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

0.13 MB·English

by  Guo-Li Wang

The Decays of B+ D¯0 + D+ (2S) and B+ D¯0 + D+ (1D)
sJ sJ
→ →
Guo-Li Wang∗, Jin-Mei Zhang and Zhi-Hui Wang
Department of Physics, Harbin Institute of Technology, Harbin 150001,China
0
1
0
2

n
a ABSTRACT
J

3
1
We analyzed the decays of B+ D¯0 + D+(2S) and B+ D¯0 + D+(1D) by naive fac-
] → sJ → sJ
h
p torization method and model dependent calculation based on the Bethe-Salpeter method,
-
p
the branching ratios are Br(D¯0D+(2S)) = (0.72 0.12)% and Br(D¯0D+(1D)) = (0.027
e sJ ± sJ ±
h
[ 0.007)%. The branching ratio of decay B+ D¯0D+(2S) D¯0D0K+ consist with the data
→ sJ →

1
v of Belle Collaboration, so we conclude that the new state D+(2700) is the first excited state
sJ
5
3 D+(2S).
0 sJ
2
.
1
0
0
1
:
v
i
X
r
a
∗gl [email protected]
1
BABAR Collaboration reported the observation of a new charmed meson called D+(2690) with a
sJ
mass 2688 4 3 MeV and with a broad width Γ = 112 7 36 MeV [1], Belle Collaboration also
± ± ± ±
reported a charmed state D+(2700), M = 2715 11+11 MeV with width Γ = 115 20+36 MeV [2], later
sJ ± −14 ± −32
modified to M = 2708 9+11 MeV and Γ = 108 23+36 MeV [3]. Most authors believe that these two
± −10 ± −31
states should beone state, and the most possiblecandidate is the 2S or the S D (2 3S 1 3D ) mixing
1 1
− −
−
cs¯1 state [4, 5, 6, 7, 8, 9, 10, 11].
Recently, we have analyzed the decays of the D+(2S) and D+(1D), we concluded that one can not
sJ sJ
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+ D¯0 +
→
D+(2S) and B+ D¯0 +D+(1D), we find that we can separate them, and according to the current
sJ → sJ
experimental data of Belle Collaboration, we conclude that the new state D+(2700) is the first excited
sJ
state D+(2S). These channels have also been considered in literatures, for example, Close et al [4, 23]
sJ
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+ D¯0+D+ can be described in the naive factorization approach:
→ sJ
G
T = FV V a D+ J 0 D¯0 Jµ B+ , (1)
cs cb 1 µ
√2 h sJ| | ih | | i
where the CKM matrix element V = 0.97334 and V = 0.0415 [13], since we focus on the difference
cs cb
between D+(2S) and D+(1D), not on the careful study, we have chosen a = c +1c = 1, where c and
sJ sJ 1 1 3 2 1
c are the Wilson coefficients [14]. We also delete other higher order contributions like the contributions
2
from penguin operators. And
D+ J 0 = iF∗M ǫλ , (2)
h sJ| µ| i V Ds+J µ
F and ǫλ are the decay constant and polarization vector of the meson D+, respectively.
V µ sJ
We have solved the exact instantaneous Bethe-Salpeter equations [15] (or the Salpeter equations [16])
− −
for1 states, thegeneral formfor therelativistic Salpeter wave function of vector 1 state can bewritten
as [17, 18]:
ϕλ1−(q⊥) = q⊥·ǫλ⊥(cid:20)f1(q⊥)+ M6P f2(q⊥)+ 6Mq⊥f3(q⊥)+ 6PM6q2⊥f4(q⊥)(cid:21)+M6ǫλ⊥f5(q⊥)
1
+ǫλ⊥Pf6(q⊥)+(q⊥ǫλ⊥ q⊥ ǫλ⊥)f7(q⊥)+ (Pǫλ⊥q⊥ Pq⊥ ǫλ⊥)f8(q⊥), (3)
6 6 6 6 − · M 6 6 6 −6 ·
1
wherethe P, q andǫλ arethe momentum, relative innermomentum and polarization vector of thevector
⊥
meson, respectively; fi(q⊥)is the functionof q⊥2, andwe have usedthenotation q⊥µ qµ (P q/M2)Pµ
− ≡ − ·
(which is (0, ~q) in the center of mass system).
The 8 wave functions f are not independent, there are only 4 independent wave functions [18], and
i
we have the relations
q⊥2f3(q⊥)+M2f5(q⊥) (m1m2 ω1ω2+q⊥2) f5(q⊥)M( m1m2+ω1ω2+q⊥2)
f1(q⊥)= (cid:2) M(m +(cid:3)m )q2 − , f7(q⊥)= (−m m )q2 ,
1 2 ⊥ 1 2 ⊥
−
q⊥2f4(q⊥)+M2f6(q⊥) (m1ω2 m2ω1) f6(q⊥)M(m1ω2 m2ω1)
f2(q⊥)= (cid:2)− M(ω +ω (cid:3))q2 − , f8(q⊥)= (ω ω )q−2 .
1 2 ⊥ 1 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 JP = 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 and f are dominate, and they are S wave, but there is little D wave mixing in these
5 6
two states which come from the terms f and f . But for the third state, we labeled as 1D state, the
3 4
terms f and f are dominate, but these two terms are not pure D wave, they will give contribute as a
3 4
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 and f ), while the 1D state is a D wave dominate state (f and f ),
3 4 3 4
mixed with a valuable part of S wave (still come from the terms f and f ).
3 4
−
For cs¯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.
InRef. [18],weonlygivetheleadingordercalculationfordecayconstant,whichisF = 4√N dq~ f (~q),
V c (2π)3 5
R
the whole equation should be
dq~ ~q2f
3
F = 4 N (f ), (4)
V cZ (2π)3 5− 3M2
p
and our results of the decay constants for vector cs¯system are:
F (1S) = 353 21 MeV,
V
±
F (2S) = 295 13 MeV,
V
±
F (1D) = 57.1 5.1 MeV.
V
±
Where the uncertainties are given by varying all the input parameters simultaneously within 5% of
±
the central values in our model, and we will calculate all the uncertainties this way in this letter. The
2
7
) 6
2 2 2
- V q f (|q|)/M
3
e
G 2 2
( 5 q f4(|q|)/M
e
at f5(|q|)
t
s 4
S -f6(|q|)
1

r
o
f 3

s
n
o
nti 2
c
u
f

e
v 1
a
w
0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
|q| GeV
∗
Figure 1: wave functions of D (1S).
s
10
)
-2 V 8 q2f (|q|)/M2
e 3
G 2 2
( q f (|q|)/M
4
e
t 6
a f (|q|)
t 5
s

S -f (|q|)
6
2

r 4
o
f

s
n
o
cti 2
n
u
f

e
v
a 0
w
-2
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
|q| GeV
∗
Figure 2: wave functions of D (2S).
s
3
6
)
2 2 2
- V q f (|q|)/M
e 3
G 5 2 2
( q f4(|q|)/M
e
t
a f (|q|)
st 4 5

D -f (|q|)
6
1

r
o
f 3

s
n
o
ti
c
n 2
u
f

e
v
a 1
w
0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
|q| GeV
∗
Figure 3: wave functions of D (1D).
s
center values of S waves are little smaller than the estimates in Ref. [18], where F (1S) = 375 24 MeV
V
±
and F (2S) = 312 17 MeV, but the value of D wave decay constant is much smaller than the predict
V
±
in Ref. [18] where we did not shown it, this is because we should not ignore the contribution from the
term of f when consider a D wave state. Our estimate of F (1S) = 353 23 MeV is close to the
3 V
±
newer result FD∗ = 268 290 MeV by Choi [19]. Our estimate of FV(2S) = 312 17 MeV is a little
s ∼ ±
larger than the estimate F (2S) = 243 41 MeV in Ref. [7], where they extracted it from the decay
V
±
B+ D¯0+D+(2700) assuming the new state D+(2700) is the first radial excited state D+(2S).
→ sJ sJ sJ
According to the Mandelstam formalism [20], at the leading order, the transition amplitude for
B+ D¯0 can be written as [21]:
→
dq~ m P
D¯0 Jµ B+ = Tr ϕ¯++(~q u ~r)6 ϕ++(~q)γµ(1 γ ) , (5)
h | | i Z (2π)3 (cid:20) D¯0 − mc+mu M B+ − 5 (cid:21)
where~r is the recoil three dimensional momentum of the final state D¯0 meson, ϕ++ is called the positive
energy wave function, and ϕ¯++ = γ (ϕ++)+γ .
0 0
D¯0 D¯0
The wave function forms for pseudoscalar B+ and D¯0 are similar, for example, the wave function for
B+ can be written as [22]
→ M → → m +m
ϕ++(q) = B+ ϕ (q)+ϕ (q) u b
B+ 2 (cid:18) 1 2 ωu+ωb (cid:19)
4
ω +ω ~q(m m ) ~qγ (ω +ω )
u b +γ 6 u − b + 6 0 u b γ , (6)
×(cid:20)m +m 0 − m ω +m ω (m ω +m ω )(cid:21) 5
u b b u u b b u u b
where ω = m2 +~q2 and ω = m2+~q2; ϕ (~q), ϕ (~q) are the radial part wave functions, and their
u u b b 1 2
q
p
−
numerical values can be obtained by solving the full Salpter equation of 0 state [22].
The decay width is:
1 p~
Γ = | f2| T 2
8π M2 | |
B
1 p~ G2V2V2
= | f2| F cs bcF2M2 X, (7)
8π M2 2 V f2
B
where p~ and M are the three dimensional momentum and mass of the final new state D∗+. M2
f2 f2 sJ f2
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:
Γ(B+ D¯0D∗+(2S)) = F2(2S) (3.50 0.38) 10−14 GeV, (8)
s V
→ × ± ×
Γ(B+ D¯0D∗+(1D)) = F2(1D) (3.25 0.30) 10−14 GeV. (9)
s V
→ × ± ×
If we ignore the mass difference between D∗+(2S) and D∗+(1D), and use 2700 MeV as input, the results
s s
become
Γ(B+ D¯0D∗+(2S)) = F2(2S) (3.36 0.25) 10−14 GeV, (10)
s V
→ × ± ×
Γ(B+ D¯0D∗+(1D)) = F2(1D) (3.36 0.25) 10−14 GeV. (11)
s V
→ × ± ×
So the difference between this two channel mainly come from the difference of decay constants. Then our
predictions of branching ratios are:
Br(B+ D¯0D∗+(2S)) =(0.72 0.12)%, (12)
s
→ ±
Br(B+ D¯0D∗+(1D)) = (0.027 0.007)%. (13)
s
→ ±
In Ref. [12], we have calculated the main decay channels of D+(2S) and D+(1D), and we have the
sJ sJ
following estimates:
Br(D+(2S) D0K+) = 0.20 0.03
sJ → ±
5
and
Br(D+(1D) D0K+) = 0.32 0.04,
sJ → ±
so we obtain:
Br(B+ D¯0D+(2S)) Br(D+(2S) D0K+) = (1.4 0.5) 10−3 (14)
→ sJ × sJ → ± ×
and
Br(B+ D¯0D+(1D)) Br(D+(1D) D0K+) = (0.9 0.3) 10−4. (15)
→ sJ × sJ → ± ×
Our estimate of Br(D+(2S) D0K+) 0.20 is larger than than the estimate 0.11 in Ref. [4] and
sJ → ≃
the estimate 0.05 in Ref. [5], if we use their value as input, we will obtain a smaller value of Br(B+
→
D¯0D+(2S)) Br(D+(2S) D0K+). But our estimate of Br(D+(1D) D0K+) 0.32 consist with
sJ × sJ → sJ → ≃
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+(2S) and
sJ
D+(1D). In Ref. [7], they assumed that the new state D+(2700) is D+(2S), and they extracted decay
sJ sJ sJ
constant of D+(2S): F = 243 41 MeV.
sJ Ds+J(2S) ±
The Belle Collaboration have the data [3, 13]
Br(B+ D¯0D+(2700)) Br(D+(2700) D0K+) = (1.13 +0.26) 10−3.
→ sJ × sJ → −0.36 ×
Because we only calculated the decay widths of six main channels for D+(2S) (D+(1D)), and based
sJ sJ
on the summed width of these six channels, not full width, we give a relative larger branching ratio
Br(D+(2S) D0K+) and Br(D+(1D) D0K+), the real branching ratios should be smaller than
sJ → sJ →
our estimates, so our estimate of B+ decay to D+(2S) is close to the data, while the estimate of B+
sJ
decay to D+(1D) is much smaller than the data, then we can draw a conclusion that the new state
sJ
D+(2700) is D+(2S).
sJ sJ
We have another method to estimate the branching ratio, because the mass of B+ is much heavier
than the mass of D¯0, and the mass of D+(2700) is close to the mass of D∗+, as a rough estimate, we
sJ s
ignore the mass difference of D+(2700) and D∗+(2112), from Eq.(1) and Eq.(2), then we have
sJ s
Br(B+ D¯0D+(2S)) F2(2S)
→ sJ V , (16)
Br(B+ D¯0D∗+(1S)) ≃ F2(1S)
s V
→
and from Particle Data Group [13], the branching ratio of B+ D¯0+D+(1S):
→ sJ
Br(B+ D¯0D∗+(1S)) = (7.8 1.6) 10−3.
s
→ ± ×
6
So we have:
F2(2S)
Br(B+ D¯0D∗+(2S)) V (7.8 1.6) 10−3 (5.4 1.7) 10−3, (17)
→ s ≃ F2(1S) × ± × ≃ ± ×
V
F2(1D)
Br(B+ D¯0D∗+(1D)) V (7.8 1.6) 10−3 (2.0 1.0) 10−4. (18)
→ s ≃ F2(1S) × ± × ≃ ± ×
V
This rough estimate results are very close to our calculations (Eq.(12) and Eq.(13)), so we have the same
conclusion that D+(2700) is D+(2S).
sJ sJ
In Ref. [12], we estimate the full widths of D+(2S) and D+(1D) by six main decay channels, the
sJ sJ
estimated full widths are 46.4 6.2 MeV for D+(2S), 73.0 10.4 MeV for D+(1D), comparing with
± sJ ± sJ
experimental data, Γ = 112 7 36 MeV for D+(2690) and Γ = 108 23+36 MeV for D+(2700), there
± ± sJ ± −31 sJ
is still the possible that there are two states around 2700 MeV, one is the S wave dominate D+(2S), the
sJ
other is D wave dominate D+(1D).
sJ
As summary, from the decays B+ D¯0+D+(2S) and B+ D¯0+D+(1D), we conclude that the
→ sJ → sJ
new state D+(2700) is the first radial excited state D+(2S), and there may exist another state around
sJ sJ
2700, D+(1D), with a mass around 2718 MeV, and a width 73.0 10.4 MeV.
sJ ±
Acknowledgements
This work was supportedin partby theNational Natural Science Foundation of China(NSFC) under
Grant No. 10875032, and in part by SRF for ROCS, SEM.
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