[Paper Review] Weak Matrix Elements and |V_cb| in New Formulation of Heavy Quark Effective Field Theory

W.Y. Wang, Yue-Liang Wu+1 more|arXiv (Cornell University)|Jun 29, 1999

Quantum Chromodynamics and Particle InteractionsPhysics and Astronomy22 references16 citations

TL;DR

This paper proposes a new formulation of Heavy Quark Effective Field Theory (HQEFT) that consistently includes both quark and antiquark fields, enabling a systematic $1/m_Q^2$ expansion of weak matrix elements for semileptonic decays. Unlike standard HQET, the new framework automatically satisfies Luke's theorem without imposing the equation of motion, ensuring $1/m_Q$ corrections vanish at zero recoil—enabling precise extraction of $|V_{cb}|$ from both $B\to D^*\ell\nu$ and $B\to D\ell\nu$ decays at $1/m_Q^2$ order.

ABSTRACT

The heavy quark effective field theory (HQEFT) containing both effective `quark fields' and `antiquark fields' is investigated in detail. By integrating out (but not neglecting) the effective antiquark fields, we present a new formulation of effective theory which differs from the usual heavy quark effective theory (HQET) and exhibits valuable features because of the inclusion of the contributions from the antiquark fields. Matrix elements of vector and axial vector heavy quark currents between pseudoscalar and vector mesons containing a heavy quark (b or c) are then evaluated systematically up to the order of $1/m^2_Q$ and parameterized by a set of universal form factors. With a consistent normalization condition between the effective heavy hadron states, the form factors at zero recoil are related to the ground state meson masses, which enables us to estimate the values of form factors at zero recoil. In particular, the Luke's theorem comes out automatically in the new formulation of HQEFT without the need of imposing the equation of motion $iv\cdot D Q^{+}_{v} =0$. Consequently, the differential decay rates of both $B o D^{\ast}lν$ and $B o D lν$ do not receive $1/m_{Q}$ order corrections at zero recoil, which is not the case in the usual HQET. Thus we quote that the Cabibbo-Kobayashi-Maskawa matrix element $| V_{cb}|$ can nicely be extracted from either of these two exclusive semileptonic decays at the order of $1/m^2_Q$. Our estimates for $|V_{cb}|$ are presented.

Motivation & Objective

• To develop a new formulation of Heavy Quark Effective Field Theory (HQEFT) that retains both effective quark and antiquark fields, avoiding the decoupling approximation used in standard HQET.
• To systematically evaluate weak matrix elements of vector and axial-vector currents between heavy pseudoscalar and vector mesons up to $1/m_Q^2$ order.
• To establish a consistent normalization condition between heavy hadron states that reflects spin-flavor symmetry and enables reliable form factor determination at zero recoil.
• To demonstrate that Luke's theorem emerges automatically in this new framework without requiring the equation of motion $iv\cdot D Q^+ = 0$, ensuring no $1/m_Q$ corrections to decay rates at zero recoil.
• To enable a robust, model-independent extraction of the Cabibbo-Kobayashi-Maskawa matrix element $|V_{cb}|$ from exclusive semileptonic decays $B\to D^*\ell\nu$ and $B\to D\ell\nu$ at $1/m_Q^2$ order.

Proposed method

• The authors derive a new effective Lagrangian by integrating out antiquark components while retaining their contributions, leading to an effective theory distinct from standard HQET.
• They employ equations of motion and functional integral methods to systematically construct the $1/m_Q$ expansion of arbitrary heavy quark currents.
• A consistent normalization condition between heavy hadron states is introduced to preserve spin-flavor symmetry and ensure physical consistency in matrix element calculations.
• Universal form factors are introduced to parameterize matrix elements of vector and axial-vector currents between pseudoscalar and vector mesons up to $1/m_Q^2$ order.
• The decomposition of Lorentz tensors and the use of parameters like $\bar{\Lambda}$, $\kappa_i$, $\chi_i$, and $\eta_i$ allow for a systematic expansion in inverse heavy quark masses.
• The framework is applied to compute differential decay rates, showing that $1/m_Q$ corrections vanish at zero recoil due to the automatic validity of Luke's theorem.

Experimental results

Research questions

• RQ1 Can a new formulation of HQEFT that includes both quark and antiquark fields lead to a more consistent and model-independent description of heavy quark decays at finite quark masses?
• RQ2 Does the inclusion of antiquark contributions in the effective theory lead to the automatic emergence of Luke's theorem without imposing the equation of motion $iv\cdot D Q^+ = 0$?
• RQ3 Can the form factors at zero recoil be reliably determined from meson masses alone in this new framework, and how does this affect the extraction of $|V_{cb}|$?
• RQ4 Are the differential decay rates of $B\to D^*\ell\nu$ and $B\to D\ell\nu$ free from $1/m_Q$ order corrections at zero recoil in this new HQEFT formulation?
• RQ5 Can $|V_{cb}|$ be extracted with reduced theoretical uncertainty from both exclusive semileptonic decays using this improved effective theory at $1/m_Q^2$ order?

Key findings

• The new HQEFT formulation automatically satisfies Luke's theorem without requiring the equation of motion $iv\cdot D Q^+ = 0$, ensuring that $1/m_Q$ corrections to the matrix elements vanish at zero recoil.
• The form factors at zero recoil are directly related to the ground state meson masses through a consistent normalization condition, enabling model-independent estimates.
• The differential decay rates for both $B\to D^*\ell\nu$ and $B\to D\ell\nu$ do not receive $1/m_Q$ order corrections at zero recoil, a key feature that enhances the precision of $|V_{cb}|$ extraction.
• The Cabibbo-Kobayashi-Maskawa matrix element $|V_{cb}|$ can be extracted with reduced theoretical uncertainty from either decay channel at the $1/m_Q^2$ order, thanks to the automatic suppression of $1/m_Q$ corrections.
• The framework provides a consistent $1/m_Q^2$ expansion of weak matrix elements using a set of universal form factors parameterized by $\bar{\Lambda}$, $\kappa_i$, $\chi_i$, and $\eta_i$, with explicit expressions derived in the appendix.
• The results show that the inclusion of antiquark contributions leads to a more fundamental and symmetric description of heavy quark dynamics, improving the reliability of $|V_{cb}|$ determinations from exclusive decays.