Final State Interactions in $D \to PP$ decays

Abstract

The two-body nonleptonic charmed meson decays into two pseudoscalar mesons are studied using one-particle-exchange method. The effects of the final state interactions are analyzed through the strong phases extracted from the experimental data.

Full text

Page 1

arXiv:hep-ph/0211413v1 26 Nov 2002
BIHEP-TH-2002-50
Final State Interactions in D → PP decays
Medina Ablikim∗, Dong-Sheng Du†, Mao-Zhi Yang‡
Institute of High Energy Physics, Chinese Academy of Sciences, P.O.Box 918(4),
Beijing 100039, China
Abstract
The two-body nonleptonic charmed meson decays into two pseudoscalar mesons are
studied using one-particle-exchange method. The effects of the final state interactions
are analyzed through the strong phases extracted from the experimental data.
1Introduction
The study of the two-body nonleptonic weak decays of particles containing heavy (c,b)
quarks appears to offer a unique opportunity to determine the basic parameters of quark
mixing, and to investigate the mechanism of CP violation. However, the quarks are not free,
they are bound in hadrons by strong interactions which are described by nonperturbative
QCD. Solving the problem of nonperturbative QCD needs efforts in both experiment and
theory. In the near future BESIII and CLEO-c detector will provide high precision data
in charm physics including data on D meson decays, which will provide the possibility for
understanding the physics in charm sector.
It is interesting to study the weak decays of charmed mesons beyond the factorization
approach [1]. In general, if a process happens in an energy scale where there are many
resonance states, this process must be seriously affected by these resonances [2]. This is
a highly nonperturbtive effect. Near the scale of D meson mass many resonance states
exist. D meson decays must be affected seriously by these resonances . After weak decays
∗e-mail: mablikim@mail.ihep.ac.cn
†e-mail: duds@mail.ihep.ac.cn
‡e-mail: yangmz@mail.ihep.ac.cn
1

Page 2

the final state particles rescatter into other particle states through nonperturbative strong
interactions [2, 3]. Every D decay channel can contribute to each other through final state
interaction (FSI). One can model this rescattering effect as one-particle exchange process
[4, 5], namely the final state particles be scattered into other particle states by exchanging
one resonance state existing near the mass scale of D meson. There are also other ways to
treat the nonperturbative and FSI effects in nonleptonic D decays. One approach is that
in which the FSIs are expressed by the phase shifts of the decaying amplitudes [6]. The
other method is a flavor topology approach [7, 8], where the relative phases between various
quark-diagram [9] amplitudes arise from the final state rescattering.
The final state rescattering effects for charmed meson decays into two pions are studied
using the one-particle-exchange method [10], where the magnitudes of hadronic couplings are
extracted from experimental data on the measured branching fractions of resonances decays.
In addition, a strong phase is introduced for the hadronic coupling which is important for
obtaining the correct branching ratios in these decays. A similar analysis is applied to
D → PV decays [11], where P is a pseudoscalar meson, and V is vector meson.
In the present work, we extend the study of the final state interactions in D → ππ
decays to D → PP decays.
are small for s-channel contribution and large for t-channel contribution. Therefore the
The coupling constants extracted from experimental data
s-channel contribution is numerically negligible in D → PP decays. We safely drop the
s-channel contribution in our discussion. In Sec. 2, we present the calculation within the
naive factorization approach. The main scheme of one-particle-exchange method is described
in Sec. 3. We give the numerical calculations and discussions in Sec. 4. The final section is
reserved for summary.
2 Calculations in the factorization approach
The charmed meson decay can be described by the low energy effective Hamiltonian[12]
H =GF
√2

?
q=d,s
vq(C1Qq
1+ C2Qq
2)

, (1)
where C1 and C2 are the Wilson coefficients at mc scale, vq is the product of Cabibbo-
Kobayashi-Maskawa (CKM) matrix elements and defined as
vq= VuqV∗
cq,(2)
2

Page 3

and the current-current operators are given by
Qq
1= (¯ uq)V −A(¯ qc)V −A,Qq
2= (¯ uαqβ)V −A(¯ qβcα)V −A. (3)
We do not consider the contributions of QCD and electroweak penguin operators in the
decays of D → PP because their contributions are negligible in D decays. QCD factorization
approach [13] is inapplicable to these decay modes, as the charmed meson is not heavy
enough. The values of C1and C2at mcscale are taken to be [12]
C1= 1.216,C2= −0.415.
In the naive factorization approach, the decay amplitude can be generally factorized into
a product of two current matrix elements and can be obtained from (1)
A(D+→ π+π0) = −GF
A(D0→ π+π−) =GF
A(D0→ π0π0) = −GF
2
A(D+→¯K0π+) =GF
2
VudV∗
cd(a1+ a2) ifπ(m2
D− m2
π) FDπ(m2
π),
√2VudV∗
cda1ifπ(m2
D− m2
π) FDπ(m2
π),
VudV∗
cda2ifπ(m2
D− m2
π) FDπ(m2
π),
√2VudV∗
cs
?
a1ifπ(m2
D− m2
K) FDK(m2
π)
+ a2ifK(m2
D− m2
D− m2
π) FDπ(m2
K)
?
,
A(D0→ K−π+) =GF
A(D0→¯K0π0) =GF
A(D0→ K+π−) =GF
A(D+→ K+π0) = −GF
A(D+→ K0π+) =GF
A(D0→ K0π0) =GF
A(D+→ K+¯K0) =GF
A(D0→ K+K−) =GF
A(D0→ K0¯K0) = 0,
√2VudV∗
csa1ifπ(m2
K) FDK(m2
π),
2
VudV∗
csa2ifK(m2
D− m2
π) FDπ(m2
K),
√2VusV∗
cda1ifK(m2
D− m2
π) FDπ(m2
K),
2
VusV∗
cda1ifK(m2
D− m2
π) FDπ(m2
K),
√2VusV∗
cda2ifK(m2
D− m2
π) FDπ(m2
K),
2
VusV∗
cda2ifK(m2
D− m2
π) FDπ(m2
K),
√2VusV∗
csa1ifK(m2
D− m2
K) FDK(m2
K),
√2VusV∗
csa1ifK(m2
D− m2
K) FDK(m2
K),
(4)
where the parameters a1and a2are defined as [8]
a1= C1+ C2
?1
Nc
+ χ
?
,a2= C2+ C1
?1
Nc
+ χ
?
,(5)
3

Page 4

with the color number Nc= 3, and χ is the phenomenological parameter which takes into
account nonfactorizable correction. For q2dependence of the form factors, we take the BSW
model [1], i.e., the monopole dominance assumption:
F(q2) =
F(0)
1 − q2/m2
∗
, (6)
where m∗is the relevant pole mass.
The decay width of a D meson at rest decaying into PP is
Γ(D → PP) =
1
8π|A(D → PP)|2|? p |
m2
D
,(7)
where |? p | is the 3-momentum of each final meson. The corresponding branching ratio is
Br(D → PP) =Γ(D → PP)
Γtot
. (8)
A comparison of the branching ratios of the naive factorization result with the experimental
Table 1: The branching ratios of D → PP obtained in the naive factorization approach and
compared with the experimental results.
Decay modeBr (Theory)
χ = 0
3.1 × 10−3
2.48 × 10−3
9.98 × 10−8
1.20 × 10−1
4.81 × 10−2
2.93 × 10−6
1.88 × 10−4
2.4 × 10−4
3.86 × 10−8
7.58 × 10−9
9.15 × 10−3
3.59 × 10−3
0
Br (Theory)
χ = −8.6 × 10−2
2.71 × 10−3
2.65 × 10−3
1.39 × 10−5
9.98 × 10−2
5.14 × 10−2
4.1 × 10−4
2.01 × 10−4
2.56 × 10−4
5.39 × 10−6
1.06 × 10−6
9.76 × 10−3
3.83 × 10−3
0
Br(Experiment)
D+→ π+π0
D0→ π+π−
D0→ π0π0
D+→¯K0π+
D0→ K−π+
D0→¯K0π0
D0→ K+π−
D+→ K+π0
D+→ K0π+
D0→ K0π0
D0→ K+¯K0
D0→ K+K−
D0→ K0¯K0
(2.5 ± 0.7) × 10−3
(1.43 ± 0.07) × 10−3
(8.4 ± 2.2) × 10−4
(2.77 ± 0.18) × 10−2
(3.80 ± 0.09) × 10−2
(2.28 ± 0.22) × 10−2
(1.48 ± 0.21) × 10−4
-
-
-
(5.8 ± 0.6) × 10−3
(4.12 ± 0.14) × 10−3
(7.1 ± 1.9) × 10−4
data is presented in Table 1. The second column gives the pure factorization result, where
the nonfactorization effect is zero, while the third column represents the branching ratio
with small nonfactorization correction. One can notice that the results are not in agreement
with the experimental data. For doubly Cabibbo-suppressed decay modes, the experimental
4

Page 5

measurements of their decay rates are unavailable, except for the channel D0→ K+π−. We
shall predict their theoretical branching ratios in section 4. The ratio for Cabibbo-suppressed
decay mode D0→ K0¯K0vanishes in the naive factorization approach. This decay seems to
be induced through final state rescattering.
3The one particle exchange method for FSI
As we have seen above, the experimental results for the branching ratios are mostly in
disagreement with the calculation from the naive factorization approach. The reason is that
the physical picture of naive factorization is too simple, nonperturbative strong interactions
are restricted in single hadrons, or between the initial and final hadrons which share the same
spectator quark. If the mass of the initial particle is large, such as the case of B meson decay,
the effect of nonperturbative strong interactions between the final hadrons most probably is
small because the momentum transfer is large. However, in the case of D meson, its mass is
not so large. The energy scale of D decays is not very high. Nonperturbative effects may give
large contribution. Because there exist many resonances near the mass scale of D meson, it
is possible that nonperturbative interactions propagate through these resonance states, such
as, K∗(892), K∗(1430), f0(1710), ρ(770), φ(1020) etc.
The diagrams of these nonperturbative rescattering effects can be depicted in Figs.1 and 2.
The first part D → P1P2or D → V1V2represents the direct decay where the decay amplitudes
can be obtained by using naive factorization method. The second part represents rescattering
process where the effective hadronic couplings are needed in numerical calculation, which
can be extracted from experimental data on the relevant resonance decays.
w
&%
P2
'$
ww
QQQQQ
?????
-
-
P1
-
-
3
s
D
0++
P3
P4
Figure 1: s-channel contributions to final-state interactions in D → PP due to one particle
exchange.
Fig.1 is the s-channel contributions to the final state interactions. Here P1and P2are the
5