High Precision Test of the Standard Model using $\varepsilon_K$ and $V_{cb}$ in Lattice QCD

Abstract

In the Standard Model (SM), CP violation is described by the single Kobayashi- Maskawa (KM) phase. By the generalization of the Cabibbo mixing angle for the first and second quark generations, the Cabibbo-Kobayashi-Maskawa (CKM) matrix $V_{CKM}$ defines quark avor mixing amplitudes in the electroweak processes among the three quark generations. The precise determination of the CKM matrix elements is a central part of the SM avor physics, because it allows a test of the SM by observing CP violating processes. Moreover, some of electroweak processes with the W boson exchange are sensitive to the new physics (NP) effects and they can be used to constrain the model space of the NP beyond the SM. By the a lot of dedicated works both from theory and experiment, testing the CKM mechanism enters into the precision era. Among progresses in the theory such as the higher order perturbative calculations, the lattice QCD plays the essential role by providing precise values for the nonperturbative hadronic matrix elements. In the first part of this thesis, the current status of the indirect CP violation parameter in neutral kaon system, $varepsilon_K$, in the SM is discussed in detail. The SM evaluation of the $varepsilon_K$ uses inputs from lattice QCD: the kaon bag parameter $\hat{B}_K$, $\xi_0$, $

V_{us}

$ from the $K\ell3$ and $K\mu2$ decays, and $

V_{cb}

$ from the axial current form factor for the exclusive decay $\bar{B}\to D^{\ast}\ell\nu$ at zero-recoil. The theoretical expression for $varepsilon_K$ is thoroughly reviewed to give an estimate of the size of the neglected corrections, including long distance effects. The Wolfenstein parametrization ($

V_{cb}

$; $\lambda$, $\bar{\rho}$, $\bar{\eta}$) is adopted for CKM matrix elements which enter through the short-distance contribution of the box diagrams. It is found that the SM prediction of $varepsilon_K$ with exclusive $V_{cb}$ and the Unitarity Triangle apex ($\bar{eta}$, $\bar{\rho}$) from the angle-only fit is lower than the experimental value by 3.4$\sigma$. However, with inclusive $V_{cb}, there is no gap between the SM prediction of $varepsilon_K$ and its experimental value. The importance of a specific CKM matrix element $V_{cb}$ is reemphasized in this context. The determination of $V_{cb}$ in a sub-precent level is needed for a decisive test of the SM with $varepsilon_K$. To meet the target precision for the $V_{cb}$, the heavy quark discretization effect on the lattice, which dominates the error for form factor calculations for the $b\to c$ transition, should be controlled with a similar precision. Thus, in the second part, a discussion on the treatment of the heavy quarks on the lattice follows. A numerical test of a highly improved lattice action for heavy quarks, so-called the Oktay-Kronfeld (OK) action, is performed by assessing improvements on the meson spectrums of heavy-strange systems and quarkonia. The OK action is an extension to higher order of the Fermilab improvement program for massive Wilson fermions, a lattice description of the fermion fields. The OK action includes dimension-six and -seven operators necessary for tree-level matching to QCD through order O($\Lambda^3/m_Q^3$) for heavy-light mesons and O($v^6$) for quarkonium, or, with Symanzik power counting, O($a^2$) with some O($a^3$) terms. Data is generated with the tadpole-improved Fermilab and OK actions on 500 gauge configurations from a MILC coarse (a $\approx$ 0.12 fm) $N_f$ = 2 + 1 flavors, asqtad staggered ensemble. From the analysis of the inconsistency parameter and the hyperfine splittings for the rest and kinetic masses, it is clearly shown that, with one exception, the results obtained with the tree-level matched OK action are significantly closer to the continuum limits than the results obtained with the Fermilab action. The exception occurs for the hyperfine splitting of the bottom strange system, where statistics are too low to draw a firm conclusion, though a similar improvement is expected. An optimization of the conjugate gradient inverter code for the OK action is also discussed. It promotes the OK action to the practical level of use.