aainstitutetext: Theory Center, IPNS, KEK, Tsukuba 305-0801,
Japanbbinstitutetext: The Graduate University of Advanced Studies (Sokendai),
Tsukuba 305-0801, Japanccinstitutetext: Kavli Institute for the Physics and
Mathematics of the Universe, University of Tokyo, Kashiwa 277-8583, Japan

# Revisiting rescattering contributions to $\overline{B}_{(s)}\to
D^{(*)}_{(s)}M$ decays

Motoi Endo c Syuhei Iguro a and Satoshi Mishima<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Motivated by the reported discrepancies between experimental data and Standard
Model predictions for the branching ratios of the color-allowed decays
$\overline{B}^{0}\to D^{(*)+}K^{-}$ and $\overline{B}^{0}_{s}\to
D^{(*)+}_{s}\pi^{-}$, we study final-state rescattering effects on
$\overline{B}_{(s)}\to D^{(*)}_{(s)}P$ and $\overline{B}_{(s)}\to D_{(s)}V$,
where $P$ ($V$) is a light pseudoscalar (vector) meson. We consider quasi-
elastic rescatterings in the framework of $SU(3)$ and $U(3)$ flavor
symmetries, and find that the effects cannot explain the measured branching
ratios of the color-allowed and color-suppressed decays simultaneously. We
also perform global fits to the $\overline{B}_{(s)}\to D^{(*)}_{(s)}P$ and
$\overline{B}_{(s)}\to D_{(s)}V$ data, allowing for new physics contributions
to the Wilson coefficient $a_{1}$ associated with the color-allowed tree
amplitudes. It is shown that the fits prefer a downward shift of
$\mathcal{O}(10\%)$ in $a_{1}$ even in the presence of the quasi-elastic
rescattering contributions.

††preprint: IPMU21–0057, KEK–TH–2348

##  1 Introduction

Two-body hadronic $B$ decays into a heavy-light final state
$\overline{B}_{(s)}\to D_{(s)}^{(*)}M$, where $M$ is a light meson, have
played an important role in testing the factorization hypothesis
Bjorken:1988kk . In $\overline{B}_{(s)}^{0}\to D_{(s)}^{+}\pi^{-}$ and
$\overline{B}_{(s)}^{0}\to D_{(s)}^{+}K^{-}$ decays, which are dominated by
the so-called color-allowed tree amplitude, the hypothesis is expected to work
well, since contributions from a gluon exchange between the light meson
emitted from the weak vertex and the soft spectator quark in a heavy meson are
power suppressed in the heavy-quark limit. Moreover, they have no penguin
contribution. For these decays the factorization theorem was proved at the
leading power in the heavy-quark expansion with the QCD factorization (QCDF)
approach Beneke:2000ry  and with the soft-collinear effective theory (SCET)
Bauer:2001cu . There are great improvements in theoretical calculations of the
color-allowed tree amplitude in recent years, which enable us to study
possible contributions to these decays from new-physics (NP) beyond the
Standard Model (SM).

The branching ratios of $\overline{B}_{(s)}^{0}\to D_{(s)}^{(*)+}\pi^{-}$ and
$\overline{B}_{(s)}^{0}\to D_{(s)}^{(*)+}K^{-}$ are calculated in QCDF to
next-to-next-to-leading order (NNLO) accuracy Huber:2016xod . They have
recently been updated in ref. Bordone:2020gao  with new determinations of
$\overline{B}_{(s)}\to D_{(s)}^{(*)}$ form factors in the heavy-quark
expansion up to $\mathcal{O}(\Lambda^{2}_{\mathrm{QCD}}/m_{c}^{2})$
Bordone:2019guc . The updated predictions for the branching ratios show a
tension with the experimental measurements with a significance of the
$4.4\hskip 0.80002pt\sigma$ level Bordone:2020gao .111Similar results are
obtained in ref. Cai:2021mlt  with different inputs.

The tension requires a downward shift of 10% in the $b\to c\bar{u}q$ amplitude
with $q$ being $d$ or $s$, which might be caused by NP. It is shown in ref.
Iguro:2020ndk  that the tension can be reduced partly by introducing a new
vector boson $W^{\prime}$ without conflicting with narrow resonance searches
at the Large Hadron Collider (LHC).222A model-independent NP analysis has also
been carried out in ref. Cai:2021mlt .

On the other hand, the experimental data for the branching ratios of
$\overline{B}^{0}\to D^{(*)+}\pi^{-}$, $\overline{B}^{0}\to D^{(*)0}\pi^{0}$
and $B^{-}\to D^{(*)0}\pi^{-}$ indicate a failure of the naive factorization
Cheng:2001sc ; Neubert:2001sj .333A different conclusion was obtained in refs.
Xing:2001nj ; Xing:2003fe . Here $\overline{B}^{0}\to D^{(*)0}\pi^{0}$ is
dominated by the color-suppressed tree amplitude, while $B^{-}\to
D^{(*)0}\pi^{-}$ has both the color-allowed and color-suppressed tree
amplitudes. For $a_{1}$ and $a_{2}$ being the Wilson coefficients associated
with the color-allowed and color-suppressed tree amplitudes, respectively, a
fit to the data prefers a larger $|a_{2}|$ compared to those extracted from
$\overline{B}\to J/\psi\overline{K}^{(*)}$ and
$\overline{B}\to\psi(2S)\overline{K}^{(*)}$ with a sizable relative phase
between $a_{1}$ and $a_{2}$. Similar results can also be obtained from other
$\overline{B}_{(s)}\to D_{(s)}^{(*)}M$ processes Chiang:2002tv ; Kim:2004hx ;
Chiang:2007bd . The relative phase between $a_{1}$ and $a_{2}$ implies that
final-state interactions (FSI) are significant in the color-suppressed
modes.444In the PQCD approach based on the $k_{T}$ factorization, the larger
$|a_{2}|$ with the sizable phase is explained by spectator-scattering
contributions Keum:2003js ; Li:2008ts .

It is however difficult to calculate the FSI effects in $\overline{B}_{(s)}\to
D_{(s)}^{(*)}M$ from the first principle. In SCET, the FSI contributions in
the color-suppressed decays are factorized into soft factors Mantry:2003uz ;
Blechman:2004vc , which have to be calculated in non-perturbative methods or
extracted from data. Non-factorizable contributions from a single soft-gluon
exchange are estimated with the light-cone QCD sum rules (LCSR) for the color-
allowed decays Bordone:2020gao  and the color-suppressed decays
Halperin:1994hg ; Cui:2004jc , and the size of them for the former (latter) is
found to be insignificant (significant) compared to the factorizable ones.
Also the rescattering of the final-state mesons are studied with various
models Donoghue:1996hz ; Suzuki:1999uc ; Chua:2001br ; Fayyazuddin:2002aa ;
Wolfenstein:2003pc ; Calderon:2003vc ; Fayyazuddin:2004ac ; Wolfenstein:2004km
; Cheng:2004ru ; Chua:2005dt ; Chua:2007qw ; Suzuki:2007je .

It is pointed out in ref. Donoghue:1996hz  using the Regge approach that the
final-state rescattering effects do not vanish in the large $b$-quark mass
limit. On the other hand, in the QCDF approach, the rescattering effects are
found to be suppressed in the heavy-quark limit Beneke:2000ry . Hence the
suppression could be understood by a systematic cancellation among all
intermediate hadronic states Suzuki:1999uc ; Beneke:2000ry . Nonetheless, the
rescattering effects may remain sizable, since the $b$-quark mass is not
infinite in reality.

In this paper we study the quasi-elastic rescattering contributions to the
$\overline{B}_{(s)}\to D_{(s)}^{(*)}M$ decays, following the framework
developed in refs. Chua:2001br ; Chua:2005dt ; Chua:2007qw . The elastic
$D\pi\to D\pi$ rescattering picture is extended to quasi-elastic $D^{(*)}P\to
D^{(*)}P$ and $DV\to DV$ rescatterings by based on $SU(3)$ and $U(3)$ flavor
symmetries, where $P$ and $V$ represent the pseudoscalar $SU(3)$ octet and the
vector $U(3)$ nonet, respectively. We assume that inelastic-rescattering
contributions are suppressed, and examine the question whether the quasi-
elastic rescattering contributions can resolve the tensions between the
theoretical predictions and the experimental data for the color-allowed
decays. The analyses in refs. Chua:2001br ; Chua:2005dt ; Chua:2007qw  are
updated with latest theoretical and experimental information. In particular,
the Wilson coefficient $a_{1}$ at NNLO Huber:2016xod  and the
$\overline{B}_{(s)}\to D_{(s)}^{(*)}$ form factors at
$\mathcal{O}(\Lambda^{2}_{\mathrm{QCD}}/m_{c}^{2})$ Iguro:2020cpg  are
employed to make the calculation of the color-allowed tree amplitude more
reliable.555It is assumed that the $SU(3)$ flavor symmetry holds between the
$\overline{B}\to D^{(*)}$ and $\overline{B}_{s}\to D_{s}^{(*)}$ form factors
as suggested in the result of refs. Kobach:2019kfb ; Bordone:2019guc .

This paper is organized as follows. In section 2 the theoretical framework for
calculating decay amplitudes is described under the presence of the quasi-
elastic rescattering. In section 3 numerical analysis of the rescattering
effects in the color-allowed decays is presented. Finally, the section 4 is
devoted to conclusions and discussion. Besides, appendix A contains some
details of numerical results.

##  2 Theoretical framework

In this section the theoretical framework for calculating decay amplitudes is
briefly introduced with quasi-elastic rescattering contributions. The decays
of $\overline{B}\to DP$, $\overline{B}\to D^{*}P$, and $\overline{B}\to DV$
with $b\to c\bar{u}q$ transitions are studied, where $q$ is $d,s$, and $P$
($V$) stands for a light pseudoscalar (vector) meson.666We do not consider
$\overline{B}\to D^{*}V$ in this paper, since experimental uncertainties are
large PDG:2020 .

###  2.1 Decay amplitudes

The weak effective Hamiltonian relevant for the current study consists only of
the following tree operators:

$\displaystyle\mathcal{H}_{W}=\frac{4\hskip
0.80002ptG_{F}}{\sqrt{2}}\,\sum_{q=d,s}V_{cb}V_{uq}^{*}\big{(}C_{1}\mathcal{O}_{1}^{\hskip
0.27995ptq}+C_{2}\mathcal{O}_{2}^{\hskip 0.27995ptq}\big{)}+\mathrm{h.c.},$
(1)

where $V_{ij}$ are the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements
Cabibbo:1963yz ; Kobayashi:1973fv . The local operators are defined in the
Chetyrkin-Misiak-Münz (CMM) basis Chetyrkin:1996vx ; Chetyrkin:1997gb  as

$\displaystyle\mathcal{O}_{1}^{\hskip 0.27995ptq}$
$\displaystyle=\big{(}\bar{c}\hskip 0.80002pt\gamma^{\mu}T^{a}P_{L}\hskip
0.39993ptb\big{)}\big{(}\bar{q}\hskip
0.39993pt\gamma_{\mu}T^{a}P_{L}u\big{)}\,,$ (2)
$\displaystyle\mathcal{O}_{2}^{\hskip 0.27995ptq}$
$\displaystyle=\big{(}\bar{c}\hskip 0.80002pt\gamma^{\mu}P_{L}\hskip
0.39993ptb\big{)}\big{(}\bar{q}\hskip 0.39993pt\gamma_{\mu}P_{L}u\big{)}\,,$
(3)

where $P_{L}=(1-\gamma_{5})/2$, and $T^{a}$ is a generator of $SU(3)$. Suppose
that there is no physical CP-violating phase in $\mathcal{H}_{W}$, the time-
reversal invariance of $\mathcal{H}_{W}$ leads to

$\displaystyle\langle
i;\mathrm{out}|\mathcal{H}_{W}|\overline{B}\rangle^{*}=\sum_{j}S_{ji}^{*}\hskip
0.80002pt\langle j;\mathrm{out}|\mathcal{H}_{W}|\overline{B}\rangle\hskip
0.39993pt,$ (4)

where the strong scattering $S$-matrix element $S_{ji}$ is defined in terms of
in and out states as $S_{ji}=\langle j;\mathrm{out}|i;\mathrm{in}\rangle$. The
solution of this relation is given by $\langle
i;\mathrm{out}|\mathcal{H}_{W}|\overline{B}\rangle=\sum_{j}S_{ij}^{1/2}\mathcal{A}_{j}^{\hskip
0.27995pt0}$, where $\mathcal{A}_{j}^{\hskip 0.27995pt0}$ is real
Suzuki:1999uc . A decay amplitude with FSI, denoted as
$\mathcal{A}_{i}^{\hskip 0.27995pt\mathrm{FSI}}=\langle
i;\mathrm{out}|\mathcal{H}_{W}|\overline{B}\rangle$, is given by

$\displaystyle\mathcal{A}_{i}^{\hskip
0.27995pt\mathrm{FSI}}=\sum_{j}S_{ij}^{1/2}\mathcal{A}_{j}^{\hskip
0.27995pt0}\hskip 0.39993pt,$ (5)

where $A_{j}^{0}$ represents the corresponding decay amplitude without FSI.
Following the procedure described in ref. Chua:2018ikx , the scattering matrix
is decomposed into a product of two matrices,
$S_{ij}^{1/2}=\sum_{k}(S_{\mathrm{res}}^{1/2})_{ik}(S_{2}^{1/2})_{kj}$. Then
the amplitude is written as

$\displaystyle\mathcal{A}_{i}^{\hskip
0.27995pt\mathrm{FSI}}=\sum_{k}(S_{\mathrm{res}}^{1/2})_{ik}\hskip
0.80002pt\mathcal{A}_{k}^{\hskip 0.56001pt\mathrm{fact}}\hskip
0.39993pt,\qquad\mathcal{A}_{k}^{\hskip
0.56001pt\mathrm{fact}}=\sum_{j}(S_{2}^{1/2})_{kj}\hskip
0.80002pt\mathcal{A}_{j}^{\hskip 0.27995pt0}\hskip 0.39993pt,$ (6)

where $S_{\mathrm{res}}^{1/2}$ is a rescattering matrix, and
$\mathcal{A}_{k}^{\hskip 0.56001pt\mathrm{fact}}$ is a decay amplitude which
incorporates a part of FSI effects. For the color-allowed tree amplitude,
$\mathcal{A}_{k}^{\hskip 0.56001pt\mathrm{fact}}$ is assumed to correspond to
the amplitude calculated in the QCDF approach.

The decay amplitude $\mathcal{A}_{k}^{\hskip 0.56001pt\mathrm{fact}}$ can be
expressed with topological amplitudes in the flavor-diagram approach based on
the $SU(3)$ flavor symmetry Zeppenfeld:1980ex ; Savage:1989ub ; Gronau:1995hm
; Chiang:2002tv ; Kim:2004hx ; Colangelo:2005hh ; Chiang:2007bd ;
Fleischer:2010ca . There are three types of amplitudes for
$\overline{B}_{(s)}\to D_{(s)}^{(*)}M$ decays: the color-allowed tree
amplitude $T$, the color-suppressed tree amplitude $C$, and the $W$-exchange
amplitude $E$, as shown in figure 1.

Figure 1: Representative diagrams of the $b\to c\bar{u}d$ decays: the color-
allowed tree diagram $T$ (left), the color-suppressed tree diagram $C$
(center), and the $W$-exchange diagram $E$ (right). A pair of blobs denotes a
four-quark operator originating through the weak interaction.

For example, the decay amplitude for $B^{-}\to D^{0}\pi^{-}$ is written as

$\displaystyle\mathcal{A}(B^{-}\to D^{0}\pi^{-})=T_{D}+C_{P}\hskip 0.39993pt,$
(7)

where the subscript represents the final-state meson that involves the
spectator quark. The topological amplitudes for other channels are summarized
in tables 1, 2 and 3, where $s_{\phi}$ and $c_{\phi}$ are the sine and cosine
of the mixing angle $\phi_{\eta}$ in the $\eta$-$\eta^{\prime}$ system. The
definition of the mixing angle is based on ref. Escribano:2015yup , and the
single angle hypothesis $\phi_{\eta}\equiv\phi_{q}=\phi_{s}$ is adopted.

Transition | $\\{S,I_{z}\\}$ |  Mode | Amplitude | Data
---|---|---|---|---
$b\to c\bar{u}d$ | $\\{1,-1\\}$ | $\overline{B}_{s}\to D_{s}^{+}\pi^{-}$ | $T_{D}$ | $30.0\pm 2.3$
|  | $\overline{B}_{s}\to D^{0}K^{0}$ | $C_{P}$ | $4.3\pm 0.9$
| $\\{0,-3/2\\}$ | $B^{-}\to D^{0}\pi^{-}$ | $T_{D}+C_{P}$ | $46.8\pm 1.3$
| $\\{0,-1/2\\}$ | $\overline{B}^{0}\to D^{+}\pi^{-}$ | $T_{D}+E$ | $25.2\pm 1.3$
|  | $\overline{B}^{0}\to D^{0}\pi^{0}$ | $\frac{1}{\sqrt{2}}(-C_{P}+E)$ | $2.63\pm 0.14$
|  | $\overline{B}^{0}\to D^{0}\eta$ | $\frac{c_{\phi}}{\sqrt{2}}(C_{P}+E)$ | $2.36\pm 0.32$
|  | $\overline{B}^{0}\to D^{0}\eta^{\prime}$ | $\frac{s_{\phi}}{\sqrt{2}}(C_{P}+E)$ | $1.38\pm 0.16$
|  | $\overline{B}^{0}\to D_{s}^{+}K^{-}$ | $E$ | $0.27\pm 0.05$
$b\to c\bar{u}s$ | $\\{-1,0\\}$ | $\overline{B}^{0}\to D^{+}K^{-}$ | $T_{D}$ | $1.86\pm 0.20$
|  | $\overline{B}^{0}\to D^{0}\overline{K}^{0}$ | $C_{P}$ | $0.52\pm 0.07$
| $\\{-1,-1\\}$ | $B^{-}\to D^{0}K^{-}$ | $T_{D}+C_{P}$ | $3.63\pm 0.12$
| $\\{0,-1/2\\}$ | $\overline{B}_{s}\to D_{s}^{+}K^{-}$ | $T_{D}+E$ | $2.27\pm 0.19\,{}^{\dagger}$
|  | $\overline{B}_{s}\to D^{0}\eta$ | $-s_{\phi}\,C_{P}+\frac{c_{\phi}}{\sqrt{2}}\hskip 0.80002ptE$ | —
|  | $\overline{B}_{s}\to D^{0}\eta^{\prime}$ | $c_{\phi}\,C_{P}+\frac{s_{\phi}}{\sqrt{2}}\hskip 0.80002ptE$ | —
|  | $\overline{B}_{s}\to D^{0}\pi^{0}$ | $\frac{1}{\sqrt{2}}E$ | —
|  | $\overline{B}_{s}\to D^{+}\pi^{-}$ | $E$ | —
Table 1: List of $\overline{B}\to DP$ decays with $b\to c\bar{u}d$ and $b\to c\bar{u}s$ transitions. The values in the second column are the strangeness and the $z$ component of isospin of the final state. The fourth column shows the $SU(3)$ decomposition of the decay amplitudes. The experimental data for the CP-averaged branching ratios (in units of $10^{-4}$) in the fifth column correspond to the PDG values PDG:2020 , which are consistent with the HFLAV ones HFLAV:2019otj . †The data for $\overline{B}_{s}\to D_{s}^{+}K^{-}$ represents a sum of CP-averaged branching ratios, $\mathcal{B}(\overline{B}_{s}\to D_{s}^{+}K^{-})$ and $\mathcal{B}(\overline{B}_{s}\to D_{s}^{-}K^{+})$.  Transition | $\\{S,I_{z}\\}$ |  Mode | Amplitude | Data
---|---|---|---|---
$b\to c\bar{u}d$ | $\\{1,-1\\}$ | $\overline{B}_{s}\to D_{s}^{*+}\pi^{-}$ | $T_{D}$ | $20\pm 5$
|  | $\overline{B}_{s}\to D^{*0}K^{0}$ | $C_{P}$ | $2.8\pm 1.1$
| $\\{0,-3/2\\}$ | $B^{-}\to D^{*0}\pi^{-}$ | $T_{D^{*}}+C_{P}$ | $49.0\pm 1.7$
| $\\{0,-1/2\\}$ | $\overline{B}^{0}\to D^{*+}\pi^{-}$ | $T_{D^{*}}+E$ | $27.4\pm 1.3$
|  | $\overline{B}^{0}\to D^{*0}\pi^{0}$ | $\frac{1}{\sqrt{2}}(-C_{P}+E)$ | $2.2\pm 0.6$
|  | $\overline{B}^{0}\to D^{*0}\eta$ | $\frac{c_{\phi}}{\sqrt{2}}(C_{P}+E)$ | $2.3\pm 0.6$
|  | $\overline{B}^{0}\to D^{*0}\eta^{\prime}$ | $\frac{s_{\phi}}{\sqrt{2}}(C_{P}+E)$ | $1.40\pm 0.22$
|  | $\overline{B}^{0}\to D_{s}^{*+}K^{-}$ | $E$ | $0.22\pm 0.03$
$b\to c\bar{u}s$ | $\\{-1,0\\}$ | $\overline{B}^{0}\to D^{*+}K^{-}$ | $T_{D^{*}}$ | $2.12\pm 0.15$
|  | $\overline{B}^{0}\to D^{*0}\overline{K}^{0}$ | $C_{P}$ | $0.36\pm 0.12$
| $\\{-1,-1\\}$ | $B^{-}\to D^{*0}K^{-}$ | $T_{D^{*}}+C_{P}$ | $3.97^{+0.31}_{-0.28}$
| $\\{0,-1/2\\}$ | $\overline{B}_{s}\to D_{s}^{*+}K^{-}$ | $T_{D^{*}}+E$ | $1.33\pm 0.35\,{}^{\dagger}$
|  | $\overline{B}_{s}\to D^{*0}\eta$ | $-s_{\phi}\,C_{P}+\frac{c_{\phi}}{\sqrt{2}}\hskip 0.80002ptE$ | —
|  | $\overline{B}_{s}\to D^{*0}\eta^{\prime}$ | $c_{\phi}\,C_{P}+\frac{s_{\phi}}{\sqrt{2}}\hskip 0.80002ptE$ | —
|  | $\overline{B}_{s}\to D^{*0}\pi^{0}$ | $\frac{1}{\sqrt{2}}E$ | —
|  | $\overline{B}_{s}\to D^{*+}\pi^{-}$ | $E$ | —
Table 2: List of $\overline{B}\to D^{*}P$ decays with $b\to c\bar{u}d$ and $b\to c\bar{u}s$ transitions. See also the caption of table 1. †The data for $\overline{B}_{s}\to D_{s}^{*+}K^{-}$ represents a sum of CP-averaged branching ratios $\mathcal{B}(\overline{B}_{s}\to D_{s}^{*+}K^{-})$ and $\mathcal{B}(\overline{B}_{s}\to D_{s}^{*-}K^{+})$.  Transition | $\\{S,I_{z}\\}$ |  Mode | Amplitude | Data
---|---|---|---|---
$b\to c\bar{u}d$ | $\\{1,-1\\}$ | $\overline{B}_{s}\to D_{s}^{+}\rho^{-}$ | $T_{D}$ | $69\pm 14$
|  | $\overline{B}_{s}\to D^{0}K^{*0}$ | $C_{V}$ | $4.4\pm 0.6$
| $\\{0,-3/2\\}$ | $B^{-}\to D^{0}\rho^{-}$ | $T_{D}+C_{V}$ | $134\pm 18$
| $\\{0,-1/2\\}$ | $\overline{B}^{0}\to D^{+}\rho^{-}$ | $T_{D}+E$ | $76\pm 12$
|  | $\overline{B}^{0}\to D^{0}\rho^{0}$ | $\frac{1}{\sqrt{2}}(-C_{V}+E)$ | $3.21\pm 0.21$
|  | $\overline{B}^{0}\to D^{0}\omega$ | $\frac{1}{\sqrt{2}}(C_{V}+E)$ | $2.54\pm 0.16$
|  | $\overline{B}^{0}\to D^{0}\phi$ | $0$ | —
|  | $\overline{B}^{0}\to D_{s}^{+}K^{*-}$ | $E$ | $0.35\pm 0.10$
$b\to c\bar{u}s$ | $\\{-1,0\\}$ | $\overline{B}^{0}\to D^{+}K^{*-}$ | $T_{D}$ | $4.5\pm 0.7$
|  | $\overline{B}^{0}\to D^{0}\overline{K}^{\hskip 0.56001pt*0}$ | $C_{V}$ | $0.45\pm 0.06$
| $\\{-1,-1\\}$ | $B^{-}\to D^{0}K^{*-}$ | $T_{D}+C_{V}$ | $5.3\pm 0.4$
| $\\{0,-1/2\\}$ | $\overline{B}_{s}\to D_{s}^{+}K^{*-}$ | $T_{D}+E$ | —
|  | $\overline{B}_{s}\to D^{0}\phi$ | $-C_{V}$ | $0.30\pm 0.05$
|  | $\overline{B}_{s}\to D^{0}\omega$ | $\frac{1}{\sqrt{2}}E$ | —
|  | $\overline{B}_{s}\to D^{0}\rho^{0}$ | $\frac{1}{\sqrt{2}}E$ | —
|  | $\overline{B}_{s}\to D^{+}\rho^{-}$ | $E$ | —
Table 3: List of $\overline{B}\to DV$ decays with $b\to c\bar{u}d$ and $b\to
c\bar{u}s$ transitions. See also the caption of table 1.

In the factorization approach, the amplitudes $T_{D}$ and $C_{P}$ are given by

$\displaystyle T_{D}$ $\displaystyle=\frac{G_{F}}{\sqrt{2}}\hskip
0.80002pta_{1}(\mu)\hskip 0.80002pt(m_{B}^{2}-m_{D}^{2})\hskip
0.80002ptf_{P}\hskip 0.80002ptf_{0}^{B\to
D}(m_{P}^{2})\quad\qquad\mathrm{for}\ \overline{B}\to DP\hskip 0.39993pt,$ (8)
$\displaystyle C_{P}$ $\displaystyle=\frac{G_{F}}{\sqrt{2}}\hskip
0.80002pta_{2}(\mu)\hskip 0.80002pt(m_{B}^{2}-m_{P}^{2})\hskip
0.80002ptf_{D}\hskip 0.80002ptf_{0}^{B\to
P}(m_{D}^{2})\quad\qquad\mathrm{for}\ \overline{B}\to DP\hskip 0.39993pt,$ (9)
$\displaystyle T_{D^{*}}$ $\displaystyle=\frac{G_{F}}{\sqrt{2}}\hskip
0.80002pta_{1}(\mu)\,2\hskip 0.80002ptm_{D^{*}}(\epsilon^{*}\cdot p_{B})\hskip
0.80002ptf_{P}\hskip 0.80002ptA_{0}^{B\to D}(m_{P}^{2})\quad\quad\mathrm{for}\
\overline{B}\to D^{*}P\hskip 0.39993pt,$ (10) $\displaystyle C_{P}$
$\displaystyle=\frac{G_{F}}{\sqrt{2}}\hskip 0.80002pta_{2}(\mu)\,2\hskip
0.80002ptm_{D^{*}}(\epsilon^{*}\cdot p_{B})\hskip 0.80002ptf_{D^{*}}\hskip
0.80002ptf_{+}^{B\to P}(m_{D^{*}}^{2})\quad\,\mathrm{for}\ \overline{B}\to
D^{*}P\hskip 0.39993pt,$ (11) $\displaystyle T_{D}$
$\displaystyle=\frac{G_{F}}{\sqrt{2}}\hskip 0.80002pta_{1}(\mu)\,2\hskip
0.80002ptm_{V}(\epsilon^{*}\cdot p_{B})\hskip 0.80002ptf_{V}\hskip
0.80002ptf_{+}^{B\to D}(m_{V}^{2})\qquad\,\,\mathrm{for}\ \overline{B}\to
DV\hskip 0.39993pt,$ (12) $\displaystyle C_{V}$
$\displaystyle=\frac{G_{F}}{\sqrt{2}}\hskip 0.80002pta_{2}(\mu)\,2\hskip
0.80002ptm_{V}(\epsilon^{*}\cdot p_{B})\hskip 0.80002ptf_{D}\hskip
0.80002ptA_{0}^{B\to V}(m_{D}^{2})\qquad\,\,\mathrm{for}\ \overline{B}\to
DV\hskip 0.39993pt,$ (13)

where $\epsilon^{*\mu}$ is the polarization vector of $D^{*}$ or $V$. The
definitions of the decay constants $f_{P}$, $f_{V}$ and $f_{D^{(*)}}$ and the
transition form factors $f_{0}^{B\to D}$, $f_{0}^{B\to P}$, $f_{+}^{B\to P}$,
$f_{+}^{B\to D}$, $A_{0}^{B\to D}$ and $A_{0}^{B\to V}$ are found in ref.
Gubernari:2018wyi . The coefficients $a_{1}(\mu)$ and $a_{2}(\mu)$ are
evaluated at the scale $\mu=m_{b}$, where the leading-order contributions are
given by $a_{1}(\mu)=C_{2}(\mu)$ and $a_{2}(\mu)=4\hskip
0.80002ptC_{1}(\mu)/9+C_{2}(\mu)/3$ in terms of the Wilson coefficients in eq.
(1).

The coefficient $a_{1}(\mu)$ associated with the color-allowed tree amplitude
is calculated for different processes in the framework of QCDF to the NNLO
accuracy Huber:2016xod :

$\displaystyle
a_{1}(m_{b})=\begin{cases}\big{(}1.069^{+0.009}_{-0.012}\big{)}+\big{(}0.046^{+0.023}_{-0.015}\big{)}\hskip
0.80002pti&\mathrm{for}\ \overline{B}^{0}\to D^{+}K^{-},\\\
\big{(}1.072^{+0.011}_{-0.013}\big{)}+\big{(}0.044^{+0.022}_{-0.014}\big{)}\hskip
0.80002pti&\mathrm{for}\ \overline{B}^{0}\to D^{+}\pi^{-},\\\
\big{(}1.068^{+0.010}_{-0.012}\big{)}+\big{(}0.034^{+0.017}_{-0.011}\big{)}\hskip
0.80002pti&\mathrm{for}\ \overline{B}^{0}\to D^{*+}K^{-},\\\
\big{(}1.071^{+0.012}_{-0.013}\big{)}+\big{(}0.032^{+0.016}_{-0.010}\big{)}\hskip
0.80002pti&\mathrm{for}\ \overline{B}^{0}\to D^{*+}\pi^{-}.\end{cases}$ (14)

It is noticed that these are almost universal, and hence a common value of
$a_{1}(m_{b})$ is used in our numerical analysis:

$\displaystyle a_{1}(m_{b})=1.070\pm 0.012\,.$ (15)

We set $a_{1}$ to be real by absorbing its phase in $a_{2}$ and $E$, which are
complex parameters in the following analysis. On the other hand, it is hard to
evaluate the color-suppressed tree amplitudes reliably in QCDF, because
subleading-power and non-factorizable contributions are expected to be
significant. In our analysis the effective coefficient $a_{2}^{\mathrm{eff}}$
is considered to include these contributions and taken to be a free parameter.
Also, $SU(3)$ breaking effects in the amplitudes are simply assumed to be
described by the factorization form as in eqs. (9), (11) and (13).
Furthermore, the $W$-exchange amplitude $E$, which is power suppressed
compared to the color-allowed amplitude $T$, also cannot be calculated in the
QCDF approach Beneke:2000ry . Therefore $E$ is taken as another free parameter
in the analysis with assuming the $SU(3)$ flavor symmetry.

Finally, the partial decay widths for the $b\to c\bar{u}q$ channels ($q=d,s$)
are expressed with the decay amplitudes as

$\displaystyle\Gamma=\frac{p_{\mathrm{cm}}}{8\hskip 0.39993pt\pi\hskip
0.39993ptm_{B}^{2}}\hskip
0.80002pt\big{|}V_{cb}V_{uq}^{*}\,\mathcal{A}\,\big{|}^{2},$ (16)

where $p_{\mathrm{cm}}$ is the magnitude of the three momentum of a final-
state meson in the center-of-mass frame.

###  2.2 Quasi-elastic rescattering

The framework of the quasi-elastic rescattering in $\overline{B}_{(s)}\to
D_{(s)}^{(*)}M$ was developed in refs. Chua:2001br ; Chua:2005dt ; Chua:2007qw
. In the quasi-elastic picture, the rescattering occurs among different final
states carrying the same quantum numbers. Since QCD respects the $SU(3)$
flavor symmetry to some extent, the rescattering is considered among the
states that reside in the same multiplet under the symmetry. In the case of
$\overline{B}\to DV$, the $U(3)$ symmetry instead of $SU(3)$ can be applied,
since there is no $U(1)$ axial anomaly for the vector mesons. Moreover,
following the procedure in ref. Chua:2018ikx , we include a part of the
$SU(3)$ breaking effects through the meson decay constants.

Under the $SU(3)$ flavor symmetry the $D^{(*)}P$ final states are decomposed
into one $\bm{\overline{15}}$, one $\bm{6}$ and two $\bm{\overline{3}}$
representations, since the light pseudo-scalar mesons transform as ${\bm{8}}$
and ${\bm{1}}$, and the charm mesons $D^{(*)0}$, $D^{(*)+}$ and $D_{s}^{(*)+}$
form a $\bm{\overline{3}}$ representation. Therefore, the rescattering matrix
in eq. (6) can be written in the $SU(3)$ basis as Chua:2001br

$\displaystyle S_{\mathrm{res}}^{1/2}=e^{i\hskip
0.27995pt\delta_{\bm{\overline{15}}}}\sum_{a=1}^{15}\big{|}{\bm{\overline{15}}};a\big{\rangle}\big{\langle}{\bm{\overline{15}}};a\big{|}+e^{i\hskip
0.27995pt\delta_{\bm{6}}}\sum_{b=1}^{6}\big{|}{\bm{6}};b\big{\rangle}\big{\langle}{\bm{6}};b\big{|}+\sum_{m,n={\bm{\bar{3}}},{\bm{\bar{3}^{\prime}}}}\sum_{c=1}^{3}\big{|}m;c\big{\rangle}\,\mathcal{U}^{1/2}_{mn}\,\big{\langle}n;c\big{|}\hskip
0.39993pt,$ (17)

where $\mathcal{U}^{1/2}$ is a $2\times 2$ symmetric unitary matrix. Since the
overall phase is not physical, the phase difference,

$\displaystyle\delta^{\prime}\equiv\delta_{\bm{6}}-\delta_{\bm{\overline{15}}}\hskip
0.39993pt,$ (18)

is defined, and $\mathcal{U}^{1/2}$ is parameterized in terms of three
parameters $\theta$, $\sigma$ and $\tau$ (see, e.g., ref. Suzuki:2007je ):

$\displaystyle\mathcal{U}^{\hskip
0.56001pt\prime}_{mn}\equiv\mathcal{U}^{1/2}e^{-i\hskip
0.27995pt\delta_{\bm{\overline{15}}}}=\left(\\!\begin{array}[]{cc}\cos\tau&\sin\tau\\\
-\sin\tau&\cos\tau\end{array}\\!\right)\left(\\!\begin{array}[]{cc}e^{i\hskip
0.27995pt\theta}&0\\\ 0&e^{i\hskip
0.27995pt\sigma}\end{array}\\!\right)\left(\\!\begin{array}[]{cc}\cos\tau&-\sin\tau\\\
\sin\tau&\cos\tau\end{array}\\!\right).$ (25)

Thus the rescattering matrix is parameterized by the four parameters
$\delta^{\prime}$, $\theta$, $\sigma$ and $\tau$. Note that the rescattering
effects are turned off at $\delta^{\prime}=\theta=\sigma=\tau=0$.

The rescattering in the $D^{(*)}V$ final states can be parameterized in a
similar way. The light vector mesons form a $U(3)$ nonet instead of a $SU(3)$
octet. In this case the rescattering parameters have specific values as
follows, which are called “solution 1” and “solution 2” in ref. Chua:2001br :

$\displaystyle\bullet\ \mathrm{Solution}~{}1:$
$\displaystyle\quad\delta^{\prime}=\theta,\quad\sigma=0\hskip
0.39993pt,\quad\sin\tau=\sqrt{\frac{1}{3}}\hskip
0.39993pt,\quad\cos\tau=\sqrt{\frac{2}{3}}\hskip 0.39993pt,$ (26)
$\displaystyle\bullet\ \mathrm{Solution}~{}2:$
$\displaystyle\quad\delta^{\prime}=\theta=0,\quad\sin\tau=\frac{2\sqrt{2}}{3}\hskip
0.39993pt,\quad\cos\tau=\frac{1}{3}\hskip 0.39993pt.$ (27)

Hence only a single parameter is left free for each solution.

In the following the rescattering formulae are presented for the
$\overline{B}\to DP$ decays. The corresponding formulae for $\overline{B}\to
D^{*}P$ and $\overline{B}\to DV$ can be obtained by substituting the final-
state mesons appropriately.

The final states are classified by the strangeness and the isospin. There are
three sets for the decays with the $b\to c\bar{u}d$ transition:
$\\{S,I_{z}\\}=\\{0,-3/2\\}$, $\\{0,-1/2\\}$ and $\\{1,-1\\}$. For
$\overline{B}\to DP$, $\\{S,I_{z}\\}=\\{0,-3/2\\}$ corresponds only to the
single process $\overline{B}\to D^{0}\pi^{-}$. Then the elastic rescattering
causes only an unphysical overall phase shift, and thus, does not alter the
branching ratio of this channel. For the other sets the decay amplitudes are
assembled as

$\displaystyle\mathcal{A}_{d,0,-1/2}=\begin{pmatrix}\mathcal{A}(\overline{B}^{0}\to
D^{+}\pi^{-})\\\\[2.84526pt] \mathcal{A}(\overline{B}^{0}\to
D^{0}\pi^{0})\\\\[2.84526pt] \mathcal{A}(\overline{B}^{0}\to
D_{s}^{+}K^{-})\\\\[2.84526pt] \mathcal{A}(\overline{B}^{0}\to
D^{0}\eta_{8})\\\\[2.84526pt] \mathcal{A}(\overline{B}^{0}\to
D^{0}\eta_{1})\end{pmatrix},\qquad\mathcal{A}_{d,1,-1}=\begin{pmatrix}\mathcal{A}(\overline{B}_{s}\to
D_{s}^{+}\pi^{-})\\\\[2.84526pt] \mathcal{A}(\overline{B}_{s}\to
D^{0}K^{0})\end{pmatrix},$ (28)

where the subscripts denote the flavor of the down-type quark, the strangeness
and the isospin of the final states. The physical states, $\eta$ and
$\eta^{\prime}$, are given in terms of the octet state $\eta_{8}$ and the
singlet one $\eta_{1}$ as
$\eta=\eta_{8}\cos\theta_{\eta}-\eta_{1}\sin\theta_{\eta}$ and
$\eta^{\prime}=\eta_{8}\sin\theta_{\eta}+\eta_{1}\cos\theta_{\eta}$ with
$\theta_{\eta}=\phi_{\eta}-\mathrm{arctan}\sqrt{2}$.

Similarly, the final states in the processes with the $b\to c\bar{u}s$
transition are classified into $\\{S,I_{z}\\}=\\{-1,-1\\}$, $\\{-1,0\\}$ and
$\\{0,-1/2\\}$. For $\\{S,I_{z}\\}=\\{-1,-1\\}$, there is only the single
process $B^{-}\to D^{0}K^{-}$. The decay amplitudes for the others are written
as

$\displaystyle\mathcal{A}_{s,-1,0}=\begin{pmatrix}\mathcal{A}(\overline{B}^{0}\to
D^{+}K^{-})\\\\[2.84526pt] \mathcal{A}(\overline{B}^{0}\to
D^{0}\overline{K}^{0})\end{pmatrix},\qquad\mathcal{A}_{s,0,-1/2}=\begin{pmatrix}\mathcal{A}(\overline{B}_{s}\to
D^{+}\pi^{-})\\\\[2.84526pt] \mathcal{A}(\overline{B}_{s}\to
D^{0}\pi^{0})\\\\[2.84526pt] \mathcal{A}(\overline{B}_{s}\to
D_{s}^{+}K^{-})\\\\[2.84526pt] \mathcal{A}(\overline{B}_{s}\to
D^{0}\eta_{8})\\\\[2.84526pt] \mathcal{A}(\overline{B}_{s}\to
D^{0}\eta_{1})\end{pmatrix}.$ (29)

According to eq. (6), the rescattering is incorporated into the decay
amplitudes as

$\displaystyle\mathcal{A}_{q,S,I_{z}}^{\hskip
0.27995pt\mathrm{FSI}}=V_{S,I_{z}}^{-1}\,\mathcal{S}_{S,I_{z}}^{1/2}\,V_{S,I_{z}}\,\mathcal{A}_{q,S,I_{z}}^{\hskip
0.56001pt\mathrm{fact}}\hskip 0.39993pt,$ (30)

where $\mathcal{S}_{0,-1/2}^{1/2}$ is the rescattering matrix, and the matrix
$V_{S,I_{z}}$ represents the $SU(3)$ breaking effects Chua:2018ikx . It is
noticed that the breaking effects are removed from
$\mathcal{A}_{q,S,I_{z}}^{\hskip 0.56001pt\mathrm{fact}}$ before the
rescattering by applying $V_{S,I_{z}}$, and then put them back after the
rescattering by $V_{S,I_{z}}^{-1}$ Chua:2018ikx , since the rescattering is
considered to work under the $SU(3)$ symmetry.

The explicit expressions of the symmetric matrices
$\mathcal{S}_{S,I_{z}}^{1/2}$ are given by Chua:2001br ; Chua:2005dt ;
Chua:2007qw

$\displaystyle\mathcal{S}_{1,-1}^{1/2}$
$\displaystyle=\mathcal{S}_{-1,0}^{1/2}=\frac{e^{i\hskip
0.27995pt\delta_{\bm{\overline{15}}}}}{2}\begin{pmatrix}1+e^{i\hskip
0.27995pt\delta^{\prime}}&1-e^{i\hskip
0.27995pt\delta^{\prime}}\\\\[2.84526pt] &1+e^{i\hskip
0.27995pt\delta^{\prime}}\end{pmatrix},$ (31)
$\displaystyle\mathcal{S}_{0,-1/2}^{1/2}$ $\displaystyle=\frac{e^{i\hskip
0.27995pt\delta_{\bm{\overline{15}}}}}{8}\begin{pmatrix}3+u_{+}&-\frac{1}{\sqrt{2}}\big{(}5-u_{+}\big{)}&-1-u_{-}&\sqrt{\frac{3}{2}}\big{(}1-v_{-}\big{)}&\sqrt{6}\hskip
0.80002ptu^{\prime}\\\\[5.69054pt]
&\frac{1}{2}\big{(}11+u_{+}\big{)}&-\frac{1}{\sqrt{2}}\big{(}1+u_{-}\big{)}&\frac{\sqrt{3}}{2}\big{(}1-v_{-}\big{)}&\sqrt{3}\hskip
0.80002ptu^{\prime}\\\\[5.69054pt]
&&3+u_{+}&-\sqrt{\frac{3}{2}}\big{(}3-v_{+}\big{)}&\sqrt{6}\hskip
0.80002ptu^{\prime}\\\\[5.69054pt]
&&&\frac{1}{2}\big{(}9+w_{+}\big{)}&u^{\prime}\\\\[5.69054pt]
&&&&u^{\prime\prime}\end{pmatrix},$ (32)

where $u_{\pm}=2\hskip 0.80002pte^{i\hskip 0.27995pt\delta^{\prime}}\pm
3\,\mathcal{U}^{\hskip 0.56001pt\prime}_{\hskip
0.56001pt{\bm{\bar{3}}}{\bm{\bar{3}}}}$, $v_{\pm}=2\hskip 0.80002pte^{i\hskip
0.27995pt\delta^{\prime}}\pm\mathcal{U}^{\hskip 0.56001pt\prime}_{\hskip
0.56001pt{\bm{\bar{3}}}{\bm{\bar{3}}}}$, $w_{+}=6\hskip 0.80002pte^{i\hskip
0.27995pt\delta^{\prime}}+\mathcal{U}^{\hskip 0.56001pt\prime}_{\hskip
0.56001pt{\bm{\bar{3}}}{\bm{\bar{3}}}}$, $u^{\prime}=2\,\mathcal{U}^{\hskip
0.56001pt\prime}_{\hskip 0.56001pt{\bm{\bar{3}}}{\bm{\bar{3}^{\prime}}}}$ and
$u^{\prime\prime}=8\,\mathcal{U}^{\hskip 0.56001pt\prime}_{\hskip
0.56001pt{\bm{\bar{3}^{\prime}}}{\bm{\bar{3}^{\prime}}}}$, and the lower
components are omitted for simplicity. The matrix $V_{-1,0}$ is a unit matrix,
and the others are modeled by the ratios of the meson decay constants (see
ref. Chua:2018ikx ):

$\displaystyle V_{1,-1}$ $\displaystyle=\mathrm{diag}\bigg{(}1\hskip
0.39993pt,\ \frac{f_{D_{s}}f_{\pi}}{f_{D}f_{K}}\bigg{)}\hskip 0.39993pt,$ (33)
$\displaystyle V_{0,-1/2}$ $\displaystyle=\mathrm{diag}\bigg{(}1\hskip
0.39993pt,\ 1\hskip 0.39993pt,\ \frac{f_{D}f_{\pi}}{f_{D_{s}}f_{K}}\hskip
0.39993pt,\ \frac{f_{D}f_{\pi}}{f_{D}f_{\eta_{8}}}\hskip 0.39993pt,\
\frac{f_{D}f_{\pi}}{f_{D}f_{\eta_{1}}}\bigg{)}\hskip 0.39993pt.$ (34)

For the $\overline{B}\to DV$ decays with $\\{S,I_{z}\\}=\\{0,-1/2\\}$, we
calculate the amplitudes $\mathcal{A}_{q,0,-1/2}^{\hskip
0.56001pt\mathrm{fact}}$ in the basis with $\phi$ and $\omega$ instead of the
$SU(3)$ states:

$\displaystyle\mathcal{A}_{d,0,-1/2}=\begin{pmatrix}\mathcal{A}(\overline{B}^{0}\to
D^{+}\rho^{-})\\\\[2.84526pt] \mathcal{A}(\overline{B}^{0}\to
D^{0}\rho^{0})\\\\[2.84526pt] \mathcal{A}(\overline{B}^{0}\to
D_{s}^{+}K^{*-})\\\\[2.84526pt] \mathcal{A}(\overline{B}^{0}\to
D^{0}\phi)\\\\[2.84526pt] \mathcal{A}(\overline{B}^{0}\to
D^{0}\omega)\end{pmatrix},\qquad\mathcal{A}_{s,0,-1/2}=\begin{pmatrix}\mathcal{A}(\overline{B}_{s}\to
D^{+}\rho^{-})\\\\[2.84526pt] \mathcal{A}(\overline{B}_{s}\to
D^{0}\rho^{0})\\\\[2.84526pt] \mathcal{A}(\overline{B}_{s}\to
D_{s}^{+}K^{*-})\\\\[2.84526pt] \mathcal{A}(\overline{B}_{s}\to
D^{0}\phi)\\\\[2.84526pt] \mathcal{A}(\overline{B}_{s}\to
D^{0}\omega)\end{pmatrix}.$ (35)

In this case the rescattering formula includes the mixing of the states as

$\displaystyle\mathcal{A}_{q,0,-1/2}^{\hskip 0.27995pt\mathrm{FSI}}$
$\displaystyle=(V_{0,-1/2}^{DV})^{-1}\hskip
0.80002pt(U^{DV})^{\dagger}\,\mathcal{S}_{0,-1/2}^{1/2}\,U^{DV}V_{0,-1/2}^{DV}\,\mathcal{A}_{q,0,-1/2}^{\hskip
0.56001pt\mathrm{fact}}\hskip 0.39993pt,$ (36)

where $U^{DV}$ and $V_{2}^{DV}$ are defined by

$\displaystyle U^{DV}$ $\displaystyle=\left(\begin{array}[]{ccccc}1&0&0&0&0\\\
0&1&0&0&0\\\ 0&0&1&0&0\\\ 0&0&0&\cos\theta_{V}&\sin\theta_{V}\\\
0&0&0&-\sin\theta_{V}&\cos\theta_{V}\end{array}\right),$ (42) $\displaystyle
V_{0,-1/2}^{DV}$ $\displaystyle=\mathrm{diag}\bigg{(}1\hskip 0.39993pt,\
1\hskip 0.39993pt,\ \frac{f_{D}f_{\rho}}{f_{D_{s}}f_{K^{*}}}\hskip 0.39993pt,\
\frac{f_{D}f_{\rho}}{f_{D}f_{\phi}}\hskip 0.39993pt,\
\frac{f_{D}f_{\rho}}{f_{D}f_{\omega}}\bigg{)}\hskip 0.39993pt.$ (43)

The angle $\theta_{V}$ represents the mixing of the octet and singlet states:
$\phi=\omega_{8}\cos\theta_{V}-\omega_{1}\sin\theta_{V}$ and
$\omega=\omega_{8}\sin\theta_{V}+\omega_{1}\cos\theta_{V}$ with
$\cos\theta_{V}=\sqrt{2/3}$ and $\sin\theta_{V}=1/\sqrt{3}$.

##  3 Numerical analysis

The input parameters for our numerical analysis are summarized in table 4. The
decay constants and the mixing angle for the octet and singlet states of the
pseudo-scalar mesons are chosen as $f_{8}=1.27\hskip 0.80002ptf_{\pi}$,
$f_{1}=1.14\hskip 0.80002ptf_{\pi}$, and
$\theta_{\eta}=\phi_{\eta}-\mathrm{arctan}\sqrt{2}$ with $\phi_{\eta}=0.7036$
rad Escribano:2015yup . Other parameters, such as the Fermi constant $G_{F}$
and the masses and the lifetimes of the mesons, are taken from ref. PDG:2020 .
Note that the value of $V_{cb}$ in this section corresponds to the so-called
exclusive $V_{cb}$. If the inclusive $V_{cb}$, which is larger than the
exclusive one by about 5$\%$ PDG:2020 , is used, the tensions between the
experimental data and the theoretical predictions for the color-allowed decays
are enhanced.

Parameter | Value | Parameter | Value | Parameter | Value
---|---|---|---|---|---
$|V_{ud}|$ | 0.97370 PDG:2020  | $|V_{us}|$ | 0.2245 PDG:2020  | $|V_{cb}|$ | 0.0397 (6) Iguro:2020cpg
$f_{D}$ | 0.2127 Bazavov:2017lyh  | $f_{\pi}$ | 0.1302 (8) Aoki:2019cca  | $f_{K}$ | 0.1557 (3) Aoki:2019cca
$f_{D_{s}}$ | 0.2499 Bazavov:2017lyh  | $f_{\rho}$ | 0.213 (5) Straub:2015ica  | $f_{K^{*}}$ | 0.204 (7) Straub:2015ica
$f_{D^{*}}$ | 0.249 Gubernari:2018wyi  | $f_{\phi}$ | 0.233 (4) Straub:2015ica  | $f_{\omega}$ | 0.197 (8) Straub:2015ica
$f_{D_{s}^{*}}$ | 0.293 Gelhausen:2013wia  |  |  |  |
$F_{0}^{BD}(m_{\pi}^{2})$ | 0.669 (10) Iguro:2020cpg  | $F_{0}^{BD}(m_{K}^{2})$ | 0.672 (10) Iguro:2020cpg  | $F_{1}^{BD}(m_{\rho}^{2})$ | 0.686 (10) Iguro:2020cpg
$F_{1}^{BD}(m_{K^{*}}^{2})$ | 0.692 (10) Iguro:2020cpg  |  |  |  |
$A_{0}^{BD^{*}}(m_{\pi}^{2})$ | 0.725 (14) Iguro:2020cpg  | $A_{0}^{BD^{*}}(m_{K}^{2})$ | 0.732 (14) Iguro:2020cpg  | $A_{0}^{BD^{*}}(m_{\eta}^{2})$ | 0.734 (14) Iguro:2020cpg
$A_{0}^{BD^{*}}(m_{\eta^{\prime}}^{2})$ | 0.754 (14) Iguro:2020cpg  |  |  |  |
$F_{0}^{B\pi}(m_{D}^{2})$ | 0.288 Ball:2004ye  | $F_{0}^{BK}(m_{D}^{2})$ | 0.364 Ball:2004ye  | $F_{0}^{B\eta}(m_{D}^{2})$ | 0.192 Duplancic:2015zna
$F_{0}^{B\eta^{\prime}}(m_{D}^{2})$ | 0.148 Duplancic:2015zna  | $F_{0}^{B_{s}K}(m_{D}^{2})$ | 0.310 Ball:2004ye ; Khodjamirian:2017fxg  | $F_{0}^{B_{s}\eta}(m_{D}^{2})$ |  $-0.243$ Duplancic:2015zna
$F_{0}^{B_{s}\eta^{\prime}}(m_{D}^{2})$ | 0.290 Duplancic:2015zna  |  |  |  |
$F_{1}^{B\pi}(m_{D^{*}}^{2})$ | 0.328 Ball:2004ye  | $F_{1}^{BK}(m_{D^{*}}^{2})$ | 0.420 Ball:2004ye  | $F_{1}^{B\eta}(m_{D^{*}}^{2})$ | 0.210 Duplancic:2015zna
$F_{1}^{B\eta^{\prime}}(m_{D^{*}}^{2})$ | 0.162 Duplancic:2015zna  | $F_{1}^{B_{s}K}(m_{D^{*}}^{2})$ | 0.357 Ball:2004ye ; Khodjamirian:2017fxg  | $F_{1}^{B_{s}\eta}(m_{D^{*}}^{2})$ |  $-0.264$ Duplancic:2015zna
$F_{1}^{B_{s}\eta^{\prime}}(m_{D^{*}}^{2})$ | 0.311 Duplancic:2015zna  |  |  |  |
$A_{0}^{B\rho}(m_{D}^{2})$ | 0.432 Straub:2015ica  | $A_{0}^{B\omega}(m_{D}^{2})$ | 0.399 Straub:2015ica  | $A_{0}^{BK^{*}}(m_{D}^{2})$ | 0.458 Straub:2015ica
$A_{0}^{B_{s}K^{*}}(m_{D}^{2})$ | 0.438 Straub:2015ica  | $A_{0}^{B_{s}\phi}(m_{D}^{2})$ | 0.515 Straub:2015ica  |  |
Table 4: The CKM matrix elements, decay constants in units of GeV,
$\overline{B}\to D^{(*)}$ form factors, $\overline{B}_{(s)}\to P$ form
factors, and $\overline{B}_{(s)}\to V$ form factors. The definitions of the
$\overline{B}_{(s)}\to\eta^{(\prime)}$ form factors include the mixing angle
and the Clebsch-Gordan coefficients. We assume that the values of the
$\overline{B}_{s}\to D_{s}^{(*)}$ form factors are identical to those of the
$\overline{B}\to D^{(*)}$ form factors Kobach:2019kfb ; Bordone:2019guc .

###  3.1 Color-allowed decays

As pointed out in ref. Bordone:2020gao , the theoretical predictions for the
branching ratios of the color-allowed $\overline{B}\to DP$ and
$\overline{B}\to D^{*}P$ decays are universally larger than the measured
values. Let us first update the predictions in ref. Bordone:2020gao  by
adopting the latest results of the $\overline{B}\to D^{(*)}$ form factors
given in ref. Iguro:2020cpg . The $\overline{B}\to D^{*}$ form factors in ref.
Iguro:2020cpg  have smaller uncertainties compared to those in ref.
Bordone:2019guc  because they include the full distribution data reported from
Belle Abdesselam:2017kjf ; Waheed:2018djm  and QCD sum rule constraints on the
parameters appearing in higher order corrections. It is assumed that the
values of the $\overline{B}_{s}\to D_{s}^{(*)}$ form factors are identical to
those of the $\overline{B}\to D^{(*)}$ form factors, because the $SU(3)$
breaking effects between them are found to be insignificant Kobach:2019kfb ;
Bordone:2019guc . Then, ignoring the rescattering contributions, the following
predictions are obtained for the CP-averaged branching ratios of the color-
allowed tree channels:

$\displaystyle\mathcal{B}(\overline{B}_{s}\to D_{s}^{+}\pi^{-})$
$\displaystyle=\big{(}40.9\pm 2.1\big{)}\times 10^{-4},$ (44)
$\displaystyle\mathcal{B}(\overline{B}^{0}\to D^{+}K^{-})$
$\displaystyle=\big{(}3.03\pm 0.15\big{)}\times 10^{-4},$ (45)
$\displaystyle\mathcal{B}(\overline{B}_{s}\to D_{s}^{*+}\pi^{-})$
$\displaystyle=\big{(}44.6\pm 2.2\big{)}\times 10^{-4},$ (46)
$\displaystyle\mathcal{B}(\overline{B}^{0}\to D^{*+}K^{-})$
$\displaystyle=\big{(}3.27\pm 0.16\big{)}\times 10^{-4},$ (47)
$\displaystyle\mathcal{B}(\overline{B}_{s}\to D_{s}^{+}\rho^{-})$
$\displaystyle=\big{(}104.5\pm 7.1\big{)}\times 10^{-4},$ (48)
$\displaystyle\mathcal{B}(\overline{B}^{0}\to D^{+}K^{*-})$
$\displaystyle=\big{(}4.91\pm 0.41\big{)}\times 10^{-4}.$ (49)

The uncertainties come mainly from $|V_{cb}|$, $f_{P(V)}$ and the $B_{(s)}\to
D_{(s)}^{(*)}$ form factors in table 4 as well as from $a_{1}$ in eq. (15),
where the correlations between $|V_{cb}|$ and the $B_{(s)}\to D_{(s)}^{(*)}$
form factors are included. The combined uncertainties are at the level of
$5\%$ for $\overline{B}\to DP$ and $\overline{B}\to D^{*}P$ and of $7$ to
$8\%$ for $\overline{B}\to DV$. Consequently, the above results deviate from
the measured values by $3.5\hskip 0.80002pt\sigma$, $4.7\hskip
0.80002pt\sigma$, $4.5\hskip 0.80002pt\sigma$, $5.3\hskip 0.80002pt\sigma$,
$2.3\hskip 0.80002pt\sigma$ and $0.5\hskip 0.80002pt\sigma$, respectively. It
is noted that the discrepancies are milder in the case of $\overline{B}\to
DV$. Especially, the prediction for $\mathcal{B}(\overline{B}^{0}\to
D^{+}K^{*-})$ is consistent with the data.

The above predictions are very clean theoretically. There is neither penguin
nor annihilation contribution to these processes. As a result, there is no
chirally-enhanced hard-scattering contributions at
$\mathcal{O}(\Lambda_{\mathrm{QCD}}/m_{B})$. Moreover, power corrections at
$\mathcal{O}(\Lambda_{\mathrm{QCD}}/m_{B})$, including twist-3 two-particle
contributions of light-meson light-cone distribution amplitudes, a hard-
collinear gluon exchange between $b$ (or $c$) and the light meson, and a soft
gluon exchange between the $B\to D$ system and the light meson, are expected
to be less than a percent Bordone:2020gao . Besides, the QCD$\times$QED
factorization is studied recently in ref. Beneke:2021jhp , where QED
contributions to the color-allowed tree amplitudes are found to be destructive
and at the sub-percent level, though ultrasoft photons may correct the
measured decay rates up to a few percent.

###  3.2 Rescattering between color-allowed and color-suppressed decays

The rescattering effects modify the predictions for the color-allowed tree
channels. In this subsection we investigate whether the quasi-elastic
rescattering contributions based on the $SU(3)$ and $U(3)$ flavor symmetries
can explain the discrepancies between the theoretical prediction and the
experimental data discussed in the last subsection.

As explained in section 2.2, the color-allowed tree decay $\overline{B}_{s}\to
D_{s}^{+}\pi^{-}$ receives the quasi-elastic rescattering through the process
$\overline{B}_{s}\to D^{0}K^{0}\to D_{s}^{+}\pi^{-}$, where
$\overline{B}_{s}\to D^{0}K^{0}$ is a color-suppressed tree decay. The
correlations between the color-allowed and the color-suppressed decays are
shown in figure 2. Here the color-suppressed tree amplitude is parameterized
with a single complex parameter $a_{2}^{\mathrm{eff}}$, which is substituted
for $a_{2}(\mu)$ in eqs. (9), (11) and (13) and includes the factorizable and
non-factorizable contributions. Then, $|a_{2}^{\mathrm{eff}}|$ and
$\mathrm{arg}(a_{2}^{\mathrm{eff}})$ are taken as free parameters. Also, the
rescattering contributions are parameterized by a single parameter
$\delta^{\prime}$ as in eq. (31). In the numerical analyses, the CP-conserving
strong phase $\mathrm{arg}(a_{2}^{\mathrm{eff}})$ and the rescattering phase
$\delta^{\prime}$ are scanned from $0$ to $2\pi$. In particular, for
$\overline{B}\to DV$, the solution 1 in eq. (26) is adopted because
$\delta^{\prime}=0$ is set in the solution 2. The uncertainties of the input
parameters in table 4 are ignored here.

Figure 2: Contours of the effective coefficient $|a_{2}^{\mathrm{eff}}|$ for
the color-suppressed tree amplitude in the plane of the branching ratios for
the color-allowed-tree and color-suppressed-tree channels, where the
rescattering phase $\delta^{\prime}$ and the relative phase between
$a_{1}(m_{b})$ and $a_{2}^{\mathrm{eff}}$ are taken to be free. The vertical
(yellow) and horizontal (green) bands show the experimental constraints on the
branching ratios at the $1\hskip 0.80002pt\sigma$ (darker) and $2\hskip
0.80002pt\sigma$ (lighter) levels. The vertical dotted line (red) corresponds
to the case of no rescattering, i.e., $\delta^{\prime}=0$.

In figure 2, the solid black lines denote the contours for
$|a_{2}^{\mathrm{eff}}|=0.0,\,0.2,\,0.4,\,0.6$ and $0.8$. For example, in the
case of $\overline{B}_{s}\to D_{s}^{+}\pi^{-}$ and $\overline{B}_{s}\to
D^{0}K^{0}$, the branching ratios satisfy the following relation:

$\displaystyle\frac{\mathcal{B}(\overline{B}_{s}\to
D^{0}K^{0})-\mathcal{B}(\overline{B}_{s}\to
D^{0}K^{0})\big{|}_{\delta^{\prime}=0}}{\mathcal{B}(\overline{B}_{s}\to
D_{s}^{+}\pi^{-})-\mathcal{B}(\overline{B}_{s}\to
D_{s}^{+}\pi^{-})\big{|}_{\delta^{\prime}=0}}=-\frac{p_{\mathrm{cm},D^{0}K^{0}}}{p_{\mathrm{cm},D_{s}^{+}\pi^{-}}}\left(\frac{f_{D}f_{K}}{f_{D_{s}}f_{\pi}}\right)^{2},$
(50)

where $p_{\mathrm{cm},D_{s}^{+}\pi^{-}}$ and $p_{\mathrm{cm},D^{0}K^{0}}$ are
the magnitude of the three momentum of a final-state meson in the center-of-
mass frame in each process. If the rescattering contribution is absent, there
are tensions between the theoretical predictions (denoted by a red dotted
line) and the experimental data (denoted by yellow vertical bands) for the
color-allowed tree decay. Their branching ratios can be reduced by including
the rescattering contribution, while the ratios for the color-suppressed
channels increase at the same time. Although smaller $|a_{2}^{\mathrm{eff}}|$
is preferred from the data, even with $|a_{2}^{\mathrm{eff}}|=0.0$, there
still remain a discrepancy between the theoretical results and the
experimental data for $\overline{B}\to D^{(*)}P$. Therefore, we conclude that
the experimental data cannot be fully explained even if the quasi-elastic
rescattering contributions are included.

The discrepancy may be explained by introducing large power corrections in the
QCDF approach, rescattering contributions beyond the quasi-elastic
approximation and/or NP contributions. Here the last case is studied by
regarding the coefficient $a_{1}(m_{b})$ as an additional free parameter.

Figure 3: The favored region in $a_{1}(m_{b})$ vs $|a_{2}^{\mathrm{eff}}|$
plane considering color-allowed and color-suppressed channels simultaneously.
The orange regions show the experimental constraints at $1\hskip
0.80002pt\sigma$ (darker) and $2\hskip 0.80002pt\sigma$ (lighter) levels
including the rescattering. The cross in the band shows the result calculated
without the rescattering. Black solid and dashed bands correspond to the QCDF
prediction at the $1\hskip 0.80002pt\sigma$ and $2\hskip 0.80002pt\sigma$
levels.

In figure 3 the region of $a_{1}(m_{b})$ and $|a_{2}^{\mathrm{eff}}|$ that
accommodate the experimental data for the color-allowed and color-suppressed
channels simultaneously is presented. For example, the colored regions in the
top-left plot are calculated with the relation,

$\displaystyle\alpha\hskip 0.80002ptr\hskip
0.80002pt\big{|}a_{1}(m_{b})\big{|}^{2}+\beta\hskip
0.80002pt\big{|}a_{2}^{\mathrm{eff}}\big{|}^{2}=r\,\mathcal{B}(\overline{B}_{s}\to
D_{s}^{+}\pi^{-})+\mathcal{B}(\overline{B}_{s}\to D^{0}K^{0})\,,$ (51)

where
$r=(p_{\mathrm{cm},D^{0}K^{0}}/p_{\mathrm{cm},D_{s}^{+}\pi^{-}})(f_{D}f_{K}/f_{D_{s}}/f_{\pi})^{2}$,
$\alpha=\mathcal{B}(\overline{B}_{s}\to
D_{s}^{+}\pi^{-})\big{|}_{a_{1}(m_{b})=1,\delta^{\prime}=0}$ and
$\beta=\mathcal{B}(\overline{B}_{s}\to
D^{0}K^{0})\big{|}_{a_{2}^{\mathrm{eff}}=1,\delta^{\prime}=0}$. The orange
regions show the experimental constraints on the branching ratios at the
$1\hskip 0.80002pt\sigma$ (darker) and $2\hskip 0.80002pt\sigma$ (lighter)
levels, where $\mathrm{arg}(a_{2}^{\mathrm{eff}})$ and $\delta^{\prime}$ are
scanned from $0$ to $2\pi$. Here and hereafter, new CP-violating phases are
assumed to be absent for simplicity. It is observed that universal downward
shifts of $\mathcal{O}(10\%)$ in $a_{1}(m_{b})$ improve the overall
consistency between the theoretical and experimental results. It is noted that
the values of $|a_{2}^{\mathrm{eff}}|$ from each channel can be different from
each other due to differences in the non-factorizable contributions among
$\overline{B}\to DP$, $\overline{B}\to D^{*}P$ and $\overline{B}\to DV$ as
well as to the $SU(3)$ breaking effects.

When the rescattering effects are neglected, one can determine $a_{1}(m_{b})$
and $|a_{2}^{\mathrm{eff}}|$ individually from the branching ratios for the
color-allowed and color-suppressed channels, respectively. The result is
denoted by the cross in the figure. It is found that there are mild tensions
among the values of $a_{1}(m_{b})$ determined from each channel. However,
these tensions can be reduced by including the quasi-elastic rescattering.

###  3.3 Global analysis

As shown in the last subsection, the shift of $a_{1}(m_{b})$ away from the
QCDF prediction depends on the values of $|a_{2}^{\mathrm{eff}}|$,
$\mathrm{arg}(a_{2}^{\mathrm{eff}})$ and $\delta^{\prime}$. The other decay
channels listed in tables 1, 2 and 3 provide constraints on these parameters
under the $SU(3)$ and $U(3)$ flavor symmetries. In particular, the decays with
the amplitude $T+C$, such as $B^{-}\to D^{0}\pi^{-}$, give an information on
correlations among $a_{1}(m_{b})$, $|a_{2}^{\mathrm{eff}}|$ and
$\mathrm{arg}(a_{2}^{\mathrm{eff}})$. It is noted that they do not receive
rescattering contribution.

In the analysis, the global-fitting package HEPfit deBlas:2019okz  is used
with our own implementation of the $\overline{B}_{(s)}\to D^{(*)}_{(s)}M$
observables to investigate the constraints from the available data. The
statistical analysis of the package is based on the BAT library
Caldwell:2008fw , which allows us to evaluate the posterior probability
distributions of the parameters based on the Bayesian statistical inference
with the Markov Chain Monte Carlo (MCMC).

The following two cases are considered: (i) a fit with only the six decay
modes that have the amplitude $T$, $C$ or $T+C$, and (ii) a fit with all the
modes for which the experimental data are available. The four parameters
$a_{1}(m_{b})$, Re$(a_{2}^{\mathrm{eff}})$, Im$(a_{2}^{\mathrm{eff}})$ and
$\delta^{\prime}$ are floated in the former case, while the nine parameters
$a_{1}(m_{b})$, Re$(a_{2}^{\mathrm{eff}})$, Im$(a_{2}^{\mathrm{eff}})$,
Re$(E)$, Im$(E)$, $\delta^{\prime}$, $\theta$, $\sigma$ and $\tau$ are floated
in the latter. Flat priors are assumed for the fit parameters. Other
parameters are fixed to be constant in the fits. The experimental constraints
are listed in tables 1, 2 and 3.777A recent data $\mathcal{B}(B^{-}\to
D^{*0}\pi^{-})=(53.5\pm 2.2)\times 10^{-4}$ from LHCb LHCb:2020hdx  is not
included. The data for $\mathcal{B}(\overline{B}_{s}\to D_{s}^{(*)+}K^{-})$
are not included in the fits, because they are the sum of the branching ratios
of $b\to c\bar{u}s$ and $b\to u\bar{c}s$ transitions. The branching ratio for
$\overline{B}_{s}\to D_{s}^{(*)-}K^{+}$ is expected to be of the same order of
magnitude as that for $\overline{B}_{s}\to D_{s}^{(*)+}K^{-}$.

Figure 4: Two-dimensional probability distributions of $a_{1}(m_{b})$ and
$|a_{2}^{\mathrm{eff}}|$. The darker (lighter) regions correspond to the 68%
(95%) probability. For the the rescattering parameters in $\overline{B}\to
DV$, the solution 1 is adopted.

In figure 4 the two-dimensional probability distributions of $a_{1}(m_{b})$
and $|a_{2}^{\mathrm{eff}}|$ are presented for the fit (i) in the left plots
and for the fit (ii) in the right plots. The solution 1 in eq. (26) is adopted
for $\overline{B}\to DV$. The results for the solution 2 in eq. (27), which
are not presented in figure 4, become similar to those for the solution 1
without the rescattering, since the rescattering parameter $\delta^{\prime}$
is fixed to zero. In each plot, the blue (red) region shows the fit result
with (without) the rescattering. Here and in the following figures the darker
(lighter) region corresponds to the 68% (95%) probability. By comparing the
left plots in figure 4 with the plots in figure 3, it is found that the
preferred regions are diminished partly by including the $T+C$ modes. The
regions in the right plots are basically similar to those in the left plots
except for those of $\overline{B}\to DV$. Thus, the fit is not improved even
if the decay modes with the exchange amplitude $E$ are included, because the
additional five fit parameters Re$(E)$, Im$(E)$, $\theta$, $\sigma$ and $\tau$
are introduced in the fit of $\overline{B}\to DP$ and $D^{*}P$, while the
experimental data are not precise enough to constrain those parameters. The
full one- and two-dimensional probability distributions obtained from the fits
are summarized in appendix A. In the fit results of $\overline{B}\to DP$ and
$D^{*}P$, it is observed that there are significant tensions between the
theoretical predictions and the experimental data even with including the
possible quasi-elastic rescattering effects.

Figure 5: Probability distributions of $a_{1}(m_{b})$. The darker (lighter)
regions correspond to the 68% (95%) probability. The blue vertical band
represents the SM value of $a_{1}(m_{b})$ at the $2\hskip 0.80002pt\sigma$
level.

The marginalized one-dimensional probability distributions of $a_{1}(m_{b})$
in the fits (i) and (ii) are presented in figure 5. Here the result for the
solution 2 of $\overline{B}\to DV$ is also presented. In each plot the black
dotted line shows the result for the fit (i). The distributions for
$\overline{B}\to DP$ and $D^{*}P$ have also a peak in the low $a_{1}(m_{b})$
region, where $|a_{2}^{\mathrm{eff}}|$ is larger than $a_{1}(m_{b})$ as shown
in figure 4. The probability in the low $a_{1}(m_{b})$ region is suppressed in
the fit (ii) as presented by the red-colored distribution. It is found that
the preferred regions of $a_{1}(m_{b})$ differs from the SM value by more than
$2\hskip 0.80002pt\sigma$ in $\overline{B}\to DP$ and $D^{*}P$, and $1\hskip
0.80002pt\sigma$ in $\overline{B}\to DV$.

The NP interpretations of the deviation in $a_{1}(m_{b})$ are scrutinized in
refs. Bordone:2020gao ; Iguro:2020ndk ; Cai:2021mlt ; Bordone:2021cca . The NP
contributions of $\mathcal{O}(10\%)$ to the Wilson coefficients $C_{1}$ and
$C_{2}$ have not been excluded by the current measurements of other
observables Bobeth:2014rda ; Brod:2014bfa ; Lenz:2019lvd . Numerically, the NP
contribution to $a_{1}(m_{b})$ is given in terms of the coefficients at the NP
scale, which is taken to be 1 TeV as a reference Iguro:2020ndk :

$\displaystyle a_{1}^{\mathrm{NP}}(m_{b})=$
$\displaystyle\,-0.19\,C_{1}^{\mathrm{NP}}(1\,\mathrm{TeV})+1.07\,C_{2}^{\mathrm{NP}}(1\,\mathrm{TeV})\,,$
(52)

where the renormalization-group evolution is calculated in the leading-
logarithmic approximation. In order to achieve
$a_{1}^{\mathrm{NP}}(m_{b})\sim-0.1$, it is estimated that
$C_{2}^{\mathrm{NP}}(1\,\mathrm{TeV})\sim-0.1$ is necessary, which is about
10% of the SM value. Since the SM contribution to $C_{2}$ is generated through
a tree-level $W$-boson exchange, the requested NP particles must have strong
couplings to the light quarks. Such scenarios confront with constraints from
LHC, meson mixings and/or $K$ and also other $B$ decays Brod:2014bfa ;
Iguro:2020ndk , and thus, it is challenging in general to construct a viable
NP scenario.

##  4 Summary

Recent developments in the study of the $\overline{B}_{(s)}\to D_{(s)}$
transition form factors have shed light on the discrepancies between the
theoretical predictions and the experimental data for the color-allowed tree
decays $B(\overline{B}^{0}\to D^{(*)+}K^{-})$ and $B(\overline{B}^{0}_{s}\to
D^{(*)+}_{s}\pi^{-})$. The theoretical predictions for these branching ratios
are universally larger than the measured values. On the other hand, it has
been known that the decays with the color-suppressed tree amplitude have
smaller branching ratios in the factorization approaches than the experimental
data. This motivated us to revisit the final-state rescattering effects in the
$\overline{B}\to DP$, $\overline{B}\to D^{*}P$ and $\overline{B}\to DV$
channels.

We have examined whether the discrepancies can be explained by the quasi-
elastic rescattering contributions based on the $SU(3)$ and $U(3)$ flavor
symmetries. From the analysis of the color-allowed modes $\overline{B}^{0}\to
D^{(*)+}K^{-}$ and $\overline{B}^{0}_{s}\to D^{(*)+}_{s}\pi^{-}$ and the
color-suppressed modes $\overline{B}^{0}\to D^{(*)0}\overline{K}^{0}$ and
$\overline{B}^{0}_{s}\to D^{(*)0}K^{0}$, we have concluded that the
rescattering effects cannot account for the discrepancies. Hence there should
exist additional contributions such as large power corrections in the QCDF
approach, inelastic rescattering contributions beyond the quasi-elastic
picture, and/or NP contributions to the short-distance Wilson coefficients. We
have studied the last possibility by taking the coefficient $a_{1}(m_{b})$ as
a fit parameter. The preferred range of the NP contribution to $a_{1}(m_{b})$
has been determined from the fits to all the available data for
$\overline{B}_{(s)}\to D^{(*)}_{(s)}M$, where the fits have been performed
separately for $\overline{B}\to DP$, $\overline{B}\to D^{*}P$ and
$\overline{B}\to DV$. The results point to a downward shift of
$\mathcal{O}(10\%)$ in $a_{1}(m_{b})$ compared to the SM contribution.
Remarkably, all the fits are consistent with each other.

As argued in detail in ref. Bordone:2020gao , it is hard to resolve the
discrepancies by (unknown) higher-order or higher-power corrections to the
decay amplitudes in the QCDF approach. Effects of the inelastic rescattering
might be significant, though it is beyond the scope of this paper. On the
other hand, it is challenging to construct viable NP models that realizes the
NP contribution of $\mathcal{O}(10\%)$. Therefore, further theoretical and
experimental studies are needed to clarify this issue.

## Acknowledgements

We are grateful to Marzia Bordone, Nico Gubernari, Martin Jung and Danny van
Dyk for helpful information on their calculation of the $\overline{B}_{(s)}\to
D_{(s)}$ form factors. We also thank Teppei Kitahara for fruitful discussion.
This work is supported in part by the Japan Society for the Promotion of
Science under the Grant-in-Aid for Scientific Research on Innovative Areas
(No. 21H00086 [ME]), the Grant-in-Aid for Scientific Research B (No. 21H01086
[ME]), the Grant-in-Aid for Early-Career Scientists (No. 16K17681 [ME]),
Research Fellowships for Young Scientists (No. 19J10980 [SI]), Core-to-Core
Program (No. JPJSCCA20200002 [SI]), the World Premier International Research
Center Initiative [SI], and the Grant-in-Aid for Scientific Research C (No.
17K05429 [SM]). SI would like to thank the warm hospitality at the KEK theory
center, where he stayed during the initial stage of this project.

##  Appendix A Fit results

In this appendix we present some details of the fit results in section 3.3.

Figures 6, 7, 8, and 9 show the one-dimensional and two-dimensional
probability distributions of the parameters in the fit (ii), which uses all
the available experimental data for $\overline{B}\to DP$, $\overline{B}\to
D^{*}P$, $\overline{B}\to DV$ with the solution 1 in eq. (26), and
$\overline{B}\to DV$ with the solution 2 in eq. (27), respectively. In each
plot the darker (lighter) regions correspond to the smallest region of the 68%
(95%) probability. There is only a single rescattering parameter
$\delta^{\prime}$ or $\sigma$ in the fits to the $\overline{B}\to DV$ data,
because the $U(3)$ flavor symmetry is adopted instead of $SU(3)$. The results
for $\overline{B}\to DP$ and $\overline{B}\to D^{*}P$ have a long tail in the
one-dimensional distributions of $a_{1}(m_{b})$, which corresponds to a large
value of $a_{2}^{\mathrm{eff}}$.

Figure 6: Probability distributions of the parameters in $\overline{B}\to DP$.
The darker (lighter) regions correspond to the 68% (95%) probability. Figure
7: Same as figure 6, but for $\overline{B}\to D^{*}P$. Figure 8: Same as
figure 6, but for $\overline{B}\to DV$ with the solution 1 in eq. (26). Figure
9: Same as figure 6, but for $\overline{B}\to DV$ with the solution 2 in eq.
(27).

Figure 10: Probability distributions of $|E/T_{\mathrm{SM}}|$. The colored
regions correspond to 68% probability.

In addition, the one-dimensional distributions of $|E/T_{\mathrm{SM}}|$ are
shown in figure 10, where $T_{\mathrm{SM}}$ is the SM value of the color-
allowed tree amplitude for $\overline{B}^{0}\to D^{+}K^{-}$,
$\overline{B}^{0}\to D^{*+}K^{-}$ or $\overline{B}^{0}\to D^{+}K^{*-}$. Here
the results in the fits with (without) the rescattering are presented as the
distributions in yellow (blue). It is noted that the exchange amplitude $E$ is
power-suppressed compared to the color-allowed tree amplitude Beneke:2000ry .
The ranges of $|E/T_{\mathrm{SM}}|$ in the fits without the rescattering are
consistent with the naive power-counting expectation, while a mildly larger
value of $|E/T_{\mathrm{SM}}|$ is allowed in the fits with the rescattering.

Further experimental information on the $\overline{B}_{s}$ decays with the
$b\to c\bar{u}s$ transition listed in tables 1, 2 and 3 is helpful to obtain
stronger constraints on the fit parameters, though it would be challenging to
measure those decays at the LHCb experiment in most cases. Theoretical studies
of the color-suppressed-tree and the exchange amplitudes would also be helpful
to set stronger constraints.

## References

  * (1) J.D. Bjorken, _Topics in B Physics_ , _Nucl. Phys. B Proc. Suppl._ 11 (1989) 325.
  * (2) M. Beneke, G. Buchalla, M. Neubert and C.T. Sachrajda, _QCD factorization for exclusive, nonleptonic $B$ meson decays: General arguments and the case of heavy light final states_, _Nucl. Phys. B_ 591 (2000) 313 [hep-ph/0006124].
  * (3) C.W. Bauer, D. Pirjol and I.W. Stewart, _A Proof of factorization for $B\to D\pi$_, _Phys. Rev. Lett._ 87 (2001) 201806 [hep-ph/0107002].
  * (4) T. Huber, S. Kränkl and X.-Q. Li, _Two-body non-leptonic heavy-to-heavy decays at NNLO in QCD factorization_ , _JHEP_ 09 (2016) 112 [1606.02888].
  * (5) M. Bordone, N. Gubernari, T. Huber, M. Jung and D. van Dyk, _A puzzle in $\bar{B}_{(s)}^{0}\to D_{(s)}^{+}\\{\pi^{-},K^{-}\\}$ decays and extraction of the $f_{s}/f_{d}$ fragmentation fraction_, _Eur. Phys. J. C_ 80 (2020) 951 [2007.10338].
  * (6) M. Bordone, N. Gubernari, D. van Dyk and M. Jung, _Heavy-Quark expansion for ${{\bar{B}}_{s}\rightarrow D^{(*)}_{s}}$ form factors and unitarity bounds beyond the ${SU(3)_{F}}$ limit_, _Eur. Phys. J. C_ 80 (2020) 347 [1912.09335].
  * (7) F.-M. Cai, W.-J. Deng, X.-Q. Li and Y.-D. Yang, _Probing new physics in class-I $B$-meson decays into heavy-light final states_, 2103.04138.
  * (8) S. Iguro and T. Kitahara, _Implications for new physics from a novel puzzle in $\bar{B}_{(s)}^{0}\to D^{(\ast)+}_{(s)}\\{\pi^{-},K^{-}\\}$ decays_, _Phys. Rev. D_ 102 (2020) 071701 [2008.01086].
  * (9) H.-Y. Cheng, _Implications of recent $\overline{B}^{0}\to D^{(*)0}X^{0}$ measurements_, _Phys. Rev. D_ 65 (2002) 094012 [hep-ph/0108096].
  * (10) M. Neubert and A.A. Petrov, _Comments on color suppressed hadronic $B$ decays_, _Phys. Lett. B_ 519 (2001) 50 [hep-ph/0108103].
  * (11) Z.-z. Xing, _Determining the factorization parameter and strong phase differences in $B\to D^{(*)}\pi$ decays_, _HEPNP_ 26 (2002) 100 [hep-ph/0107257].
  * (12) Z.-z. Xing, _Final state rescattering and SU(3) symmetry breaking in $B\to DK$ and $B\to DK^{*}$ decays_, _Eur. Phys. J. C_ 28 (2003) 63 [hep-ph/0301024].
  * (13) C.-W. Chiang and J.L. Rosner, _Final state phases in $B\to D\pi,D^{*}\pi$, and $D\rho$ decays_, _Phys. Rev. D_ 67 (2003) 074013 [hep-ph/0212274].
  * (14) C.S. Kim, S. Oh and C. Yu, _Strong phase shifts and color-suppressed tree amplitudes in $B\to DK^{(*)}$ and $B\to D\pi$, $D\rho$ decays_, _Phys. Lett. B_ 621 (2005) 259 [hep-ph/0412418].
  * (15) C.-W. Chiang and E. Senaha, _Update analysis of two-body charmed $B$ meson decays_, _Phys. Rev. D_ 75 (2007) 074021 [hep-ph/0702007].
  * (16) Y.-Y. Keum, T. Kurimoto, H.N. Li, C.-D. Lu and A.I. Sanda, _Nonfactorizable contributions to $B\to D^{(*)}M$ decays_, _Phys. Rev. D_ 69 (2004) 094018 [hep-ph/0305335].
  * (17) R.-H. Li, C.-D. Lu and H. Zou, _The $B(B_{s})\to D_{s}P,D_{s}V,D^{*}_{s}P$ and $D^{*}_{s}V$ decays in the perturbative QCD approach_, _Phys. Rev. D_ 78 (2008) 014018 [0803.1073].
  * (18) S. Mantry, D. Pirjol and I.W. Stewart, _Strong phases and factorization for color suppressed decays_ , _Phys. Rev. D_ 68 (2003) 114009 [hep-ph/0306254].
  * (19) A.E. Blechman, S. Mantry and I.W. Stewart, _Heavy quark symmetry in isosinglet nonleptonic B-decays_ , _Phys. Lett. B_ 608 (2005) 77 [hep-ph/0410312].
  * (20) I.E. Halperin, _Soft gluon suppression of $1/N_{c}$ contributions in color suppressed heavy meson decays_, _Phys. Lett. B_ 349 (1995) 548 [hep-ph/9411422].
  * (21) J.-Y. Cui and Z.-H. Li, _Soft nonfactorizable contribution to $\bar{B}^{0}\to D^{0}\pi^{0}$_, _Eur. Phys. J. C_ 38 (2004) 187 [hep-ph/0410029].
  * (22) J.F. Donoghue, E. Golowich, A.A. Petrov and J.M. Soares, _Systematics of soft final state interactions in B decay_ , _Phys. Rev. Lett._ 77 (1996) 2178 [hep-ph/9604283].
  * (23) M. Suzuki and L. Wolfenstein, _Final state interaction phase in $B$ decays_, _Phys. Rev. D_ 60 (1999) 074019 [hep-ph/9903477].
  * (24) C.-K. Chua, W.-S. Hou and K.-C. Yang, _Final state rescattering and color suppressed $\overline{B}^{0}\to D^{(*)0}h^{0}$ decays_, _Phys. Rev. D_ 65 (2002) 096007 [hep-ph/0112148].
  * (25) Fayyazuddin, _Final state interactions and $\Delta S=-1$, $\Delta C=\pm 1$ $B$ decays_, _JHEP_ 09 (2002) 055 [hep-ph/0206164].
  * (26) L. Wolfenstein, _Strong phases in the decays $B\to D\pi$_, _Phys. Rev. D_ 69 (2004) 016006 [hep-ph/0309166].
  * (27) G. Calderon, J.M. Gerard, J. Pestieau and J. Weyers, _Relating final state interactions in $B\to D\pi$ and $B\to DK$_, _Phys. Lett. B_ 588 (2004) 81 [hep-ph/0311163].
  * (28) Fayyazuddin, _Final state phases in $B\to D\pi$, $\bar{D}\pi$ decays and CP asymmetry_, _Phys. Rev. D_ 70 (2004) 114018 [hep-ph/0402189].
  * (29) L. Wolfenstein, _Regge amplitudes and final state phases in the decays $B\to D\pi$_, hep-ph/0407344.
  * (30) H.-Y. Cheng, C.-K. Chua and A. Soni, _Final state interactions in hadronic $B$ decays_, _Phys. Rev. D_ 71 (2005) 014030 [hep-ph/0409317].
  * (31) C.-K. Chua and W.-S. Hou, _Implications of $\overline{B}\to D^{0}h^{0}$ decays on $\overline{B}\to D\overline{K}$, $\overline{D}\overline{K}$ decays_, _Phys. Rev. D_ 72 (2005) 036002 [hep-ph/0504084].
  * (32) C.-K. Chua and W.-S. Hou, _Rescattering effects in $\overline{B}_{u,d,s}\to DP$, $\overline{D}P$ decays_, _Phys. Rev. D_ 77 (2008) 116001 [0712.1882].
  * (33) M. Suzuki, _Inelastic final-state interaction_ , _Phys. Rev. D_ 77 (2008) 054021 [0710.5534].
  * (34) S. Iguro and R. Watanabe, _Bayesian fit analysis to full distribution data of $\overline{B}\to{D}^{\left(\ast\right)}\ell\overline{\nu}:\left|{V}_{cb}\right|$ determination and new physics constraints_, _JHEP_ 08 (2020) 006 [2004.10208].
  * (35) A. Kobach, _Continuity and semileptonic $B_{(s)}\rightarrow D_{(s)}$ form factors_, _Phys. Lett. B_ 809 (2020) 135708 [1910.13024].
  * (36) Particle Data Group collaboration, _Review of Particle Physics_ , _PTEP_ 2020 (2020) 083C01.
  * (37) N. Cabibbo, _Unitary Symmetry and Leptonic Decays_ , _Phys. Rev. Lett._ 10 (1963) 531.
  * (38) M. Kobayashi and T. Maskawa, _CP Violation in the Renormalizable Theory of Weak Interaction_ , _Prog. Theor. Phys._ 49 (1973) 652.
  * (39) K.G. Chetyrkin, M. Misiak and M. Munz, _Weak radiative B meson decay beyond leading logarithms_ , _Phys. Lett. B_ 400 (1997) 206 [hep-ph/9612313].
  * (40) K.G. Chetyrkin, M. Misiak and M. Munz, _$|\Delta F|=1$ nonleptonic effective Hamiltonian in a simpler scheme_, _Nucl. Phys. B_ 520 (1998) 279 [hep-ph/9711280].
  * (41) C.-K. Chua, _Revisiting final state interaction in charmless $B_{q}\to PP$ decays_, _Phys. Rev. D_ 97 (2018) 093004 [1802.00155].
  * (42) D. Zeppenfeld, _SU(3) Relations for B Meson Decays_ , _Z. Phys. C_ 8 (1981) 77.
  * (43) M.J. Savage and M.B. Wise, _SU(3) Predictions for Nonleptonic B Meson Decays_ , _Phys. Rev. D_ 39 (1989) 3346.
  * (44) M. Gronau, O.F. Hernandez, D. London and J.L. Rosner, _Broken SU(3) symmetry in two-body $B$ decays_, _Phys. Rev. D_ 52 (1995) 6356 [hep-ph/9504326].
  * (45) P. Colangelo and R. Ferrandes, _Model independent analysis of a class of $\overline{B}_{s}^{0}$ decay modes_, _Phys. Lett. B_ 627 (2005) 77 [hep-ph/0508033].
  * (46) R. Fleischer, N. Serra and N. Tuning, _Tests of Factorization and $SU(3)$ Relations in $B$ Decays into Heavy-Light Final States_, _Phys. Rev. D_ 83 (2011) 014017 [1012.2784].
  * (47) R. Escribano, S. Gonzàlez-Solís, P. Masjuan and P. Sanchez-Puertas, _$\eta^{\prime}$ transition form factor from space- and timelike experimental data_, _Phys. Rev. D_ 94 (2016) 054033 [1512.07520].
  * (48) HFLAV collaboration, _Averages of b-hadron, c-hadron, and $\tau$-lepton properties as of 2018_, _Eur. Phys. J. C_ 81 (2021) 226 [1909.12524].
  * (49) N. Gubernari, A. Kokulu and D. van Dyk, _$B\to P$ and $B\to V$ Form Factors from $B$-Meson Light-Cone Sum Rules beyond Leading Twist_, _JHEP_ 01 (2019) 150 [1811.00983].
  * (50) A. Bazavov et al., _$B$ \- and $D$-meson leptonic decay constants from four-flavor lattice QCD_, _Phys. Rev. D_ 98 (2018) 074512 [1712.09262].
  * (51) Flavour Lattice Averaging Group collaboration, _FLAG Review 2019: Flavour Lattice Averaging Group (FLAG)_ , _Eur. Phys. J. C_ 80 (2020) 113 [1902.08191].
  * (52) A. Bharucha, D.M. Straub and R. Zwicky, _$B\to V\ell^{+}\ell^{-}$ in the Standard Model from light-cone sum rules_, _JHEP_ 08 (2016) 098 [1503.05534].
  * (53) P. Gelhausen, A. Khodjamirian, A.A. Pivovarov and D. Rosenthal, _Decay constants of heavy-light vector mesons from QCD sum rules_ , _Phys. Rev. D_ 88 (2013) 014015 [1305.5432].
  * (54) P. Ball and R. Zwicky, _New results on $B\to\pi,K,\eta$ decay formfactors from light-cone sum rules_, _Phys. Rev. D_ 71 (2005) 014015 [hep-ph/0406232].
  * (55) G. Duplancic and B. Melic, _Form factors of $B,B_{s}\to\eta^{(^{\prime})}$ and $D,D_{s}\to\eta^{(^{\prime})}$ transitions from QCD light-cone sum rules_, _JHEP_ 11 (2015) 138 [1508.05287].
  * (56) A. Khodjamirian and A.V. Rusov, _$B_{s}\to K\ell\nu_{\ell}$ and $B_{(s)}\to\pi(K)\ell^{+}\ell^{-}$ decays at large recoil and CKM matrix elements_, _JHEP_ 08 (2017) 112 [1703.04765].
  * (57) Belle collaboration, _Precise determination of the CKM matrix element $\left|V_{cb}\right|$ with $\bar{B}^{0}\to D^{*\,+}\,\ell^{-}\,\bar{\nu}_{\ell}$ decays with hadronic tagging at Belle_, 1702.01521.
  * (58) Belle collaboration, _Measurement of the CKM matrix element $|V_{cb}|$ from $B^{0}\to D^{*-}\ell^{+}\nu_{\ell}$ at Belle_, _Phys. Rev. D_ 100 (2019) 052007 [1809.03290].
  * (59) M. Beneke, P. Böer, G. Finauri and K.K. Vos, _QED factorization of two-body non-leptonic and semi-leptonic $B$ to charm decays_, 2107.03819.
  * (60) J. De Blas et al., _HEPfit : a code for the combination of indirect and direct constraints on high energy physics models_, _Eur. Phys. J. C_ 80 (2020) 456 [1910.14012].
  * (61) A. Caldwell, D. Kollar and K. Kroninger, _BAT: The Bayesian Analysis Toolkit_ , _Comput. Phys. Commun._ 180 (2009) 2197 [0808.2552].
  * (62) LHCb collaboration, _Measurement of CP observables in $B^{\pm}\to D^{(*)}K^{\pm}$ and $B^{\pm}\to D^{(*)}\pi^{\pm}$ decays using two-body $D$ final states_, _JHEP_ 04 (2021) 081 [2012.09903].
  * (63) M. Bordone, A. Greljo and D. Marzocca, _Exploiting dijet resonance searches for flavor physics_ , 2103.10332.
  * (64) C. Bobeth, U. Haisch, A. Lenz, B. Pecjak and G. Tetlalmatzi-Xolocotzi, _On new physics in $\Delta\Gamma_{d}$_, _JHEP_ 06 (2014) 040 [1404.2531].
  * (65) J. Brod, A. Lenz, G. Tetlalmatzi-Xolocotzi and M. Wiebusch, _New physics effects in tree-level decays and the precision in the determination of the quark mixing angle $\gamma$_, _Phys. Rev. D_ 92 (2015) 033002 [1412.1446].
  * (66) A. Lenz and G. Tetlalmatzi-Xolocotzi, _Model-independent bounds on new physics effects in non-leptonic tree-level decays of B-mesons_ , _JHEP_ 07 (2020) 177 [1912.07621].