ISSN 00213640, JETP Letters, 2014, Vol. 100, No. 5, pp. 285–294. © Pleiades Publishing, Inc., 2014.
Final State Interaction Effects in the B+
D+K0 Decay¶
H. Mehraban, M. Borhani, and A. Asadi* Physics Department, Semnan University, Semnan, 35195363 Iran * email:
[email protected] Received July 4, 2014
The exclusive decay of B+ D+K0 is calculated by the QCD factorization method (QCDF) and final state + interaction (FSI). First, the B D+K0 decay is calculated via QCDF method. The result that is found by using the QCDF method is less than the experimental result. So FSI is considered to solve the B+ D+K0 + 0
+
0
+
0
0
decay. For this decay, the D s π , D s * ρ , D s * φ via the exchange of K , K * , D–, and D–* mesons are cho sen for the intermediate states. The above intermediate states are calculated by using the QCDF method. In the FSI effects, the results of our calculations depend on η as the phenomenological parameter. The range of this parameter is selected from 2 to 2.4. It is found that if η = 2.4 is selected, the numbers of the branching ratio are placed in the experimental range. The experimental branching ratio of this decay is less than 2.9 × 10–6 and our results calculated by QCDF and FSI are (0.16 ± 0.04) × 10–6 and (2.8 ± 0.09) × 10–6, respec tively. DOI: 10.1134/S002136401417010X
1. INTRODUCTION Nonleptonic decays of the B mesons are signifi cant for testing theoretical frameworks and searching new physics beyond the standard model. The nextto leading order lowenergy effective Hamiltonian is used for the weak interaction matrix elements and final state interaction (FSI). The importance of the FSI in hadronic processes has been identified for a long time. Recently, its applications in D and B decays have attracted extensive interests and attentions of theorists. Since the hadronic matrix elements are fully controlled by nonperturbative QCD, the most important problem is how to evaluate them properly. The factorization method enables one to separate the nonperturbative QCD effects from the perturbative parts and to calculate the latter in terms of the field theory order by order. Several factorization approaches have been proposed to analyze B meson decays, such as the naive factorization approach, QCDF approach, the perturbative QCD approach and softcollineareffective theory, but none provided an estimate of the FSI at the hadronic level. These approaches successfully explain many phenomena; however, there are still some problems which are not easy to describe within this framework. These may be some hints for the need of FSI in B decays. FSI effects are nonperturbative in nature [1]. In many decay modes, FSI may play a crucial role [2]. In this way, the CKM matrix elements and the color factor are sup pressed and the CKM’s most favored twobody inter ¶The article is published in the original.
mediate states are the only ones that have been taken into consideration [3]. The FSI can be considered as a soft rescattering style for certain intermediate two +
0
body hadronic channel B+ D s π decay [4]. Therefore, FSI are estimated via the one particle exchange processes at the hadron loop level (HLL) as explained in Section 4. We calculated the B+ D+ K0 decay according to QCDF method. This process only occurs via annihilation between b and u . We selected the leading order Wilson coefficients at the scale mb [5, 6] and obtained the BR (B+ D+K0) = (0.16 ± 0.04) × 10–6. The decay is a pure annihilation decay channel. It is therefore, expected to be very small in factoriza tion approach. The FSI can give sizable corrections. Rescattering amplitude can be derived by calculating the absorptive part of triangle diagrams. In this decay, +
+
+ 0 0 intermediate state are the D s π , D s * ρ , D s * φ . Then we calculated the B+ D+K0 decay according to HLL method. By FSI method we obtain the branching ratio of B+ D+K0 decay, (2.8 ± 0.09) × 10–6 and the experimental result of this decay is less than 2.9 × 10–6. We present the calculation of QCDF for B+ D+K0 decay in Section 2. In Section 3, we calculate the amplitudes of the intermediate states. Then we present the calculation of HLL for B+ D+K0 decay in Sec tion 4. In Section 5, we give the numerical results, and in Section 6, we have conclusion.
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Fig. 1. (Color online) Feynman annihilation diagrams for B+ D+K0 decay.
2. QCD FACTORIZATION D+K0 DECAY OF B+
1
∫ dx/x 0
1
and
+ 0
D s π decay diagrams.
Fig. 2. (Color online) B+
To compare QCDF with FSI, we explore QCDF analysis. In this case, we only have currentcurrent, penguin and electroweak penguin annihilation effects. These contributions are small, but it is interesting and necessary to discuss them. For annihilation amplitude when all the basic building blocks equations are solved it is found that weak annihilation kernels exhibit end point divergence. Divergence terms are determined by
∫ dy/y . For the liberation of the diver
There are large theoretical uncertainties related to the modeling of power corrections corresponding to weak annihilation effects, we parameterize these effects in terms of the divergent integrals XA (weak annihilation) mB iφ X A = 1 + ρe ln , Λh
ρ ≤ 1,
Λ h = 0.5 GeV.
(4)
0
gence, a small ⑀ of ΛQCD/MB order was added to the denominator. So the answer to the integral becomes ln(1 + ⑀)/⑀ form, which is shown with XA. Specifically, we treat XA as a universal parameter obtained by using ρA = 0.5 and a strong phase for PP(M1M2) case, φA = ⎯55° [7]. According to Fig. 1, we obtained the annihi lation amplitude as A(B
+
+
And for the pseudoscalar mesons of D+ and K0, the D
2
0 2m K K r X = . ( mb – md ) ( ms + md )
(1)
where fB, fD, and fK are the decay constants and V pb V *ps (p = u, c) are the CKM matrix elements and the quan tities of the b1 and b2 correspond to the current–cur rent annihilation. These quantities are given by (2)
C1,2 are the Wilson coefficients, Nc is the color number and i
0
2
0
C i B 1, 2 = FC A , 2 1, 2 1, 2 NC
K
+ 2m D D , r X = ( mb – mc ) ( md + mc )
D K )
iG = F f B f D f K { b 1 V cb V *us + b 2 V ub V *cs }, 2
+
ration r X and r X are defined as
(5)
3. AMPLITUDES OF INTERMEDIATE STATES In this section, before analyzing FSI in the B+ decay, we introduce the factorization approach in detail. The effective weak Hamiltonian for B decays consists of a sum of local operators Qi multiplied by QCDF coefficients Ci and products of elements of the quark mixing matrix [7]. The factorization approach of the heavymeson decays can be expressed in terms of different topologies of various decay mechanisms such as tree, penguin and annihilation. D+K0
i
A1 ≈ –A2 3.1. B+
π D K 2 = 6πα s 3 ⎛ X A – 4 + 2⎞ + r X r X ⎛ X A – 2X A⎞ , ⎝ ⎠ ⎝ 3⎠ 2
NC – 1 C F = . 2N C
(3)
+
0
D s π Decay
According to quark level diagrams in the next sec + + 0 0 tion, D s π and D s * ρ are produced for intermediate 0
0 states via exchange meson of K , K * , D–, and D–*.
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287
0
For B+ D s π decay, Feynman diagrams are shown in Fig. 2, and the amplitude reads A(B
+
+
0
Ds π ) (6)
iG B→π 2 2 = F f D+ F 0 ( m B – m π ) ( a 1 V ub V *cs + a 2 V cb V *us ). 2 s
Fig. 3. (Color online) B+
Here, eff 1 eff a i = c i + c i + 1 Nc
( i = odd ),
eff 1 eff a i = c i + c i – 1 Nc
( i = even ),
(7)
D* Rχ s
⊥
2m D* f D* s = s , m b f D*
(10)
s
⊥ 2m φ f φ
φ
R χ = . mb fφ
where i runs from i = 1,…, 10 and a1 and a2 are both related to the coefficients c1 and c2, which are very large compared to other Wilson coefficients. There are also similar diagrams such as B+ + 0 Ds * ρ
+ D s * φ decay diagrams.
4. FINAL STATE INTERACTION OF THE B+ D+K0 DECAY
decay. So the amplitude reads
For B+
A(B
+
+ iG 0 D s * ρ ) = F f D*s m D*s ( a 1 V ub V *cd + a 2 V cb V *us ) 2
Bρ
D+k0 decay, two body intermediate + + + 0 0 states such as D s π , D s * ρ , D s * φ are produced. We can write out the decay amplitude involving HLL con tributions with the following formula
2
× ( ε *1 ε *2 ) ( m B + m ρ )A 1 ( m Ds* )
(8) AbsM ( B ( p B )
2 Bρ 2A 2 ( m Ds* )
3
– ( ε *1 p B ) ( ε *2 p B ) . mB + mρ
× M(B
+
+
and two final state mesons exchange d quark, the D s * and φ mesons are produced for intermediate state via 0
0 exchange meson of ( K * , K ). The intermediate mesons cannot exchange d quark because B+
AbsM ( B ( p B )
M ( p 1 )M ( p 2 )
Mp 3 M ( p 4 ) )
– iG = F V CKM f V2 m 2 2 2 3
3
d p1 d p2 4 4 × 3 3 ( 2π ) δ ( p B – p 1 – p 2 ) 2E 1 ( 2π ) 2E 2 ( 2π )
∫
BM
where Vol. 100
M 3 M 4 ),
+
Ds * φ )
– iG = F f B f D* f φ ( b 1 V cb V *us + b 2 V ub V *cs ), s 2
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(9)
(11)
for which both intermediate mesons (M1, M2) are pseudoscalar, and the absorptive part of the HLL dia grams for PP case can be calculated by
+
D+K0 D+K0 decay is impossible. For B+ Ds * φ decay, Feynman diagrams are shown in Fig. 3, and the amplitude reads +
3
∫
When two intermediate mesons exchange s quark
A(B
M ( p 3 )M ( p 4 ) )
d p1 d p2 4 4 = 1 ( 2π ) δ ( p B – p 1 – p 2 ) 2 2E 1 ( 2π ) 3 2E 2 ( 2π ) 3
D s * φ Decay
3.2. B+
M ( p 1 )M ( p 2 )
2
× ( ε *1 ε *2 ) ( m B + m 1 )A 1 1 ( m 2 ) BM
2
2A 2 1 ( m 2 ) – ( ε *1 p B ) ( ε *2 p B ) G ( M1 M2 mB + m1
M 3 M 4 ),
(12)
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4.1. Final State Interaction + 0 in the B+ Ds π D+K0 Decay +
0
The quark model diagram for B+ Ds π + 0 + D K via the exchange of D is shown in Fig. 4. And the hadronic level diagrams are shown in Fig. 5. +
The amplitude of the mode B0 D s (p1)π0(p2) D+(p3)K0(p4) via the exchange of D+ are given by
Fig. 4. (Color online) Quark level diagram for B+ D+K0.
1
3
3
d p1 d p2 4 4 Abs(5a) = 1 3 3 ( 2π ) δ ( p B – p 1 – p 2 ) 2 2E 1 ( 2π ) 2E 2 ( 2π ) –1
∫
× M(B
+
+
0
D s π ) ( – ig D+ D*K ) [ ε D* ( p 1 + p 3 ) ] s
2 ·2 2 F ( q m D* ) × ( – ig D*Dπ ) [ – ε D* ( p 2 + p 4 ) ] T1 1
= –
∫p
1
–1
(15)
d ( cos θ ) ( g Ds D*K ) ( g D*Dπ ) 16πm B
× H1 M ( B
2 ·2 2 + 0 F ( q m D* ) D s π ) . T1
+
Here, Fig. 5. HLL diagrams for long distance tchannel contri bution to B+ D+K0.
where M(B M1M2) is the amplitude of B M1M2 decay that calculated via QCDF method, and G(M1M2 M3M4) involves hadronic vertices factor defined as
μ
H1 = εμ ( p1 + p3 ) εν ( p2 + p4 )
ν
q μ q ν⎞ μ ν = ( p 1 + p 3 ) ( p 2 + p 4 ) ⎛ – g μν + 2 ⎝ m ⎠ D*
2
2
2
2
( m1 – m3 ) ( m4 – m2 ) = – ( p 1 p 2 + p 1 p 4 + p 2 p 3 + p 3 p 4 ) + , 2 m D*
0*
2
2
〈 D ( p 2 )K ( p 3, ε 3 ) i ᏸ D s ( p 1 )〉 = – ig Ds K*D ε ( p 1 + p 2 ),
T 1 = ( p 1 – p 3 ) – m D*
〈 D ( ε 2, p 2 )K ( q ) i ᏸ D *s ( ε 1, p 2 )〉
= p 1 + p 3 – 2p 1 p 3 – 2p 1 p 3 + 2p 1 ⋅ p 3 – m D* ,
0
μ
(13)
ν α β
2
2
2
2
s
2
2
2
2
The dispersive part of the rescattering amplitude can be obtained from the absorptive part via the dispersion relation [1]: ∞
∫
0 0
q = m 1 + m 3 – 2E 1 E 3 + 2 p 1 p 3 cos θ
= – ig D* D*K ε μναβ ε 1 ε *2 q p 1 .
2 1 AbsM ( s' ) DisM ( m B ) = ds' , π s' – m 2B s
0 0
(14)
where s' is the square of the momentum carried by the 2 exchanged particle and s ~ m B is the threshold of intermediate states, in this case s. Unlike the absorp tive part, the dispersive contribution suffers from the large uncertainties arising from the complicated inte gration.
0 0
= m D* + m Ds* – 2p 1 p 3 + 2 p 1 p 3 cos θ.
(16)
θ is the angle between p1 and p3, q is the momentum of 2
the exchange D* meson, and F(q2, m D* ) is the form factor defined to take care of the offshell of the exchange particles, which introduced as [8] n
2
F(q ,
2 m D* )
⎛ Λ 2 – m 2D*⎞ = ⎜ ⎟ . ⎝ Λ2 – q2 ⎠
(17)
The form factor (i.e., n = 1) normalized to unity at 2 q2 = m D* , mD* and q are the physical parameters of the exchange particle, and Λ is the phenomenological parameter. JETP LETTERS
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H2 = ( ε1 q ) ( ε2 q ) ( ε1 pB ) ( ε2 pB )
2
It is obvious that for q2
0, F(q2, m D* ) becomes
0
1
3
2
2
2
0 0
2
2
2
q = m 1 + m 3 – 2E 1 E 3 + 2 p 1 p 3 cos θ 2
Abs(5b)
2
2
(18)
where the η is the phenomenological parameter that its value in the form factor is expected to be of the order of unity and can be determined from the mea sured rates, and
0
T 2 = ( p 1 – p 3 ) – m D = p 1 + p 3 – 2p 1 p 3 + 2p 1 p 3 – m D , 2
Λ = m D* + ηΛ QCD ,
2
mB p1 ( m1 – p1 p3 ) mB p2 ( p2 p4 – m2 ) = , 2 2 m1 m2
2
a number. If Λ Ⰷ mD* then F(q2, m D* ) turns to be unity, whereas, as q2 ∞ the form factor approaches zero and the distance becomes small and the hadron inter action is no longer valid. Since Λ should not be far from mD* and q, we choose
289
2
(20)
0 0
= m Ds* + m D – 2p 1 p 3 + 2 p 1 p 3 cos θ, and 1
3
3
d p2 d p1 4 4 1 Abs(5c) = ( 2π ) δ ( p B – p 1 – p 2 ) 2 2E 1 ( 2π ) 3 2E 2 ( 2π ) 3 –1
∫
3
μ ν α β iG × ( – iq Ds*D*K ) ( ε μναβ ε 1 ε D* p 3 p 1 ) ⎛ F⎞ ⎝ 2 ⎠
d p2 d p1 4 4 = 1 3 3 ( 2π ) δ ( p B – p 1 – p 2 ) 2 2E 1 ( 2π ) 2E 2 ( 2π ) –1
∫
ρ σ
λ η
× ( – ig D*Dρ ) ( ε ρσλη ε 2 ε D* P 4 p 2 )f Ds* m Ds*
iG × ( – ig D*DK ) ( ε 1 q ) ( – ig ρDD ) ( – ε 2 q ) ⎛ F⎞ f D*s m D*s s ⎝ 2 ⎠
2
Bρ
× ( ε 1* ε 2* ) ( m B + m ρ )A 1 ( m D*s ) 2 Bρ × ( ε *1 ε *2 ) ( m B + m ρ )A 1 ( m D*s )
2
Bρ
2A 2 ( m Ds* ) – ( ε *1 p B ) ( ε *2 p B ) mB + mρ 2
2
1
× ( a 1 V ub V *cs + a 2 V cb V *us )
1 1
d( cos θ )
Bρ
2
2
1
d ( cos θ) 16πm B
2
2 2 2 2A 2 ( m Ds* ) F ( q , m D* ) 2 Bρ × ( m B – m ρ )A 1 ( m D*s )H 1 – . H 2 mB + mρ T1
2
Bρ
× ( mB +
∫p
–1
–1
2 Bρ m ρ )A 1 ( m D*s )H 1
2
iG F 2 = ( gD ) ( g D*Dρ )f Ds* m Ds* 32πm B s*D*K
iG F = f D m D ( a 1 V ub V *cs + a 2 V cb V *us ) 16πm B s* s*
∫p
2
(21)
F ( q , m D* ) × ( a 1 V ub V *cs + a 2 V cb V *us ) T1
(19)
2
2
F ( q , mD ) × (a 1 V ub V *cs + a 2 V cb V *us ) T2
× g Ds*DK g ρDD
2
Bρ
2A 2 ( m Ds* ) – ( ε *1 p B ) ( ε *2 p B ) mB + mρ
2A 2 ( m D*s ) – H2 mB + mρ
Here,
2 mD )
F (q , × . T2
μ ν α β ρ σ λ η H 1 = ε μναβ ε ρσλη ε 1 ε D* p 3 q ε 2 ε D* p 4 q ( ε *1 ε *2 )
= { ( p1 q ) [ ( p3 q ) ( p1 p4 ) – ( p3 p4 ) ( p1 q ) ]
Here,
2
2
+ ( p 1 p 3 ) [ ( p 4 q ) ( p 1 q ) – q ( p 1 p 4 ) ] }/m D* + { ( p2 q ) [ ( p4 q ) ( p2 p3 ) – ( p3 p4 ) ( p2 q ) ]
H 1 = ( ε 1 q ) ( ε 2 q ) ( ε *1 ε *2 ) 2 2
2
2
( p2 p4 – m2 ) ( m1 – p1 p3 ) 2 2 = p 1 + p 3 – 2p 1 p 3 – – 2 2 m2 m1 2
2
( m1 – p1 p3 ) ( p2 p4 – m2 ) ( p1 p2 ) + , 2 2 m1 m2 JETP LETTERS
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2
2
+ ( p 2 p 4 ) [ ( p 3 q ) ( p 2 q ) – q ( p 2 p 3 ) ] }/m D* 2
+ { ( p1 p2 ) [ ( p3 q ) ( p4 q ) – q ( p3 p4 ) ] + ( p 1 p 2 ) ( p 2 q )[ ( p 3 p 4 ) ( p 1 q ) – ( p 1 p 4 ) ( p 3 q )] 2
4
+ ( p 1 p 2 ) ( p 2 p 3 ) [ q ( p 1 p 4 ) – ( p 4 q ) ( p 1 q ) ] }/m D* ,
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The dispersion relation is ∞
2 Dis5 ( m B )
1 Abs5a ( s' ) + Abs5b ( s' ) = ds'. 2 π s' m – B s
∫
+
Ds π
The quark model diagram for B+
(23)
0
D+K0
0
decay via the exchange of K is shown in Fig. 6. And the hadronic level diagrams are shown in Fig. 7.
Fig. 6. (Color online) Quark level diagram for B+ D+K0.
And the amplitude of the mode B0 + D s (p1)π0(p2)
D+(p3)K0(p4) via the exchange of K
0
are given by Abs(7a) 1
3
3
d p2 d p1 4 4 1 = 3 ( 2π ) δ ( p B – p 1 – p 2 ) 2 2E 1 ( 2π ) 2E 2 ( 2π ) 3 –1
∫
× M(B
+
+
0
D s π ) ( – ig D+ K*D ) [ ε K* ( p 1 + p 3 ) ] s
2
Fig. 7. HLL diagrams for long distance tchannel contri bution to B+ D+K0.
1
= μ ν
α β ρ σ
λ η
2
H 2 = ε μναβ ε ρσλη ε 1 ε D* p 3 q ε 2 ε D* p 4 q ( ε *1 p B ) ( ε 2 p B ) 2
× [ ( qp B ) – q
2
2 mB ]
∫p
1
–1
= ( p 3 q ) [ m B ( p 4 q ) – ( p 4 p B ) ( qp B ) ] + ( p 3 p 4 ) 2
× H1 M ( B Here,
+ { ( qp B ) [ ( p 3 q ) ( p 1 p 4 ) – ( p 1 q ) ( p 3 p 4 ) ] 2
+ ( p3 pB ) [ ( p1 q ) ( p4 q ) – q ( p1 p4 ) ]
μ
H1 = εμ ( p1 + p3 ) εν ( p2 + p4 ) 2
+ ( p 1 p B ) [ q ( p 3 p 4 ) – ( p 3 q ) ( p 4 q ) ] } ( p 1 p B )/m D*
q μ q n u⎞ μ ν = ( p 1 + p 3 ) ( p 2 + p 4 ) ⎛ – g μν + 2 ⎝ m ⎠
+ { ( p 2 q ) [ ( p 3 q ) ( p 4 p B ) – ( p 3 p 4 ) ( qp B ) ]
K*
2
+ ( p 2 p 3 ) [ ( p 4 q ) ( qp B ) – q ( p 4 p B ) ] + ( p2 pB ) [ q ( p3 p4 ) –
(24)
2 2 2 + 0 F ( q , m K* ) D s π ) . T1
+
+ ( p 3 p B ) [ q ( p 4 p B ) – ( p 4 q ) ( qp B ) ]
2
2
d ( cos θ ) ( g D+ K*D ) ( g K*Kπ ) s 16πm B
2
2
2
F ( q , m K* ) × ( – ig K*Kπ ) [ ε K* ( p 2 + p 4 ) ] T1
= – ( p1 p2 + p1 p4 + p2 p3 + p3 p4 ) +
2 ( p 4 q ) ( p 3 q ) ] } ( p 2 p B )/m D*
2 ( m1
2 2 2 – m3 ) ( m4 – m2 ) , 2 m K*
2
+ { ( p1 p2 ) [ ( p3 q ) ( p4 q ) – q ( p3 p4 ) ]
2
2
+ ( p 2 p 3 ) [ q ( p 1 p 4 ) – ( p 1 q ) ( p 4 q ) ] } [ ( p 1 p B ) ( p 2 p B ) ]/m D* , 2
2
2
2
0 0
2
T 1 = ( p 1 – p 3 ) – m D* = p 1 + p 3 – 2p 1 p 3 + 2p 1 p 3 – m D* , 2
q =
2 m1 2
+
2 m3 2
– 2E 1 E 3 + 2 p 1 p 3 cos θ 0 0
= m Ds* + m D – 2p 1 p 3 + 2 p 1 p 3 cos θ .
2
0 0
2
= p 1 + p 3 – 2p 1 p 3 + 2p 1 p 3 – m K* ,
4
2
2
T 1 = ( p 1 – p 3 ) – m K*
+ ( p2 q ) [ ( p2 p3 ) ( p1 q ) – ( p1 p4 ) ( p3 q ) ]
2
2
2
q = m 1 + m 3 – 2E 1 E 3 + 2 p 1 p 3 cos θ 2
2
0 0
= m Ds* + m D – 2p 1 p 3 + 2 p 1 p 3 cos θ ,
(25)
(22) and JETP LETTERS
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and
Abs(7b) 1
3
3
Abs(7c)
d p2 – iG d p1 4 4 = F 3 3 ( 2π ) δ ( p B – p 1 – p 2 ) 2 2 2E 1 ( 2π ) 2E 2 ( 2π ) –1
∫
1
– iG μ ν α β × ( – i 2g D*K*D ) ⎛ F⎞ ( ε μναβ ε 1 ε K* p 3 p 1 ) s ⎝ 2 ⎠
× ( ε 1* ε 2* ) ( m B + m ρ )A 1 ( m D*s )
ρ σ
2
Bρ
2
(26)
2
2
2
d( cos θ )
– iG F 2 = ( g Ds*K*D ) ( g ρK*K )f Ds* m Ds* 32πm B
–1
2
Bρ
2
2
(28) 2
2
F ( q , m K* ) × ( a 1 V ub V *cs + a 2 V cb V *us ) T1
1
× ( mB +
2
Bρ
2A 2 ( m Ds* ) – ( ε *1 p B ) ( ε *2 p B ) mB + mρ
– iG F f D m D ( a 1 V ub V *cs + a 2 V cb V *us ) = 16πm + B s* s*
Bρ m ρ )A 1 ( m D*s )H 1
2
Bρ
× ( ε *1 ε *2 ) ( m B + m ρ )A 1 ( m D*s )
F ( q , mK ) × ( a 1 V ub V *cs + a 2 V cb V *us ) T2
1
λ η
× ( – ig ρK*K ) ( ε ρσλη ε 2 ε K* p 4 p 2 )f Ds* m Ds*
2A 2 ( m Ds* ) – ( ε *1 p B ) ( ε *2 p B ) mB + mρ
∫p
3
∫
2
Bρ
3
d p1 d p2 4 4 1 = 3 ( 2π ) δ ( p B – p 1 – p 2 ) 2 2E 1 ( 2π ) 2E 2 ( 2π ) 3 –1
– iG × ( – ig D*DK ) ( 2ε 1 p 3 ) ( – ig ρKK ) ( 2ε 2 p 4 ) ⎛ F⎞ f D*s m D*s s ⎝ 2 ⎠
× g Ds*DK g ρKK
291
2A 2 ( m Ds* ) – H2 mB + mρ
1
∫p
1
–1
d ( cos θ ) 16πm B
Bρ
2
2
2
2A 2 ( m Ds* ) 2 Bρ × ( m B + m ρ )A 1 ( m Ds* )H 1 – H2 mB + mρ
2
F ( q , mK ) × . T2
2
F ( q , m K* ) . × ( a 1 V ub V *cs + a 2 V cb V *us ) T1
Here, Here, H 1 = ( ε 1 q ) ( ε 2 q ) ( ε *1 ε *2 ) 2 2
=
2 p1
+
2 p3
μ ν α β ρ σ λ η H 1 = ε μναβ ε ρσλη ε 1 ε D* p 3 q ε 2 ε D* p 4 q ( ε *1 ε *2 )
2
2
( p2 p4 – m2 ) ( m1 – p1 p3 ) – 2p 1 p 3 – – 2 2 m2 m1 2
= { ( p1 q ) [ ( p3 q ) ( p1 p4 ) – ( p3 p4 ) ( p1 q ) ] 2
2
+ ( p 1 p 3 ) [ ( p 4 q ) ( p 1 q ) – q ( p 1 p 4 ) ] }/m K*
2
( m1 – p1 p3 ) ( p2 p4 – m2 ) ( p1 p2 ) + , 2 2 m1 m2
+ { ( p2 q ) [ ( p4 q ) ( p2 p3 ) – ( p3 p4 ) ( p2 q ) ] 2
2
+ ( p 2 p 4 ) [ ( p 3 q ) ( p 2 q ) – q ( p 2 p 3 ) ] }/m K*
H2 = ( ε1 q ) ( ε2 q ) ( ε1 pB ) ( ε2 pB ) 0
2
0
2
+ { ( p1 p2 ) [ ( p3 q ) ( p4 q ) – q ( p3 p4 ) ]
2
mB p1 ( m1 – p1 p3 ) mB p2 ( p2 p4 – m2 ) = , 2 2 m1 m2 2
+ ( p1 p2 ) ( p2 q ) [ ( p3 p4 ) ( p1 q ) – ( p1 p4 ) ( p3 q ) ]
2
T2 = ( p1 – p3 ) – mK 2
2
0 0
μ ν
2
2
2
0 0
= m Ds* + m D – 2p 1 p 3 + 2 p 1 p 3 cos θ , JETP LETTERS
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λ η
2
= ( p 3 q ) [ m B ( p 4 q ) – ( p 4 p B ) ( qp B ) ]
2
q = m 1 + m 3 – 2E 1 E 3 + 2 p 1 p 3 cos θ
α β ρ σ
H 2 = ε μναβ ε ρσλη ε 1 ε D* p 3 q ε 2 ε D* p 4 q ( ε *1 p B ) ( ε *2 p B )
2
= p 1 + p 3 – 2p 1 p 3 + 2p 1 p 3 – m K , 2
4
2
+ ( p 1 p 2 ) ( p 2 p 3 ) [ q ( p 1 p 4 ) – ( p 4 q ) ( p 1 q ) ] }/m K* ,
2
(27)
2
2
+ ( p 3 p 4 ) [ ( qp B ) – q m B ] 2
+ ( p 3 p B ) [ q ( p 4 p B ) – ( p 4 q ) ( qp 4 ) ]
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Fig. 8. (Color online) Quark level diagram for
Fig. 9. HLL diagrams for long distance tchannel contri bution to B+ D+K0.
B+
+ D s * φ.
Abs(9a) 1
+ { ( qp B ) [ ( p 3 q ) ( p 1 p 4 ) – ( p 1 q ) ( p 3 p 4 ) ]
3
3
d p2 d p1 4 4 1 = 3 ( 2π ) δ ( p B – p 1 – p 2 ) 2 2E 1 ( 2π ) 2E 2 ( 2π ) 3 –1
∫
2
+ ( p3 pB ) [ ( p1 q ) ( p4 q ) – q ( p1 p4 ) ]
× ( ε 1 q ) ( – ig D*KD ) ( – ig KKφ ) ( 2ε 2 p 4 ) s
2
2
+ ( p 1 p B ) [ q ( p 3 p 4 ) – ( p 3 q ) ( p 4 q ) ] } ( p 1 p B )m K* + { ( p 2 q ) [ ( p 3 q ) ( p 4 p B ) – ( p 3 p 4 ) ( qp B ) ]
2
× M(B
2
+ ( p 2 p 3 ) [ ( p 4 q ) ( qp B ) – q ( p 4 p B ) ] + ( p 2 p B ) [ q ( p 3 p 4 ) – ( p 4 q ) ( p 3 q ) ] } ( p 2 p B )/m K*
iG = g Ds*KD g KKφ p 1 d( cos θ ) 8 2πm B –1
∫
2
+ { ( p1 p2 ) [ ( p3 q ) ( p4 q ) – q ( p3 p4 ) ]
2
+ ( p2 q ) [ ( p2 p3 ) ( p1 q ) – ( p1 p4 ) ( p3 q ) ] 4
2
(31)
1
2
2
2
2
F ( q , mK ) + D s * φ ) T1
+
+ ( p 2 p 3 ) [ q ( p 1 p 4 ) – ( p 1 q ) ( p 4 q ) ] } [ ( p 1 p B ) ( p 2 p B ) ]/m K* ,
2
2
F ( q , mK ) × f B f Ds* f φ [ b 1 V cb V *us + b 2 V ub V *cs ] H 1 . T1 Here,
2
T1 = ( p1 – p3 ) – 2 p1
=
+
2
2
2 p3
–
0 0 2p 1 p 3
2 m K*
+ 2p 1 p 3 –
H1 = ( ε1 q ) ( ε2 p4 ) 2 m K* ,
E 3 p 1 – E 1 p 3 cos θ⎞ ⎛ E 4 p 2 – E 2 p 4 cos θ⎞ = – ⎛ – , ⎝ ⎠ ⎝ ⎠ mB p1 mB p2
2
q = m 1 + m 3 – 2E 1 E 3 + 2 p 1 p 3 cos θ =
2 m Ds*
+
2 mD
–
0 0 2p 1 p 3
(29)
+ 2 p 1 p 3 cos θ .
2
2
2
0 0
2
2
2
Abs7a ( s' ) + Abs7b ( s') ds' = 1 . 2 π – s' m B s
∫
(30)
Abs(9b)
+
D+K0 Decay +
The quark model diagram for B+ Ds * φ D+K0 decay is shown in Fig. 8. The hadronic level dia grams are shown in Fig. 9. The amplitude of the mode +
D s * (ε1, p1)φ(ε1, p2) 0
exchange of K are given by
3
3
d p2 d p1 4 4 1 = 3 3 ( 2π ) δ ( p B – p 1 – p 2 ) 2 2E 1 ( 2π ) 2E 2 ( 2π ) –1
∫
4.2. Final State Interaction Ds * φ
(32)
0 0
and
1
in the B+
2
= m Ds* + m D – 2p 1 p 3 + 2 p 1 p 3 cos θ ,
∞
B0
2
q = m 1 + m 3 – 2E 1 E 3 + 2 p 1 p 3 cos θ
The dispersion relation is 2 Dis7 ( m B )
2
2
T 1 = ( p 1 – p 3 ) – m K = p 1 + p 3 – 2p 1 p 3 + 2p 1 p 3 – m K ,
D+(p3)K0(p4) via the
× M(B
+
+
(33)
D s * φ ) ( – ig D*K*D ) ( – ig K*Kφ ) s 2
2
2
μ ν α β ρ σ λ η F ( q , m K* ) × ε μναβ ε ρσλη ε 1 ε K* p 3 p 1 ε 2 ε K* p 4 p 2 . T2
Here, 2
2
2
2
0 0
2
T 2 = ( p 1 – p 3 ) – m K* = p 1 + p 3 – 2p 1 p 3 + 2p 1 p 3 – m K* , JETP LETTERS
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FINAL STATE INTERACTION EFFECTS Branching ratio of the B+ D+K0 decay with η = 2–2.4 and experimental data (in units of 10–6) η
2
2.1
2.2
BR
1.68 ± 0.04
1.92 ± 0.05
2.20 ± 0.07
η
2.3
2.4
EXP
BR
2.49 ± 0.08
2.80 ± 0.09
<2.9
2
2
+
+
293
0
A(B D K ) = Abs(5a) + Abs(5b) + Abs(5c) (36) + Abs(7a) + Abs(7b) + Abs(7c) + Abs(9a) 2
2
+ Abs(9b) + Dis7 ( m B ) + Dis9 ( m B ). 5. NUMERICAL RESULTS Numerical values of effective coefficients ai for b d transition at Nc = 3 are given by [13]
2
q = m 1 + m 3 – 2E 1 E 3 + 2 p 1 p 3 cos θ 2
2
(34)
0 0
= m Ds* + m D – 2p 1 p 3 + 2 p 1 p 3 cos θ . The dispersion relation is
a 2 = 0.053,
a 3 = 0.0048,
a 4 = – 0.046 – 0.012i,
a 5 = – 0.0045,
a 6 = – 0.059 – 0.012i, (37)
a 7 = 0.00003 – 0.00018i,
∞
2 1 Abs9a ( s' ) + Abs9b ( s' ) Dis9 ( m B ) = . 2 2 s' m – B s
∫
The decay amplitude of B+ grams is
a 1 = 1.05,
a 8 = 0.0004 – 0.00006i,
(35)
a 9 = – 0.009 – 0.00018i, a 10 = – 0.0014 – 0.00006i.
D+K0 via the HLL dia
The relevant input parameters used as follows:
m b = 4.2 ± 0.12 GeV,
m u = 1.7–3.1 MeV,
m d = 4.1–5.7 MeV,
m B = 5279 ± 0.3 GeV,
m D = 187 ± 0.2 MeV,
m D* = 2010.2 ± 0.17 MeV,
m K = 493.6 ± 0.016 GeV,
m K* = 891 ± 0.26 MeV,
m Ds = 197 ± 0.34 MeV,
m Ds* = 2010 ± 0.17 GeV,
m π = 139.5 MeV,
m ρ = 775.4 ± 0.34 MeV,
f B = 176 ± 42 GeV,
f D = 222.6 ± 19.5 MeV,
f D* = 230 ± 20 MeV,
f K* = 217 ± 5 GeV,
f π = 130.7 ± 0.46 MeV,
f ρ = 211 MeV,
V ub = 0.0043 ± 0.0003 GeV,
f ud = 0.974 ± 0.0002 MeV,
f us = 0.2257 ± 0.002 MeV,
V cs = 0.9745 ± 0.11 GeV,
V cd = 0.30 ± 0.011 MeV,
V cb = 0.0416 ± 0.0006 MeV [9],
Bρ
Bρ
Bπ
A 1 = A 2 = 0.26,
F 0 = 0.3 [6, 13],
φ = – 55° (PP),
ρ = 0.5,
Λ QCD = 0.225 GeV,
G F = 1.166 × 10
–5
g φKK = 5.77 [7],
g φK*K = 6.48 [10],
g K*DDs = 2.59 [1],
g KD*D*s = 9.23 [11],
g πK*K = 4.6 [1],
g ρKK = 5.55,
g ρK*K = 6.48 [12],
g πD*D = 12.5 [3],
g DDρ = 2.52,
(38)
[13, 14],
g ρD*D = 2.82 [8]. By using the input parameters and according to the D+K0 decay, we get QCDF method of B+ BR ( B
+
+
–6
0
D K ) = ( 0.16 ± 0.04 ) × 10 .
(39)
We note that our estimate of branching ratio B+ D+K0 of decay according to QCDF method seems less than the experimental result. Before calculating the B+ D+K0 decay amplitude via FSI, we have to compute the intermediate state amplitude. JETP LETTERS
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We are able to calculate the branching ratio of B+ D+K0 decay with different values of η which are shown in table. 6. CONCLUSIONS In this work, we have calculated the contribution of the tchannel FSI, that is, inelastic rescattering pro D+K0 decay. For cesses to the branching ratio of B+
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evaluating the FSI effects, we have only considered the absorptive part of the HLL because both hadrons which produced via the weak interaction are on their mass shells. We have calculated the branching ratio of B+ D+K0 decay by using QCDF and FSI. The experimental result of this decay is less than 2.9 × 10–6. According to QCDF and FSI, our results are BR(B+ D+K0) = (0.16 ± 0.04) × 10–6 and (2.8 ± –6 0.09) × 10 , respectively. We have introduced the phenomenological param eter η that its value in the form factor is expected to be of the order of unity and can be determined from the measured rates. For a given exchanged particle, we have used η = 2–2.4. If η = 2.4 is selected, the branch D+K0 decay approaches to the ing ratio of the B+ experimental bound. REFERENCES 1. H. Y. Cheng, C. K. Chua, and A. Soni, Phys. Rev. D 71, 014030 (2005). 2. Y. S. Oh, T. Song, and S. H. Lee, Phys. Rev. C 63, 034901 (2001).
3. V. M. Belyaev, V. M. Braun, A. Khodjamirian, and R. Ruckl, Phys. Rev. D 51, 6177 (1995). 4. J. F. Donoghue, E. Glowich, A. A. Petrow, and J. M. Soares, Phys. Rev. Lett. 77, 2178 (1996). 5. M. Beneke, G. Buchalla, M. Neubert, and C. T. Sachra jda, Nucl. Phys. B 591, 313 (2000). 6. A. Ali and C. Greub, Phys. Rev. D 57, 2996 (1998). 7. M. Beneke, G. Buchalla, M. Neubert, and C. T. Sachra jda, Nucl. Phys. B 606, 245 (2001). 8. X. Liu, Z. T. Wei, and X. Q. Li, Eur. Phys. J. C 59, 683 (2009). 9. K. Nakamura et al. (Particle Data Group), Nucl. Part. Phys. 37, 075021 (2012). 10. X. Liu, B. Zhang, L. L. Shen, and S. L. Zhu, Phys. Rev. D 75, 074017 (2007). 11. C. D. Lu, Y. L. Shen, and W. Wang, Phys. Rev. D 73, 034005 (2006). 12. Q. Zhao and B. S. Zou, Phys. Rev. D 74, 114025 (2006). 13. A. Ali, G. Kramer, and C. D. Lu, Phys. Rev. D 58, 094009 (1998). 14. M. Beneke and M. Neubert, Nucl. Phys. B 675, 333 (2003). 15. C. W. Hwang, Phys. Rev. D 81, 114024 (2010).
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