Chiral corrections to the scalar form factor in $B_q\to D_q$ transitions

We consider chiral loop corrections to the scalar form factor in $B_q\to D_{q}l\nu$ decays. First we consider chiral corrections to the $1/m_Q$ suppressed operators and then we propose a procedure for the extraction of the relevant form factor using lattice QCD results. In particular, the extraction of a $B^{*}\to D^{*}$ matrix element on the lattice can be used to determine all the necessary $1/m_Q$ suppressed contributions to the scalar form factor in $B\to D$ transitions. In the case of $(B_s\to D_s)/(B\to D)$ decay rate ratios we speculate that effects of kinematics and chiral corrections tend to cancel for the scalar form factor contributions. In particular the $1/m_Q$ suppression of chiral corrections is compensated by the potentially large $SU(3)$ flavor symmetry breaking corrections. The calculated corrections are relevant for the precise determination of possible new physics effects in $B_q\to D_{q}l\nu$ decays.

Figure 1

Weak vertex correction sunrise diagram. Crossed box represents effective weak current operator vertex, while filled black circles represent effective strong vertices.

Figure 2

Estimate of ${\xi }_3(w)$ from lattice data on $h_{\pm }(w)$ [8, 9].