Aspects Of Heavy Quark Theory

TL;DR

The paper surveys heavy quark theory through the Wilsonian OPE lens, emphasizing the importance of short-distance mass definitions, nonperturbative parameters μ_π^2 and μ_G^2, and exact QCD inequalities that constrain heavy-flavor dynamics. It analyzes inclusive and exclusive methods for determining |V_cb|, detailing perturbative and power-suppressed corrections, the role of duality, and the significance of zero-recoil form factors like F_{D^*}(0). By combining sum rules, virial-type relations, and careful mass scheme choices, it argues for a coherent, model-independent framework that yields reliable CKM inferences while acknowledging uncertainties from duality and higher-order effects. The work highlights both the successes (e.g., stringent bounds and consistent |V_cb| determinations) and the ongoing challenges (duality violations, perturbative ambiguities) shaping future developments in heavy quark phenomenology.

Abstract

Recent achievements in the heavy quark theory are critically reviewed. The emphasis is put on those aspects which either did not attract enough attention or cause heated debates in the current literature. Among other topics we discuss (i) basic parameters of the heavy quark theory; (ii) a class of exact QCD inequalities; (iii) new heavy quark sum rules; (iv) virial theorem; (v) applications (|V_cb| from the total semileptonic width and from the B->D* transition at zero recoil). In some instances new derivations of the previously known results are given, or new aspects addressed. In particular, we dwell on the exact QCD inequalities. Furthermore, a toy model is considered that may shed light on the controversy regarding the value of the kinetic energy of heavy quarks obtained by different methods.

Figures (6)

• Figure 1: Perturbative diagrams leading to the IR renormalon uncertainty in $m_Q^{\rm pole}$ of order $\Lambda_{\rm QCD}$. The number of bubble insertions in the gluon propagator can be arbitrary.
• Figure 2: Three-point function relevant to the derivation of the virial theorem in QCD.
• Figure 3: Sum rules for the three-dimensional harmonic oscillator. The solid curve presents $\langle H\rangle$ (in units of $\frac{3}{2}\omega$) vs. $T$. The dashed curve gives $\langle E_{\rm kin} \rangle$ in units of $\frac{3}{4}\omega$, the dotted curve $\langle E_{\rm pot} \rangle$ in units of $\frac{3}{4}\omega$. The horizontal axis is $T$ in units of $\omega^{-1}$. The exact results for the ground state energy, kinetic energy and potential energy are $\frac{3}{2}\omega$, $\frac{3}{4}\omega$, $\frac{3}{4}\omega$, respectively.
• Figure 4: a) Spectator effect leading to $1/m_b$ correction in the decay width $b\rightarrow s +\gamma$. b) A different cut of the same diagram leads to an electromagnetic correction in the hadronic decay width. Terms $1/m_b$ and $1/m_b^2$ cancel out in the sum of two decay widths. The solid dot in the vertex denotes the penguin-induced $b\rightarrow s \bar{q} q$ interaction.
• Figure 5: Cuts of the transition amplitude in the complex $q_0$ plane. The physical cut for the weak decay starts at $q_0=M_B- (M_D^2+\vec{q}^{\,2})^{1/2}$ and continues towards $q_0=-\infty$. Other physical processes generate cuts starting near $q_0=\pm (M_{B_c}^2+\vec{q}^{\,2})^{1/2}$ (one pair); another pair of cuts originates at close values of $q_0$. An additional channel opens at $q_0 \mathrel{\hbox{$\sim$} \hbox{$>$}} 2m_b +m_c$.