Factorization in $B\to V\gamma$ decays

The factorization properties of the radiative decays $B\to V\gamma$ are analyzed at leading order in $1/m_b$ using the soft-collinear effective theory. It is shown that the decay amplitudes can be expressed in terms of a $B\to V$ form factor evaluated at $q^2=0$, light-cone distribution amplitudes of the $B$ and $V$ mesons, and calculable hard-scattering kernels. The renormalization-group equations in the effective theory are solved to resum perturbative logarithms of the different scales in the decay process. Phenomenological implications for the $B\to K^{*}\gamma$ branching ratio, isospin asymmetry, and $CP$ asymmetries are discussed, with particular emphasis on possible effects from physics beyond the standard model.

Figure 1

Three QCD Feynman diagrams for the contributions of $Q_8$ and their leading-order representations in the effective theory. The double line denotes the heavy-quark field. The dashed lines denote hard-collinear fields in ${\mathrm{SCET}}_{\mathrm{I}}$ and collinear fields in ${\mathrm{SCET}}_{\mathrm{II}}$. Solid lines in the effective-theory diagrams denote soft fields and the dotted line denotes a soft-collinear field.

Figure 2

Example of an exceptional momentum configuration giving rise to an interaction in ${\mathrm{SCET}}_{\mathrm{I}}$ involving $hc$ and $\overline{hc}$ fields.

Figure 3

Example of a ${\mathrm{SCET}}_{\mathrm{I}}$ diagram relevant for $B\to V\gamma$ decay with an off-shell-collinear gluon. The same contribution is also depicted in the second line of Fig. 1.

Figure 4

Leading-order QCD diagrams for the matching of $Q_1$ and $Q_8$ onto $J_i^B$. Other diagrams are power suppressed or vanish.

Figure 5

Leading-order QCD diagrams for the matching of $Q_{1,2}^p$ and $Q_8$ onto $J_i^C$.

Figure 6

Matching of $J_i^C$ onto $O_i^C$. Dashed lines denote hard-collinear fields, full lines soft fields. Note that the quark that emits the photon is described by a hard-collinear field in the $\overline{n}$-direction, while the other two quarks are collinear with the $n$-direction.

Figure 7

Leading-order ${\mathrm{SCET}}_{\mathrm{I}}$ diagrams for the matching of the operators $J^D$ and $J^E$ onto ${\mathrm{SCET}}_{\mathrm{II}}$ operators $O^D$ and $O^E$. Dashed lines denote hard-collinear fields, solid lines soft fields.