Two body $B$ decays with isosinglet final states in soft collinear effective theory

Expressions for decay amplitudes of ${\overline{B}}^0$, $B^{-}$, and ${\overline{B}}_s^0$ mesons to two light pseudoscalar or vector mesons, including isosinglet mesons $\eta$, ${\eta }^{\prime }$, $\omega$, $\varphi$, are obtained using soft collinear effective theory (SCET) at LO in $1/m_b$. These are then used to predict unmeasured branching ratios, direct, and indirect $CP$ asymmetries in ${\overline{B}}^0$, $B^{-}$, and ${\overline{B}}_s^0$ decays into two light pseudoscalars, following a determination of nonperturbative SCET parameters from existing data using a ${\chi }^2$-fit. A separate discussion of indirect $CP$ asymmetries in penguin dominated ${\overline{B}}^0\to {\eta }^{{(}^{\prime })}{K}_{S,L}$, ${\pi }^0{K}_{S,L}$ decays is given.

Figure 1

The configuration with intermediate on shell charm quarks annihilating into two collinear quarks going in the opposite directions can lead to nonperturbative long distance contributions due to soft gluon exchanges [9]. The double line denotes heavy quark, spectator quark is not shown.

Figure 2

Matching to $Q_{gs}^{(0)}$ vanishes at LO in $1/m_b$ with a representative diagram given in (a). In (b) nonzero diagrams that contribute at LO in $1/m_b$ to $Q_{gs}^{(1)}$ matching are given. The square represents insertion of $O_{8g}$. The up- (down-) going wiggly solid lines denote $\overline{n}(n)$ collinear gluons. The off shell gluon (wiggly line) in the third diagram can also attach to any of the cross vertices.

Figure 3

The power suppressed contributions with off shell charm quarks emitting collinear gluons in $\overline{n}$ direction.

Figure 4

The $T$-products of weak Hamiltonian operators (large black dot) with ${\mathcal{L}}_{{\xi }_{n}q}^{(1)}$ (20) that boosts soft quark (dashed line) to $n$ collinear quark (dark gray solid line). Two representatives of soft overlap contribution diagrams (21) with (a) outgoing $n$ collinear quarks and (b) outgoing $n$ collinear gluons are shown. Diagrams (c) and (d) give factorizable contributions corresponding to (22). The light gray solid lines denote $\overline{n}$ collinear quarks, the dark gray wiggly and solid lines the $n$ collinear gluons, while double lines denote heavy quarks. The diagrams (b) and (d) contribute only to $\eta$, ${\eta }^{\prime }$ going in $n$ direction. The ${\alpha }_S(m_b)$ suppressed diagrams with $\overline{n}$ collinear gluons are not shown.

Figure 5

The ${\mathrm{SCET}}_{\mathrm{I}}\to {\mathrm{SCET}}_{\mathrm{II}}$ matching onto operators contributing only to $B^{*}$ decays with isosinglet mesons in $\overline{n}$ direction. The weak ${\mathrm{SCET}}_{\mathrm{I}}$ operator arising at tree level from $O_{8g}$ insertion (large black dot) is enhanced due to one less $\overline{n}$ gluon field compared to operators in Fig. 4, which is compensated by additional soft gluon in the initial state. Additional diagrams similar to Fig. 4b, 4d are possible for isosinglet final state in $n$ direction.

Figure 6

The gluonic charming penguin contributions with intermediate on shell charm quarks annihilating into two collinear quarks going in the opposite directions, with $n$ collinear quark annihilating with spectator quark and producing two $n$ collinear gluons (compare also with diagrams (b) and (d) of Fig. 4).