . . . . . . 121

B.3 Gamma Matrices of OK Inverter in QOPQDP . . . . . . . . . . . . 122

B.4 Conversion From FNAL to MILC Representation . . . . . . . . . . 123

C Heavy-heavy Current Improvement 125

C.1 Convention in the Code . . . . . . . . . . . . . . . . . . . . . . . . 125

C.2 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

D Staggered Chiral Perturbation for $B\to D^\ast$ Form Factor 129

D.1 $B\to D^\ast$_x0003_ Zero Recoil . . . . . . . . . . . . . . . . . . . . . . . . . . 129

D.1.1 NLO PQ rSChPT . . . . . . . . . . . . . . . . . . . . . . . . 129

D.1.2 Inputs for NLO: Summary . . . . . . . . . . . . . . . . . . . 131

D.1.3 Beyond the NLO . . . . . . . . . . . . . . . . . . . . . . . . 131

D.2 $B\to D^\ast$ Nonzero Recoil . . . . . . . . . . . . . . . . . . . . . . . . . 132

D.2.1 NLO full QCD rSChPT . . . . . . . . . . . . . . . . . . . . . 132

D.2.2 NLO full QCD fit function . . . . . . . . . . . . . . . . . . . 133

D.2.3 Beyond the NLO full QCD fit function . . . . . . . . . . . . 134

D.2.4 NLO PQ rSChPT . . . . . . . . . . . . . . . . . . . . . . . . 134

D.2.5 Beyond the NLO PQ fit function . . . . . . . . . . . . . . . 135

D.3 Valence Spectators for the Production . . . . . . . . . . . . . . . . 135

E Computational Cost of Oktay-Kronfeld Action 137

F Decoupling of m1 139

F.1 HQET Lagrangian at the Leading Order . . . . . . . . . . . . . . 139

F.2 m1 in the HQET Hamiltonian . . . . . . . . . . . . . . . . . . . . . 139

Bibliography 143
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dc.language.isoen-
dc.publisher서울대학교 대학원-
dc.subject.ddc523.01-
dc.titleCalculation of Semileptonic Form Factor for $\bar B\to D^\ast \ell \bar \nu$ Decay Using the Oktay-Kronfeld Action-
dc.typeThesis-
dc.description.degreeDoctor-
dc.contributor.affiliation자연과학대학 물리·천문학부(물리학전공)-
dc.date.awarded2018-08-
Appears in Collections:
College of Natural Sciences (자연과학대학)Dept. of Physics and Astronomy (물리·천문학부)Physics (물리학전공)Theses (Ph.D. / Sc.D._물리학전공)
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