# The Decays of $B^{+}\rightarrow{\bar{D}^{0}}+D^{+}_{sJ}(2S)$ and
$B^{+}\rightarrow\bar{D}^{0}+D^{+}_{sJ}(1D)$

Guo-Li<EMAIL_ADDRESS>Jin-Mei Zhang and Zhi-Hui Wang

Department of Physics, Harbin Institute of Technology, Harbin 150001, China

> ABSTRACT
>
> We analyzed the decays of $B^{+}\rightarrow{\bar{D}^{0}}+D^{+}_{sJ}(2S)$ and
> $B^{+}\rightarrow\bar{D}^{0}+D^{+}_{sJ}(1D)$ by naive factorization method
> and model dependent calculation based on the Bethe-Salpeter method, the
> branching ratios are $Br({\bar{D}^{0}}D^{+}_{sJ}(2S))=(0.72\pm 0.12)\%$ and
> $Br({\bar{D}^{0}}D^{+}_{sJ}(1D))=(0.027\pm 0.007)\%$. The branching ratio of
> decay
> $B^{+}\rightarrow{\bar{D}^{0}}D^{+}_{sJ}(2S)\rightarrow{\bar{D}^{0}}D^{0}K^{+}$
> consist with the data of Belle Collaboration, so we conclude that the new
> state $D^{+}_{sJ}(2700)$ is the first excited state $D^{+}_{sJ}(2S)$.

BABAR Collaboration reported the observation of a new charmed meson called
$D^{+}_{sJ}(2690)$ with a mass $2688\pm 4\pm 3$ MeV and with a broad width
$\Gamma=112\pm 7\pm 36$ MeV [1], Belle Collaboration also reported a charmed
state $D^{+}_{sJ}(2700)$, $M=2715\pm 11^{+11}_{-14}$ MeV with width
$\Gamma=115\pm 20^{+36}_{-32}$ MeV [2], later modified to $M=2708\pm
9^{+11}_{-10}$ MeV and $\Gamma=108\pm 23^{+36}_{-31}$ MeV [3]. Most authors
believe that these two states should be one state, and the most possible
candidate is the $2S$ or the $S-D$ ($2~{}^{3}S_{1}-1~{}^{3}D_{1}$) mixing
$c\bar{s}$ $1^{-}$ state [4, 5, 6, 7, 8, 9, 10, 11].

Recently, we have analyzed the decays of the $D^{+}_{sJ}(2S)$ and
$D^{+}_{sJ}(1D)$, we concluded that one can not distinguish them from their
decays, because they have the similar decay channels and the corresponding
decay widths are comparable [12]. In this paper, we give the calculations of
the decays $B^{+}\rightarrow{\bar{D}^{0}}+D^{+}_{sJ}(2S)$ and
$B^{+}\rightarrow\bar{D}^{0}+D^{+}_{sJ}(1D)$, we find that we can separate
them, and according to the current experimental data of Belle Collaboration,
we conclude that the new state $D^{+}_{sJ}(2700)$ is the first excited state
$D^{+}_{sJ}(2S)$. These channels have also been considered in literatures, for
example, Close et al [4, 23] give branching ratios and possible mixing of $2S$
and $1D$; Colangelo et al [7] assume the new state is the $2S$ state, and
extract the decay constant.

The decay amplitude for $B^{+}\rightarrow{\bar{D}^{0}}+D^{+}_{sJ}$ can be
described in the naive factorization approach:

$\displaystyle T=\frac{G_{F}}{\sqrt{2}}V_{cs}V_{cb}a_{1}\langle
D_{{}_{sJ}}^{+}|J_{\mu}|0\rangle\langle{\bar{D}^{0}}|J^{\mu}|B^{+}\rangle,$
(1)

where the CKM matrix element $V_{cs}=0.97334$ and $V_{cb}=0.0415$ [13], since
we focus on the difference between $D^{+}_{sJ}(2S)$ and $D^{+}_{sJ}(1D)$, not
on the careful study, we have chosen $a_{1}=c_{1}+\frac{1}{3}c_{2}=1$, where
$c_{1}$ and $c_{2}$ are the Wilson coefficients [14]. We also delete other
higher order contributions like the contributions from penguin operators. And

$\displaystyle\langle
D_{sJ}^{+}|J_{\mu}|0\rangle=iF^{*}_{V}M_{{D_{{}_{sJ}}^{+}}}\epsilon^{\lambda}_{\mu}~{},$
(2)

$F_{V}$ and $\epsilon^{\lambda}_{\mu}$ are the decay constant and polarization
vector of the meson $D_{{}_{sJ}}^{+}$, respectively.

We have solved the exact instantaneous Bethe-Salpeter equations [15] (or the
Salpeter equations [16]) for $1^{-}$ states, the general form for the
relativistic Salpeter wave function of vector $1^{-}$ state can be written as
[17, 18]:

$\varphi_{1^{-}}^{\lambda}(q_{\perp})=q_{\perp}\cdot{\epsilon}^{\lambda}_{\perp}\left[f_{1}(q_{\perp})+\frac{\not\\!P}{M}f_{2}(q_{\perp})+\frac{{\not\\!q}_{\perp}}{M}f_{3}(q_{\perp})+\frac{{\not\\!P}{\not\\!q}_{\perp}}{M^{2}}f_{4}(q_{\perp})\right]+M{\not\\!\epsilon}^{\lambda}_{\perp}f_{5}(q_{\perp})$
$+{\not\\!\epsilon}^{\lambda}_{\perp}{\not\\!P}f_{6}(q_{\perp})+({\not\\!q}_{\perp}{\not\\!\epsilon}^{\lambda}_{\perp}-q_{\perp}\cdot{\epsilon}^{\lambda}_{\perp})f_{7}(q_{\perp})+\frac{1}{M}({\not\\!P}{\not\\!\epsilon}^{\lambda}_{\perp}{\not\\!q}_{\perp}-{\not\\!P}q_{\perp}\cdot{\epsilon}^{\lambda}_{\perp})f_{8}(q_{\perp}),$
(3)

where the $P$, $q$ and ${\epsilon}^{\lambda}_{\perp}$ are the momentum,
relative inner momentum and polarization vector of the vector meson,
respectively; $f_{i}(q_{\perp})$ is the function of $-q_{\perp}^{2}$, and we
have used the notation $q^{\mu}_{\perp}\equiv q^{\mu}-(P\cdot q/M^{2})P^{\mu}$
(which is $(0,~{}\vec{q})$ in the center of mass system).

The $8$ wave functions $f_{i}$ are not independent, there are only $4$
independent wave functions [18], and we have the relations

$f_{1}(q_{\perp})=\frac{\left[q_{\perp}^{2}f_{3}(q_{\perp})+M^{2}f_{5}(q_{\perp})\right](m_{1}m_{2}-\omega_{1}\omega_{2}+q_{\perp}^{2})}{M(m_{1}+m_{2})q_{\perp}^{2}},~{}~{}~{}f_{7}(q_{\perp})=\frac{f_{5}(q_{\perp})M(-m_{1}m_{2}+\omega_{1}\omega_{2}+q_{\perp}^{2})}{(m_{1}-m_{2})q_{\perp}^{2}},$
$f_{2}(q_{\perp})=\frac{\left[-q_{\perp}^{2}f_{4}(q_{\perp})+M^{2}f_{6}(q_{\perp})\right](m_{1}\omega_{2}-m_{2}\omega_{1})}{M(\omega_{1}+\omega_{2})q_{\perp}^{2}},~{}~{}~{}f_{8}(q_{\perp})=\frac{f_{6}(q_{\perp})M(m_{1}\omega_{2}-m_{2}\omega_{1})}{(\omega_{1}-\omega_{2})q_{\perp}^{2}}.$

In our method, the $S-D$ mixing automatically exist in the wave function of
$1^{-}$ state, because we give the whole wave function Eq. (3) which is
$J^{P}=1^{-}$, but some of the wave function are $S$ wave, some are $D$ wave,
for example, see Figure 1-3, we show wave functions. One can see that, for
$1S$ and $2S$, the wave function $f_{5}$ and $f_{6}$ are dominate, and they
are $S$ wave, but there is little $D$ wave mixing in these two states which
come from the terms $f_{3}$ and $f_{4}$. But for the third state, we labeled
as $1D$ state, the terms $f_{3}$ and $f_{4}$ are dominate, but these two terms
are not pure $D$ wave, they will give contribute as a $S$ wave [17]. So we
conclude that the $1S$ and $2S$ states are $S$ wave dominate states, mixed
with a little $D$ wave (come from the terms $f_{3}$ and $f_{4}$), while the
$1D$ state is a $D$ wave dominate state ($f_{3}$ and $f_{4}$), mixed with a
valuable part of $S$ wave (still come from the terms $f_{3}$ and $f_{4}$).

Figure 1: wave functions of $D_{s}^{*}(1S)$. Figure 2: wave functions of
$D_{s}^{*}(2S)$. Figure 3: wave functions of $D_{s}^{*}(1D)$.

For $c\bar{s}$ vector $1^{-}$ state, our mass prediction for the first radial
excited $2S$ state is $2673$ MeV, and for $1D$, our result is $2718$ MeV [18],
so there are two states around $2700$ MeV.

In Ref. [18], we only give the leading order calculation for decay constant,
which is
$F_{V}=4\sqrt{N_{c}}\int\frac{d{\vec{q}}}{(2\pi)^{3}}f_{5}({\vec{q}})$, the
whole equation should be

$F_{V}=4\sqrt{N_{c}}\int\frac{d{\vec{q}}}{(2\pi)^{3}}(f_{5}-\frac{{\vec{q}}^{2}f_{3}}{3M^{2}}),$
(4)

and our results of the decay constants for vector $c\bar{s}$ system are:

$F_{V}(1S)=353\pm 21~{}~{}\rm MeV,$ $F_{V}(2S)=295\pm 13~{}~{}\rm MeV,$
$F_{V}(1D)=57.1\pm 5.1~{}~{}\rm MeV.$

Where the uncertainties are given by varying all the input parameters
simultaneously within $\pm 5\%$ of the central values in our model, and we
will calculate all the uncertainties this way in this letter. The center
values of $S$ waves are little smaller than the estimates in Ref. [18], where
$F_{V}(1S)=375\pm 24~{}\rm MeV$ and $F_{V}(2S)=312\pm 17~{}\rm MeV$, but the
value of $D$ wave decay constant is much smaller than the predict in Ref. [18]
where we did not shown it, this is because we should not ignore the
contribution from the term of $f_{3}$ when consider a $D$ wave state. Our
estimate of $F_{V}(1S)=353\pm 23~{}\rm MeV$ is close to the newer result
$F_{D^{*}_{s}}=268\sim 290~{}\rm MeV$ by Choi [19]. Our estimate of
$F_{V}(2S)=312\pm 17~{}\rm MeV$ is a little larger than the estimate
$F_{V}(2S)=243\pm 41~{}\rm MeV$ in Ref. [7], where they extracted it from the
decay $B^{+}\rightarrow{\bar{D}^{0}}+D^{+}_{sJ}(2700)$ assuming the new state
$D^{+}_{sJ}(2700)$ is the first radial excited state $D^{+}_{sJ}(2S)$.

According to the Mandelstam formalism [20], at the leading order, the
transition amplitude for $B^{+}\rightarrow\bar{D^{0}}$ can be written as [21]:

$\displaystyle\langle{\bar{D}^{0}}|J^{\mu}|B^{+}\rangle=\int\frac{d{\vec{q}}}{(2\pi)^{3}}Tr\left[\bar{\varphi}^{++}_{{}_{\bar{D}^{0}}}({\vec{q}}-\frac{m_{u}}{m_{c}+m_{u}}{\vec{r}})\frac{\not\\!P}{M}{\varphi}^{++}_{{}_{B^{+}}}({\vec{q}})\gamma^{\mu}(1-\gamma_{5})\right],$
(5)

where $\vec{r}$ is the recoil three dimensional momentum of the final state
$\bar{D}^{0}$ meson, ${\varphi}^{++}$ is called the positive energy wave
function, and
$\bar{\varphi}^{++}_{{}_{\bar{D}^{0}}}=\gamma_{0}({\varphi}^{++}_{{}_{\bar{D}^{0}}})^{+}\gamma_{0}$.

The wave function forms for pseudoscalar $B^{+}$ and $\bar{D^{0}}$ are
similar, for example, the wave function for $B^{+}$ can be written as [22]

$\varphi^{++}_{{}_{B^{+}}}(\stackrel{{\scriptstyle\rightarrow}}{{q}})=\frac{M_{{B^{+}}}}{2}\left(\varphi_{1}(\stackrel{{\scriptstyle\rightarrow}}{{q}})+\varphi_{2}(\stackrel{{\scriptstyle\rightarrow}}{{q}})\frac{m_{u}+m_{b}}{\omega_{u}+\omega_{b}}\right)$
$\times\left[\frac{\omega_{u}+\omega_{b}}{m_{u}+m_{b}}+{\gamma_{{}_{0}}}-\frac{{\not\\!\vec{q}}(m_{u}-m_{b})}{m_{b}\omega_{u}+m_{u}\omega_{b}}+\frac{{\not\\!\vec{q}}\gamma_{{}_{0}}(\omega_{u}+\omega_{b})}{(m_{b}\omega_{u}+m_{u}\omega_{b})}\right]\gamma_{{}_{5}}\;,$
(6)

where $\omega_{u}=\sqrt{m_{u}^{2}+{\vec{q}}^{2}}$ and
$\omega_{b}=\sqrt{m_{b}^{2}+{\vec{q}}^{2}}$; $\varphi_{1}(\vec{q})$,
$\varphi_{2}(\vec{q})$ are the radial part wave functions, and their numerical
values can be obtained by solving the full Salpter equation of $0^{-}$ state
[22].

The decay width is:

$\Gamma=\frac{1}{8\pi}\frac{|\vec{p}_{{}_{f_{2}}}|}{M^{2}_{B}}|T|^{2}$
$\displaystyle=\frac{1}{8\pi}\frac{|\vec{p}_{{}_{f_{2}}}|}{M^{2}_{B}}\frac{G^{2}_{F}V^{2}_{cs}V^{2}_{bc}}{2}F^{2}_{V}M^{2}_{f_{2}}X,$
(7)

where $\vec{p}_{{}_{f_{2}}}$ and $M_{f_{2}}$ are the three dimensional
momentum and mass of the final new state $D^{*+}_{sJ}$. $M^{2}_{f_{2}}$ come
from the definition of decay constant in Eq.(2), but the square of
polarization vector and transition amplitude of Eq.(5) are symbolized as $X$.

So our result are:

$\displaystyle\Gamma(B^{+}\rightarrow{\bar{D}^{0}}D^{*+}_{s}(2S))=F^{2}_{V}(2S)\times(3.50\pm
0.38)\times 10^{-14}~{}\rm GeV,$ (8)
$\displaystyle\Gamma(B^{+}\rightarrow{\bar{D}^{0}}D^{*+}_{s}(1D))=F^{2}_{V}(1D)\times(3.25\pm
0.30)\times 10^{-14}~{}\rm GeV.$ (9)

If we ignore the mass difference between $D^{*+}_{s}(2S)$ and
$D^{*+}_{s}(1D)$, and use $2700$ MeV as input, the results become

$\displaystyle\Gamma(B^{+}\rightarrow{\bar{D}^{0}}D^{*+}_{s}(2S))=F^{2}_{V}(2S)\times(3.36\pm
0.25)\times 10^{-14}~{}\rm GeV,$ (10)
$\displaystyle\Gamma(B^{+}\rightarrow{\bar{D}^{0}}D^{*+}_{s}(1D))=F^{2}_{V}(1D)\times(3.36\pm
0.25)\times 10^{-14}~{}\rm GeV.$ (11)

So the difference between this two channel mainly come from the difference of
decay constants. Then our predictions of branching ratios are:

$\displaystyle Br(B^{+}\rightarrow{\bar{D}^{0}}D^{*+}_{s}(2S))=(0.72\pm
0.12)\%,$ (12) $\displaystyle
Br(B^{+}\rightarrow{\bar{D}^{0}}D^{*+}_{s}(1D))=(0.027\pm 0.007)\%.$ (13)

In Ref. [12], we have calculated the main decay channels of $D^{+}_{sJ}(2S)$
and $D^{+}_{sJ}(1D)$, and we have the following estimates:

$Br(D^{+}_{sJ}(2S)\rightarrow D^{0}K^{+})=0.20\pm 0.03$

and

$Br(D^{+}_{sJ}(1D)\rightarrow D^{0}K^{+})=0.32\pm 0.04,$

so we obtain:

$Br(B^{+}\rightarrow{\bar{D}^{0}}D^{+}_{sJ}(2S))\times
Br(D^{+}_{sJ}(2S)\rightarrow D^{0}K^{+})=(1.4\pm 0.5)\times 10^{-3}$ (14)

and

$Br(B^{+}\rightarrow{\bar{D}^{0}}D^{+}_{sJ}(1D))\times
Br(D^{+}_{sJ}(1D)\rightarrow D^{0}K^{+})=(0.9\pm 0.3)\times 10^{-4}.$ (15)

Our estimate of $Br(D^{+}_{sJ}(2S)\rightarrow D^{0}K^{+})\simeq 0.20$ is
larger than than the estimate $0.11$ in Ref. [4] and the estimate $0.05$ in
Ref. [5], if we use their value as input, we will obtain a smaller value of
$Br(B^{+}\rightarrow{\bar{D}^{0}}D^{+}_{sJ}(2S))\times
Br(D^{+}_{sJ}(2S)\rightarrow D^{0}K^{+})$. But our estimate of
$Br(D^{+}_{sJ}(1D)\rightarrow D^{0}K^{+})\simeq 0.32$ consist with the value
$0.34$ in Ref. [5]. One can also see that, the final branching ratios depend
strongly on the decay constants, but at this time few papers have calculated
the values of decay constants for $D^{+}_{sJ}(2S)$ and $D^{+}_{sJ}(1D)$. In
Ref. [7], they assumed that the new state $D^{+}_{sJ}(2700)$ is
$D^{+}_{sJ}(2S)$, and they extracted decay constant of $D^{+}_{sJ}(2S)$:
$F_{D^{+}_{sJ}(2S)}=243\pm 41$ MeV.

The Belle Collaboration have the data [3, 13]

$Br(B^{+}\rightarrow{\bar{D}^{0}}D^{+}_{sJ}(2700))\times
Br(D^{+}_{sJ}(2700)\rightarrow D^{0}K^{+})=(1.13^{~{}+0.26}_{~{}-0.36})\times
10^{-3}.$

Because we only calculated the decay widths of six main channels for
$D^{+}_{sJ}(2S)$ ($D^{+}_{sJ}(1D)$), and based on the summed width of these
six channels, not full width, we give a relative larger branching ratio
$Br(D^{+}_{sJ}(2S)\rightarrow D^{0}K^{+})$ and $Br(D^{+}_{sJ}(1D)\rightarrow
D^{0}K^{+})$, the real branching ratios should be smaller than our estimates,
so our estimate of $B^{+}$ decay to $D^{+}_{sJ}(2S)$ is close to the data,
while the estimate of $B^{+}$ decay to $D^{+}_{sJ}(1D)$ is much smaller than
the data, then we can draw a conclusion that the new state $D^{+}_{sJ}(2700)$
is $D^{+}_{sJ}(2S)$.

We have another method to estimate the branching ratio, because the mass of
$B^{+}$ is much heavier than the mass of ${\bar{D}^{0}}$, and the mass of
$D^{+}_{sJ}(2700)$ is close to the mass of $D^{*+}_{s}$, as a rough estimate,
we ignore the mass difference of $D^{+}_{sJ}(2700)$ and $D^{*+}_{s}(2112)$,
from Eq.(1) and Eq.(2), then we have

$\frac{Br(B^{+}\rightarrow{\bar{D}^{0}}D^{+}_{sJ}(2S))}{Br(B^{+}\rightarrow{\bar{D}^{0}}D^{*+}_{s}(1S))}\simeq\frac{F^{2}_{V}(2S)}{F^{2}_{V}(1S)},$
(16)

and from Particle Data Group [13], the branching ratio of
$B^{+}\rightarrow{\bar{D}^{0}}+D^{+}_{sJ}(1S)$:

$Br(B^{+}\rightarrow{\bar{D}^{0}}D^{*+}_{s}(1S))=(7.8\pm 1.6)\times 10^{-3}.$

So we have:

$Br(B^{+}\rightarrow{\bar{D}^{0}}D^{*+}_{s}(2S))\simeq\frac{F^{2}_{V}(2S)}{F^{2}_{V}(1S)}\times(7.8\pm
1.6)\times 10^{-3}\simeq(5.4\pm 1.7)\times 10^{-3},$ (17)
$Br(B^{+}\rightarrow{\bar{D}^{0}}D^{*+}_{s}(1D))\simeq\frac{F^{2}_{V}(1D)}{F^{2}_{V}(1S)}\times(7.8\pm
1.6)\times 10^{-3}\simeq(2.0\pm 1.0)\times 10^{-4}.$ (18)

This rough estimate results are very close to our calculations (Eq.(12) and
Eq.(13)), so we have the same conclusion that $D^{+}_{sJ}(2700)$ is
$D^{+}_{sJ}(2S)$.

In Ref. [12], we estimate the full widths of $D^{+}_{sJ}(2S)$ and
$D^{+}_{sJ}(1D)$ by six main decay channels, the estimated full widths are
$46.4\pm 6.2$ MeV for $D^{+}_{sJ}(2S)$, $73.0\pm 10.4$ MeV for
$D^{+}_{sJ}(1D)$, comparing with experimental data, $\Gamma=112\pm 7\pm 36$
MeV for $D^{+}_{sJ}(2690)$ and $\Gamma=108\pm 23^{+36}_{-31}$ MeV for
$D^{+}_{sJ}(2700)$, there is still the possible that there are two states
around $2700$ MeV, one is the $S$ wave dominate $D^{+}_{sJ}(2S)$, the other is
$D$ wave dominate $D^{+}_{sJ}(1D)$.

As summary, from the decays $B^{+}\rightarrow{\bar{D}^{0}}+D^{+}_{sJ}(2S)$ and
$B^{+}\rightarrow\bar{D}^{0}+D^{+}_{sJ}(1D)$, we conclude that the new state
$D^{+}_{sJ}(2700)$ is the first radial excited state $D^{+}_{sJ}(2S)$, and
there may exist another state around $2700$, $D^{+}_{sJ}(1D)$, with a mass
around $2718$ MeV, and a width $73.0\pm 10.4$ MeV.

Acknowledgements

This work was supported in part by the National Natural Science Foundation of
China (NSFC) under Grant No. 10875032, and in part by SRF for ROCS, SEM.

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