[Paper Review] The Static Approximation to B Meson Mixing using Light Domain-Wall Fermions: Perturbative Renormalization and Ground State Degeneracies

Norman H. Christ, Thomas T. Dumitrescu+2 more|ArXiv.org|Oct 28, 2007

Quantum Chromodynamics and Particle Interactions3 references3 citations

TL;DR

This paper presents a one-loop, mean-field improved perturbative matching of lattice heavy-light current and four-fermion operators to the static continuum theory in $ar{\text{MS}}$ (NDR) for B meson mixing using light domain-wall fermions and the Iwasaki gauge action. It introduces a box-source method that improves ground state overlap, enabling precise extraction of $f_B$ and $B_B$ with faster convergence than wall sources, especially on large lattices.

ABSTRACT

We discuss the theoretical input into the current RBC-UKQCD calculation of $f_{B_{d, s}}$ and $B_{B_{d, s}}$ using a smeared static heavy quark propagator, light domain-wall quarks and the Iwasaki gauge action. We present the complete one-loop, mean-field improved matching of heavy-light current and four-fermion lattice operators onto the static continuum theory renormalized in $\bar{ ext{MS}}$(NDR). The large degeneracies present in a static calculation are addressed, and a method for extracting $f_B$ and $B_B$ using only box sources is described; implications for future calculations are discussed.

Motivation & Objective

• To provide a complete one-loop, mean-field improved perturbative matching of lattice operators to the static continuum theory in $ar{\text{MS}}$ (NDR) for B meson mixing calculations.
• To address the issue of ground state degeneracy in static heavy quark effective theory that complicates the extraction of $f_B$ and $B_B$ from lattice correlation functions.
• To develop and validate a method using only box sources for projecting onto the $B$ meson ground state with improved overlap, especially for large spatial volumes.
• To enable more accurate and efficient lattice calculations of $f_{B_q}$ and $B_{B_q}$ in the infinite heavy quark mass limit, supporting precision tests of the CKM matrix.

Proposed method

• Use of a smeared static heavy quark propagator with SU(3)-projected gauge links to reduce noise in lattice correlation functions.
• Employment of light domain-wall fermions for the valence and sea quarks, with $M_5 = 1.8$ and large fifth dimension to preserve chiral symmetry.
• Application of the Iwasaki gauge action to suppress lattice artifacts and improve gauge field dynamics.
• One-loop perturbative matching of lattice operators (current and four-fermion) to the continuum static theory at scale $\mu = a^{-1} = 1.62$ GeV, using mean-field improved perturbation theory.
• Introduction of a box-source construction for both source and sink operators to enhance overlap with the $B$ meson ground state, replacing traditional wall sources.
• Use of a ratio $\mathcal{C}^{LS}(t)e^{m_B^* t/2}/\sqrt{\mathcal{C}^{\widetilde{S}\widetilde{S}}(t)}$ to extract the matrix element $\Phi_B^{\text{lat}}$ with faster plateau formation.

Experimental results

Research questions

• RQ1 How can one-loop, mean-field improved perturbative matching be performed for heavy-light current and four-fermion operators in the static approximation using light domain-wall fermions?
• RQ2 What are the implications of ground state degeneracy in the static effective theory for the extraction of $f_B$ and $B_B$ from lattice correlation functions?
• RQ3 Can a box-source method improve the overlap with the $B$ meson ground state compared to wall sources, especially on large lattices?
• RQ4 How does the use of box sources affect the convergence and noise in the ratio used to extract $f_B$ and $B_B$?
• RQ5 What is the impact of improved ground state projection on the accuracy and efficiency of $f_{B_q}$ and $B_{B_q}$ calculations in the static limit?

Key findings

• The one-loop, mean-field improved perturbative matching of lattice operators to the continuum static theory is completed and validated for the RBC-UKQCD calculation framework.
• The method successfully addresses the issue of ground state degeneracy by introducing a box-source construction that enhances overlap with the $B$ meson ground state.
• The use of box sources leads to faster convergence of the ratio $\mathcal{C}^{LS}(t)e^{m_B^* t/2}/\sqrt{\mathcal{C}^{\widetilde{S}\widetilde{S}}(t)}$ to the matrix element $\Phi_B^{\text{lat}}$, reducing the time to plateau.
• The improved ground state overlap via box sources is particularly beneficial for large spatial volumes, such as $(3\;\text{fm})^3$, where wall sources suffer from poor overlap and late plateaus.
• The calculation of $\langle \overline{B}(\vec{0})|O^\text{lat}_i(\vec{0},0)|B(\vec{0})\rangle$ using $\mathcal{C}_{O_i}(T,t)e^{m_B^* T/2}/\sqrt{\mathcal{C}^{SS}(T-t)\mathcal{C}^{SS}(t)}$ is robust and noise-reduced, enabling accurate $B_B$ extraction.
• The proposed method using only box sources is simpler and more effective than wall sources, and relies on the specific properties of the static effective theory, with potential for further improvements.