Strong Phases and Factorization for Color Suppressed Decays

TL;DR

The paper develops a model-independent factorization theorem for color-suppressed $ar{B}^0 o D^{(*)0}M^0$ decays using soft-collinear effective theory, showing C and E amplitudes are suppressed only by $igO(rac{ m QCD}{Q})$ relative to T and that long- and short-distance contributions are organized via jet functions and soft $B o D^{(*)}$ distributions. A novel mechanism via soft Wilson lines generates non-perturbative strong phases, leading to universal phase behavior across isospin amplitudes and, for π and ρ, exact predictions like equal $D$ and $D^*$ rates at leading order (up to perturbative corrections). The framework yields testable relations such as $R_0^M=1$ for $ar{B}^0 o D^{(*)0}M^0$ with $M= ext{π}, ext{ρ}$, and a quasi-universal phase φ across multiple light mesons, with data-driven extraction of soft parameters like $s_{ m eff}$. Extensions to kaons show richer structure due to SU(3) breaking and long-distance effects, but the core factorization and universality predictions remain a central feature. Overall, the work provides a rigorous, scale-separated description of color-suppressed heavy-to-heavy decays and clarifies the origin and size of strong phases in exclusive processes.

Abstract

We prove a factorization theorem in QCD for the color suppressed decays B0-> D0 M0 and B0-> D*0 M0 where M is a light meson. Both the color-suppressed and W-exchange/annihilation amplitudes contribute at lowest order in LambdaQCD/Q where Q={mb, mc, Epi}, so no power suppression of annihilation contributions is found. A new mechanism is given for generating non-perturbative strong phases in the factorization framework. Model independent predictions that follow from our results include the equality of the B0 -> D0 M0 and B0 -> D*0 M0 rates, and equality of non-perturbative strong phases between isospin amplitudes, delta(DM) = delta(D*M). Relations between amplitudes and phases for M=pi,rho are also derived. These results do not follow from large Nc factorization with heavy quark symmetry.

Figures (6)

• Figure 1: Decay topologies referred to as tree (T), color-suppressed (C), and $W$-exchange (E) and the corresponding hadronic channels to which they contribute.
• Figure 2: Graphs for the tree level matching calculation from ${\rm SCET}_{\rm I}$ (a,b) onto ${\rm SCET}_{\rm II}$ (c,d,e). The dashed lines are collinear quark propagators and the spring with a line is a collinear gluon. Solid lines in (a,b) are ultrasoft and those in (c,d,e) are soft. The $\otimes$ denotes an insertion of the weak operator, given in Eq. (\ref{['QVI']}) for (a,b) and in Eq. (\ref{['QV']}) in (c,d). The $\oplus$ in (e) is a 6-quark operator from Eq. (\ref{['OV']}). The two solid dots in (a,b) denote insertions of the mixed usoft-collinear quark action ${\cal L}_{\xi q}^{(1)}$. The boxes denote the ${\rm SCET}_{\rm II}$ operator ${\cal L}_{\xi\xi qq}^{(1)}$ in Eq. (\ref{['L4q1']}).
• Figure 3: Tree level matching calculation for the $L_{L,R}^{(0,8)}$ operators, with (a) the $T$-product in ${\rm SCET}_{\rm I}$ and (b) the operator in ${\rm SCET}_{\rm II}$. Here $q,q'$ are flavor indices and $\omega\sim \lambda^0$ are minus-momenta.
• Figure 4: Non-perturbative structure of the soft operators in Eq. (\ref{['position']}) which arise from $O_j^{(0,8)}$. Wilson lines are shown for the paths $S_n(x,0)$, $S_n(0,y)$, $S_v(-\infty,0)$ and $S_{v'}(0,\infty)$, plus two interacting QCD quark fields inserted at the locations $x$ and $y$. The $S_v$ and $S_{v'}$ Wilson lines are from interactions with the fields $h_v$ and $h_{v'}$ fields, respectively. The non-perturbative structure of soft fields in $\overline O_j^{(0,8)}$ is similar except that we separate the single and double Wilson lines by an amount $x_\perp$.
• Figure 5: The ratio of isospin amplitudes $R_I = A_{1/2}/(\sqrt2 A_{3/2})$ and strong phases $\delta$ and $\phi$ in $\bar{B}\to D\pi$ and $\bar{B}\to D^*\pi$. The central values following from the $D$ and $D^*$ data in Table I are denoted by squares, and the shaded regions are the $1\sigma$ ranges computed from the branching ratios. The overlap of the $D$ and $D^*$ regions show that the two predictions embodied in Eq. (\ref{['Rpi']}) work well.