Charmless Three-body Decays of B Mesons
July, 2007
Charmless Three-body Decays of B Mesons
Hai-Yang Cheng1, Chun-Khiang Chua2 and Amarjit Soni3
1 Institute of Physics, Academia Sinica
Taipei, Taiwan 115, Republic of China
2 Department of Physics, Chung Yuan Christian University
Chung-Li, Taiwan 320, Republic of China
3 Physics Department, Brookhaven National Laboratory
Upton, New York 11973
Abstract
An exploratory study of charmless 3-body decays of B mesons is presented using a simple model
based on the framework of the factorization approach. The nonresonant contributions arising from
B → P1P2 transitions are evaluated using heavy meson chiral perturbation theory (HMChPT).
The momentum dependence of nonresonant amplitudes is assumed to be in the exponential form
e−αNRpB ·(pi+pj) so that the HMChPT results are recovered in the soft meson limit pi, pj → 0. In
addition, we have identified another large source of the nonresonant signal in the matrix elements
of scalar densities, e.g. 〈KK|s̄s|0〉, which can be constrained from the decay B0 → KSKSKS
or B− → K−KSKS . The intermediate vector meson contributions to 3-body decays are identified
through the vector current, while the scalar meson resonances are mainly associated with the scalar
density. Their effects are described in terms of the Breit-Wigner formalism. Our main results are:
(i) All KKK modes are dominated by the nonresonant background. The predicted branching
ratios of K+K−KS(L), K
+K−K− and K−KSKS modes are consistent with the data within errors.
(ii) Although the penguin-dominated B0 → K+K−KS decay is subject to a potentially significant
tree pollution, its effective sin 2β is very similar to that of the KSKSKS mode. However, direct
CP asymmetry of the former, being of order −4%, is more prominent than the latter. (iii) For
B → Kππ decays, we found sizable nonresonant contributions in K−π+π− and K0π+π− modes,
in agreement with the Belle measurements but larger than the BaBar result. (iv) Time-dependent
CP asymmetries in KSπ
0π0, a purely CP -even state, and KSπ
+π−, an admixture of CP -even
and CP -odd components, are studied. (v) The π+π−π0 mode is found to have a rate larger than
π+π−π− even though the former involves a π0 in the final state. They are both dominated by
resonant ρ contributions. (vi) We have computed the resonant contributions to 3-body decays
and determined the rates for the quasi-two-body decays B → V P and B → SP . The predicted
ρπ, f0(980)K and f0(980)π rates are in agreement with the data, while the calculated φK, K
∗π, ρK
andK∗0 (1430)π are in general too small compared to experiment. (vii) Sizable direct CP asymmetry
is found in K+K−K− and K+K−π− modes.
http://arxiv.org/abs/0704.1049v2
I. INTRODUCTION
Three-body decays of heavy mesons are more complicated than the two-body case as they
receive resonant and nonresonant contributions and involve 3-body matrix elements. The three-
body meson decays are generally dominated by intermediate vector and scalar resonances, namely,
they proceed via quasi-two-body decays containing a resonance state and a pseudoscalar meson.
The analysis of these decays using the Dalitz plot technique enables one to study the properties
of various resonances. The nonresonant background is usually believed to be a small fraction of
the total 3-body decay rate. Experimentally, it is hard to measure the direct 3-body decays as the
interference between nonresonant and quasi-two-body amplitudes makes it difficult to disentangle
these two distinct contributions and extract the nonresonant one.
The Dalitz plot analysis of 3-body B decays provides a nice methodology for extracting infor-
mation on the unitarity triangle in the standard model (SM). For example, the Dalitz analysis
combined with isospin symmetry allows one to extract the angle α from B → ρπ → πππ [1].
Recently, a method has been proposed in [2] for determining CKM parameters in 3-body decays
B → Kππ and Bs → Kππ. This method was extended further in [3] to ∆I = 1, I(K∗π) = 1/2
amplitudes in the above decays and to I = 1 amplitudes in Bs → K∗K and Bs → K
Nonresonant 3-body decays of charmed mesons have been measured in several channels and the
nonresonant signal in charm decays are found to be less than 10% [4]. In the past few years, some
of the charmless B to 3-body decay modes have also been measured at B factories and studied
using the Dalitz plot analysis. The measured fractions and the corresponding branching ratios of
nonresonant components for some of 3-body B decay modes are listed in Table I. We see that
the nonresonant 3-body decays could play an essential role in B decays. It is now well established
that the B → KKK modes are dominated by the nonresonant background. For example, the
nonresonant fraction is about 90% in B
0 → K+K−K0 decay. While this is a surprise in view of
the rather small nonresonant contributions in 3-body charm decays, it is not entirely unexpected
because the energy release scale in weak B decays is of order 5 GeV, whereas the major resonances
lie in the energy region of 0.77 to 1.6 GeV. Consequently, it is likely that 3-body B decays may
receive sizable nonresonant contributions. At any rate, it is important to understand and identify
the underlying mechanism for nonresonant decays.
The direct nonresonant three-body decays of mesons in general receive two distinct contributions:
one from the point-like weak transition and the other from the pole diagrams that involve three-
point or four-point strong vertices. For D decays, attempts of applying the effective SU(4)×SU(4)
chiral Lagrangian to describe the DP → DP and PP → PP scattering at energies ∼ mD have
been made by several authors [18, 19, 20, 21, 22] to calculate the nonresonant D decays, though
in principle it is not justified to employ the SU(4) chiral symmetry. As shown in [21, 22], the
predictions of the nonresonant decay rates in chiral perturbation theory are in general too small
when compared with experiment. With the advent of heavy quark symmetry and its combination
with chiral symmetry [23, 24, 25], the nonresonant D decays can be studied reliably at least in the
kinematic region where the final pseuodscalar mesons are soft. Some of the direct 3-body D decays
were studied based on this approach [26, 27].
TABLE I: Branching ratios of various charmless three-body decays of B mesons. The fractions
and the corresponding branching ratios of nonresonant (NR) components are included whenever
available. The first and second entries are BaBar and Belle results, respectively.
Decay BR(10−6) BRNR(10
−6) NR fraction(%) Ref.
B− → π+π−π− 16.2 ± 1.2± 0.9 2.3± 0.9± 0.5 < 4.6 13.6 ± 6.1 [5]
– – –
B− → K−π+π− 64.1 ± 2.4± 4.0 2.87 ± 0.65 ± 0.43+0.63−0.25 4.5± 1.5 [6]
48.8 ± 1.1± 3.6 16.9 ± 1.3± 1.3+1.1−0.9 34.0 ± 2.2
−1.8 [7]
B− → K+K−K− 35.2 ± 0.9± 1.6 a 50± 6± 4 141± 16± 9 [8]
32.1 ± 1.3 ± 2.4 b 24.0 ± 1.5± 1.5 c 74.8 ± 3.6 c [9]
B− → K−KSKS 10.7 ± 1.2± 1.0 [10]
13.4 ± 1.9± 1.5 [11]
0 → K0π+π− 43.0 ± 2.3± 2.3 [12]
47.5 ± 2.4± 3.7 19.9 ± 2.5± 1.6+0.7−1.2 41.9± 5.1 ± 0.6
−2.5 [13]
0 → K−π+π0 34.9 ± 2.1± 3.9 < 4.6 [14]
36.6+4.2−4.3 ± 3.0 5.7
+2.7+0.5
−2.5−0.4 < 9.4 [15]
0 → K+K−K0 23.8 ± 2.0± 1.6 26.7 ± 4.6 112.0 ± 14.9 [16]
28.3 ± 3.3± 4.0 [11]
0 → KSKSKS 6.9+0.9−0.8 ± 0.6 [17]
4.2+1.6−1.3 ± 0.8 [11]
aWhen the intrinsic charm contribution is excluded, the charmless branching ratio will become (33.5±0.9±
1.6)× 10−6.
bWhen the contribution from B+ → χc0K+ is excluded, the charmless branching ratio will become (30.6±
1.2± 2.3)× 10−6.
cBelle found two solutions for the fractions and branching ratios. We follow Belle to use the large solution.
For the case of B mesons, consider the three-bodyB decay B → P1P2P3. Under the factorization
hypothesis, one of the nonresonant contributions arises from the transitions B → P1P2. The
nonresonant background in charmless three-body B decays due to the transition B → P1P2 has
been studied extensively [28, 29, 30, 31, 32, 33] based on heavy meson chiral perturbation theory
(HMChPT) [23, 24, 25]. However, the predicted decay rates are, in general, unexpectedly large. For
example, the branching ratio of the nonresonant decay B− → π+π−π− is predicted to be of order
10−5 in [28, 29], which is too large compared to the limit 4.6 × 10−6 set by BaBar [5]. Therefore,
it is important to reexamine and clarify the existing calculations.
The issue has to do with the applicability of HMChPT. In order to apply this approach, two
of the final-state pseudoscalars in B → P1P2 transition have to be soft. The momentum of the
soft pseudoscalar should be smaller than the chiral symmetry breaking scale of order 1 GeV. For
3-body charmless B decays, the available phase space where chiral perturbation theory is applicable
is only a small fraction of the whole Dalitz plot. Therefore, it is not justified to apply chiral and
heavy quark symmetries to a certain kinematic region and then generalize it to the region beyond
its validity. In this work we shall assume the momentum dependence of nonresonant amplitudes in
the exponential form e−αNRpB ·(pi+pj) so that the HMChPT results are recovered in the soft meson
limit pi, pj → 0. We shall see that the parameter αNR can be fixed from the tree-dominated decay
B− → π+π−π−.
However, the nonresonant background in B → P1P2 transition does not suffice to account for
the experimental observation that the penguin-dominated decay B → KKK is dominated by the
nonresonant contributions. This implies that the two-body matrix element e.g. 〈KK|s̄s|0〉 induced
by the scalar density should have a large nonresonant component. In the absence of first-principles
calculation, we will use the B
0 → KSKSKS mode in conjunction with the mass spectrum in
0 → K+K−K0 to fix the nonresonant contribution to 〈KK|s̄s|0〉.
In this work, we shall study the charmless 3-body decays of B mesons using the factorization
approach. Besides the nonresonant background as discussed above, we will also study resonant
contributions to 3-body decays. Vector meson and scalar resonances contribute to the two-body
matrix elements 〈P1P2|Vµ|0〉 and 〈P1P2|S|0〉, respectively. They can also contribute to the three-
body matrix element 〈P1P2|Vµ − Aµ|B〉. Resonant effects are described in terms of the usual
Breit-Wigner formalism. In this manner we are able to figure out the relevant resonances which
contribute to the 3-body decays of interest and compute the rates of B → V P and B → SP .
In conjunction with the nonresonant contribution, we are ready to calculate the total rates for
three-body decays.
It should be stressed from the outset that in this work we take the factorization approximation
as a working hypothesis rather than a first-principles starting point. If we start with theories such
as QCD factorization [34], or pQCD [35] or soft-collinear effective theory [36], then we can take
power corrections seriously and make an estimation. Since factorization has not been proved for
three-body B decays, we shall work in the phenomenological factorization model rather than in the
established theories such as QCDF. That is, we start with the simple idea of factorization and see
if it works for three-body decays, in the hope that it will provide a useful zeroth step for others to
try to improve.
The penguin-induced three-body decays B0 → K+K−KS and KSKSKS deserve special atten-
tion as the current measurements of the deviation of sin 2βeff in KKK modes from sin 2βJ/ψKS
may indicate New Physics in b → s penguin-induced modes. It is of great importance to examine
and estimate how much of the deviation of sin 2βeff is allowed in the SM. Owing to the presence
of color-allowed tree contributions in B0 → K+K−KS , this mode is subject to a potentially signif-
icant tree pollution and the deviation of the mixing-induced CP asymmetry from that measured
in B → J/ψKS could be as large as O(0.10). Since the tree amplitude is tied to the nonresonant
background, it is very important to understand the nonresonant contributions in order to have a
reliable estimate of sin 2βeff in KKK modes.
The layout of the present paper is as follows. In Sec. II we shall apply the factorization
approach to study B0 → K+K−KS and KSKSKS decays and discuss resonant and nonresonant
contributions. In order to set up the framework for calculations we will discuss B → KKK modes
in most details. We then turn to Kππ modes in Sec. III. The tree-dominated modes KKπ in
Sec. IV, and πππ in Sec. V. In Sec. VI, we determine the rates for B → V P and B → SP and
compare our results with the approach of QCD factorization. Sec. VII contains our conclusions.
The factorizable amplitudes of various B → P1P2P3 decays are summarized in Appendix A. The
relevant input parameters such as decay constants, form factors, etc. are collected in Appendix B.
II. B → KKK DECAYS
For 3-body B decays, the b → sqq̄ penguin transitions contribute to the final states with odd
number of kaons, namely, KKK and Kππ, while b → uqq̄ tree and b → dqq̄ penguin transitions
contribute to final states with even number of kaons, e.g. KKπ and πππ. We shall first discuss the
b→ s penguin dominated 3-body decays in details and then turn to b→ u tree dominated modes.
For B → KKK modes, we shall first consider the neutral B decays as they involve mixing-induced
CP asymmetries.
0 → KKK decays
We consider the decay B
0 → K+K−K0 as an illustration. Under the factorization approach,
the B0 → K+K−K0 decay amplitude consists of three distinct factorizable terms: (i) the current-
induced process with a meson emission, 〈B0 → K+K0〉 × 〈0 → K−〉, (ii) the transition process,
〈B0 → K0〉 × 〈0 → K+K−〉, and (iii) the annihilation process 〈B0 → 0〉 × 〈0 → K+K−K0〉, where
〈A → B〉 denotes a A → B transition matrix element. In the factorization approach, the matrix
element of the B → KKK decay amplitude is given by
〈KKK|Heff |B〉 =
p=u,c
λ(s)p 〈KKK|Tp|B〉, (2.1)
where λ
p ≡ VpbV ∗ps and the explicit expression of Tp in terms of four-quark operators is given in
Eq. (A2). The factorizable B
0 → K+K−K0 decay amplitude is given in Eq. (A4). Note that
the OZI suppressed matrix element 〈K+K−|(d̄d)V −A|0〉 is included in the factorizable amplitude
since it could be enhanced through the long-distance pole contributions via the intermediate vector
mesons such as ρ0 and ω. Likewise, the OZI-suppressed matrix elements 〈K+K−|(d̄b)
|B0〉 and
〈K+K−|d̄(1 − γ5)b|B
0〉 are included as they receive contributions from the scalar resonances like
f0(980).
For the current-induced process, the two-meson transition matrix element 〈K0K+|(ūb)V −A|B0〉
has the general expression [37]
〈K0(p1)K+(p2)|(ūb)V−A|B0〉 = ir(pB − p1 − p2)µ + iω+(p2 + p1)µ + iω−(p2 − p1)µ
+h ǫµναβp
B(p2 + p1)
α(p2 − p1)β, (2.2)
where (q̄1q2)V −A ≡ q̄1γµ(1− γ5)q2. This leads to
AHMChPTcurrent−ind ≡ 〈K−(p3)|(s̄u)V−A|0〉〈K0(p1)K+(p2)|(ūb)V −A|B0〉
2m23r + (m
B − s12 −m23)ω+ + (s23 − s13 −m22 +m21)ω−
, (2.3)
(c) (d)
FIG. 1: Point-like and pole diagrams responsible for the B
0 → K+K0 matrix element induced by
the current ūγµ(1− γ5)b, where the symbol • denotes an insertion of the current.
where sij ≡ (pi + pj)2. To compute the form factors r, ω± and h, one needs to consider not
only the point-like contact diagram, Fig. 1(a), but also various pole diagrams depicted in Fig. 1.
In principle, one can apply HMChPT to evaluate the form factors r, ω+ and ω− [37]. However,
this will lead to too large decay rates in disagreement with experiment [38]. The heavy meson
chiral Lagrangian given in [23, 24, 25] is needed to compute the strong B∗BP , B∗B∗P and BBPP
vertices. The results for the form factors are [29, 37]
ω+ = −
fB∗smB∗s
mBmB∗s
s23 −m2B∗s
1− (pB − p1) · p1
m2B∗s
fB∗smB∗s
mBmB∗s
s23 −m2B∗s
(pB − p1) · p1
m2B∗s
pB · (p2 − p1)
(pB − p1 − p2)2 −m2B
2gfB∗s
(pB − p1) · p1
s23 −m2B∗s
4g2fB
mBmB∗s
(pB − p1 − p2)2 −m2B
p1 ·p2 − p1 ·(pB − p1) p2 ·(pB − p1)/m2B∗s
s23 −m2B∗s
, (2.4)
where fπ = 132 MeV, g is a heavy-flavor independent strong coupling which can be extracted from
the CLEO measurement of the D∗+ decay width, |g| = 0.59± 0.01± 0.07 [39]. We shall follow [23]
to fix its sign to be negative. The point-like diagram Fig. 1(a) characterized by the term fB/(2f
contributes to the form factors ω+ and r, while Figs. 1(b) and 1(d) contribute to r and Fig. 1(c)
contributes to all the form factors.
A direct calculation indicates that the branching ratio of B
0 → K+K−K0 arising from the
current-induced process alone is already at the level of 77 × 10−6 which exceeds the measured
total branching ratio of 25 × 10−6 (see Table I). The issue has to do with the applicability of
HMChPT. In order to apply this approach, two of the final-state pseudoscalars (K+ and K
this example) have to be soft. The momentum of the soft pseudoscalar should be smaller than
the chiral symmetry breaking scale Λχ of order 0.83 − 1.0 GeV. For 3-body charmless B decays,
the available phase space where chiral perturbation theory is applicable is only a small fraction of
the whole Dalitz plot. Therefore, it is not justified to apply chiral and heavy quark symmetries
to a certain kinematic region and then generalize it to the region beyond its validity. If the soft
meson result is assumed to be the same in the whole Dalitz plot, the decay rate will be greatly
overestimated.
In [38, 40] we have tried to circumvent the aforementioned problem by applying HMChPT only
to the strong vertex and use the form factors to describe the weak vertex. Moreover, we introduced
a form factor to take care of the off-shell effect. For example, Fig. 1(c) can be evaluated by
considering the strong interaction B0 → K0B∗s followed by the weak transition B
s → K+ and the
result is [38]
AF ig.1(c) =
mBmB∗s
s23 −m2B∗s
F (s23,mB∗s )F
m2B − s23
FBsK1 (m
m21 +m
3 − s13 +
(s23 −m22 +m23)(m2B − s23 −m21)
2m2B∗s
, (2.5)
where FBsK0,1 are the Bs → K weak transition from factors in the standard convention [41] and we
have introduced a form factor F (s23,mB∗s ) to take into account the off-shell effect of the B
s pole [40].
It is parameterized as F (s23,mB∗s ) = (Λ
2−m2B∗s )/(Λ
2−s23) with the cut-off parameter Λ chosen to
be Λ = mB∗s +ΛQCD. Needless to say, this parametrization of the form factor is somewhat arbitrary.
Moreover, the nonresonant contribution thus calculated is too small compared to experiment.
The Dalitz plot analysis of B
0 → K+K−K0 has been recently performed by BaBar [16]. In the
BaBar analysis, a phenomenological parametrization of the non-resonant amplitudes is described
ANR = (c12e
iφ12e−αs
12 + c13e
iφ13e−αs
13 + c23e
iφ23e−αs
23)(1 + bNRe
i(β+δNR)), (2.6)
and resonant terms are described by
cr(1 + br)fre
i(φr+δr+β), ĀR =
cr(1 − br)frei(φr−δr+β). (2.7)
The BaBar results for isobar amplitudes, phases and fractions from the fit to the B0 → K+K−K0
are summarized in Table II. It is evident that this decay is dominated by the nonresonant back-
ground. For our purpose, we will parametrize the current-induced nonresonant amplitude Eq. (2.3)
Acurrent−ind = A
HMChPT
current−ind e
pB·(p1+p2)eiφ12 , (2.8)
so that the HMChPT results are recovered in the chiral limit p1, p2 → 0. That is, the nonresonant
amplitude in the soft meson region is described by HMChPT, but its energy dependence beyond
the chiral limit is governed by the exponential term e−αNRpB·(p1+p2). In what follows, we shall use
the tree-dominated B− → π+π−π− decay data to fix α
, which turns out to be
= 0.103+0.018−0.011 GeV
−2. (2.9)
This is very close to the naive expectation of α
∼ O(1/(2mBΛχ)) based on the dimensional
argument. The phase φ12 of the nonresonant amplitude in the (K
) system will be set to zero
for simplicity.
TABLE II: BaBar results for isobar amplitudes, phases, and fractions from the fit to the B0 →
K+K−K0 [16]. Three rows for non-resonant contribution correspond to coefficients of exponential
functions in Eq. (2.6), while the fraction is given for the combined amplitude. For the nonresonant
decay mode in K+K−, the amplitude c12 and the phase φ12 in Eq. (2.6) are fixed to be one and
zero, respectively. Errors are statistical only.
Decay Amplitude cr Phase φr Fraction (%)
φ(1020)K0 0.0085 ± 0.0010 −0.016 ± 0.234 12.5 ± 1.3
f0(980)K
0 0.622 ± 0.046 −0.14 ± 0.14 40.2 ± 9.6
X0(1550)K
0 0.114 ± 0.018 −0.47 ± 0.20 4.1± 1.3
(K+K−)NRK
0 1 (fixed) 0 (fixed)
(K+K0)NRK
− 0.33± 0.07 1.95 ± 0.27 112.0 ± 14.9
(K−K0)NRK
+ 0.31± 0.08 −1.34 ± 0.37
χc0(1P )K
0 0.0306 ± 0.00649 0.81−2.33 ± 0.54 3.0± 1.2
D+K− 1.11± 0.17 – 3.6± 1.5
D+s K
− 0.76± 0.14 – 1.8± 0.6
For the transition amplitude, we need to evaluate the 2-kaon creation matrix element which can
be expressed in terms of time-like kaon current form factors as
〈K+(pK+)K−(pK−)|q̄γµq|0〉 = (pK+ − pK−)µFK
〈K0(pK0)K
(pK̄0)|q̄γµq|0〉 = (pK0 − pK̄0)µFK
q . (2.10)
The weak vector form factors FK
q and F
K0K̄0
q can be related to the kaon electromagnetic (e.m.)
form factors FK
em and F
K0K̄0
em for the charged and neutral kaons, respectively. Phenomenologi-
cally, the e.m. form factors receive resonant and nonresonant contributions and can be expressed
em = Fρ + Fω + Fφ + FNR, F
K0K̄0
em = −Fρ + Fω + Fφ + F ′NR. (2.11)
It follows from Eqs. (2.10) and (2.11) that
u = F
K0K̄0
d = Fρ + 3Fω +
(3FNR − F ′NR),
d = F
K0K̄0
u = −Fρ + 3Fω,
s = F
K0K̄0
s = −3Fφ −
(3FNR + 2F
NR), (2.12)
where use of isospin symmetry has been made.
The resonant and nonresonant terms in Eq. (2.11) can be parametrized as
Fh(s23) =
m2h − s23 − imhΓh
NR(s23) =
, (2.13)
with Λ̃ ≈ 0.3 GeV. The expression for the nonresonant form factor is motivated by the asymptotic
constraint from pQCD, namely, F (t) → (1/t)[ln(t/Λ̃2)]−1 in the large t limit [42]. The unknown
parameters ch, xi and x
i are fitted from the kaon e.m. data, giving the best fit values (in units of
GeV2 for ch) [43]:
cρ = 3cω = cφ = 0.363, cρ(1450) = 7.98 × 10−3, cρ(1700) = 1.71 × 10−3,
cω(1420) = −7.64× 10−2, cω(1650) = −0.116, cφ(1680) = −2.0× 10−2,
(2.14)
x1 = −3.26 GeV2, x2 = 5.02 GeV4, x′1 = 0.47 GeV2, x′2 = 0. (2.15)
Note that the form factors Fρ,ω,φ in Eqs. (2.11) and (2.12) include the contributions from the vector
mesons ρ(770), ρ(1450), ρ(1700), ω(782), ω(1420), ω(1650), φ(1020) and φ(1680). It is interesting
to note that (i) the fitted values of cV are very close to the vector meson dominance expression
gV KK for V = ρ, ω, φ [4, 44], where gV γ is the e.m. coupling of the vector meson defined by
〈V |jem|0〉 = gV γε∗V and gV KK is the V → KK strong coupling with −gφK+K− ≃ gρK+K−/
gωK+K−/
2 ≃ 3.03, and (ii) the vector-meson pole contributions alone yield FK+K−u,s (0) ≈ 1,−1
and FK
d (0) ≈ 0 as the charged kaon does not contain the valence d quark. The matrix element
for the current-induced decay process then has the expression
〈K0(p1)|(s̄b)V −A|B0〉〈K+(p2)K−(p3)|(q̄q)V−A|0〉 = (s12 − s13)FBK1 (s23)FK
q (s23). (2.16)
We also need to specify the 2-body matrix elements 〈K+K−|s̄s|0〉〈K0|s̄b|B0〉 induced from the
scalar densities. The use of the equation of motion leads to
〈K0(p1)|s̄b|B
(pB)〉 =
m2B −m2K
mb −ms
FBK0 (s23). (2.17)
The matrix element 〈K+K−|s̄s|0〉 receives resonant and non-resonant contributions:
〈K+(p2)K−(p3)|s̄s|0〉 ≡ fK
s (s23) =
mf0i f̄
gf0i→K
m2f0i
− s23 − imf0iΓf0i
+ fNRs ,
fNRs =
(3FNR + 2F
NR) + σNRe
−α s23 , (2.18)
where f0i denote the generic f0-type scalar mesons, f0i = f0(980), f0(1370), f0(1500),X0(1550), · · ·,
the scalar decay constant f̄ sf0i
is defined by 〈f0i|s̄s|0〉 = mf0i f̄
[see Eq. (B1)], gf0i→K
+K− is the
f0i → K+K− strong coupling, and the nonresonant terms are related to those in FK
s through
the equation of motion. The presence of the nonresonant σ
term will be explained shortly. The
main scalar meson pole contributions are those that have dominant ss̄ content and large coupling
to KK. We consider the scalar mesons f0(980) and X0(1550) (denoted as fX(1500) by Belle) which
are supposed to have the largest couplings with the KK pair. Note that the nature of the broad
state X0(1550) observed by BaBar and Belle, for example, what is its relation with f0(1500), is
not clear. To proceed with the numerical calculations, we shall use gf0(980)→K
+K− = 4.3 GeV,1
1 This is different from the coupling gf0(980)→K
+K− = 1.5 GeV originally employed in [40]. The coupling
gf0(980)→π
+π− ∼ 1.33 GeV can be fixed from a recent Belle measurement of Γ(f0(980) → π+π−) [see Eq.
(3.18)]. Using the BES result (gf0(980)→KK/gf0(980)→ππ)2 = 4.21± 0.25± 0.21 [45], one can deduce that
gf0(980)→KK = 2.7± 0.6 GeV. In this work, we found that a slightly large coupling gf0(980)→KK will give
better numerical results.
gX0(1550)→K
+K− = 1.4 GeV, Γf0(980) = 80 MeV, ΓX0(1550) = 0.257 GeV [8], f̄f0(980)(µ = mb/2) ≃
0.46 GeV [46] and f̄f0(1530) ≃ 0.30 GeV. The sign of the resonant terms is fixed by fK
s (0) = v
from a chiral perturbation theory calculation (see, for example, [47]). It should be stressed that
although the nonresonant contributions to fKKs and F
s are related through the equation of
motion, the resonant ones are different and not related a priori. As stressed in [40], to apply the
equation of motion, the form factors should be away from the resonant region. In the presence
of the resonances, we thus need to introduce a nonresonant term characterized by the parameter
σNR in Eq. (2.18) which will be specified later. The parameter α appearing in the same equation
should be close to the value of α
given in Eq. (2.9). We will use the experimental measurement
α = (0.14 ± 0.02)GeV−2 [16].
As noticed before, the matrix elements 〈K+K−|(d̄b)
|B0〉 and 〈K+K−|d̄(1 − γ5)b|B
0〉 are
included in Eq. (A4) as they receive intermediate scalar pole contributions. More explicitly,
〈K+(p2)K−(p3)|(d̄b)V −A |B
0〉R =
gf0i→K
m2f0i
− s23 − imf0iΓf0i
〈f0i|(d̄b)V −A |B
0〉. (2.19)
Hence,
〈K0(p1)|(s̄d)V −A |0〉〈K
+(p2)K
−(p3)|(d̄b)V −A |B
gf0i→K
m2f0i
− s23 − imf0iΓf0i
B −m2f0i). (2.20)
The superscript u of the form factor F
0 reminds us that it is the uū quark content that gets
involved in the B to f0i form factor transition. In short, the relevant f0(980) pole contributions to
0 → K+K−K0 are
〈K0K+K−|Tp|B
0〉f0 =
gf0(980)→K
m2f0 − s23 − imf0Γf0
f̄ sf0F
)(m2B −m2K)
+ fKF
B −m2f0)
10 − (a
, (2.21)
where we have employed Eq. (2.18) and applied equations of motion to the matrix elements
〈K0|s̄γ5d|0〉〈K+K−|d̄γ5b|B
0〉. Comparing this equation with Eq. (A6) of [48], we see that the
expression inside {· · ·} is identical to that of B0 → f0(980)K
, as it should be.
We digress for a moment to discuss the wave function of the f0(980). What is the quark structure
of the light scalar mesons below or near 1 GeV has been quite controversial. In this work we shall
consider the conventional qq̄ assignment for the f0(980). In the naive quark model, the flavor wave
functions of the f0(980) and σ(600) read
(uū+ dd̄), f0 = ss̄, (2.22)
where the ideal mixing for f0 and σ has been assumed. In this picture, f0(980) is purely an ss̄
state. However, there also exist some experimental evidences indicating that f0(980) is not purely
an ss̄ state. First, the observation of Γ(J/ψ → f0ω) ≈ 12Γ(J/ψ → f0φ) [4] clearly indicates the
existence of the non-strange and strange quark content in f0(980). Second, the fact that f0(980)
and a0(980) have similar widths and that the f0 width is dominated by ππ also suggests the
composition of uū and dd̄ pairs in f0(980); that is, f0(980) → ππ should not be OZI suppressed
relative to a0(980) → πη. Therefore, isoscalars σ(600) and f0 must have a mixing
|f0(980)〉 = |ss̄〉 cos θ + |nn̄〉 sin θ, |σ(600)〉 = −|ss̄〉 sin θ + |nn̄〉 cos θ, (2.23)
with nn̄ ≡ (ūu+ d̄d)/
2. Experimental implications for the f0−σ mixing angle have been discussed
in detail in [49]. It is found that θ lies in the ranges of 25◦ < θ < 40◦ and −40◦ < θ < −15◦ (or
140◦ < θ < 165◦). Note that the phenomenological analysis of the radiative decays φ → f0(980)γ
and f0(980) → γγ favors a solution of the θ to be negative (or in the second quadrant). In this
work, we shall use θ = −25◦.
Finally, the matrix elements involving 3-kaon creation are given by [38]
〈K0(p1)K+(p2)K−(p3)|(s̄d)V −A|0〉〈0|(d̄b)V −A|B0〉 ≈ 0, (2.24)
〈K0(p1)K+(p2)K−(p3)|s̄γ5d|0〉〈0|d̄γ5b|B0〉 = v
s13 −m21 −m23
m2B −m2K
FKKK(m2B),
where
mu +ms
m2K −m2π
ms −md
, (2.25)
characterizes the quark-order parameter 〈q̄q〉 which spontaneously breaks the chiral symmetry.
Both relations in Eq. (2.24) are originally derived in the chiral limit [38] and hence the quark
masses appearing in Eq. (2.25) are referred to the scale ∼ 1 GeV . The first relation reflects
helicity suppression which is expected to be even more effective for energetic kaons. For the second
relation, we introduce the form factor FKKK to extrapolate the chiral result to the physical region.
Following [38] we shall take FKKK(q2) = 1/[1 − (q2/Λ2χ)] with Λχ = 0.83 GeV being a chiral
symmetry breaking scale.
To proceed with the numerical calculations, we need to specify the input parameters. The
relevant CKM matrix elements, decay constants, form factors, the effective Wilson coefficients
i and the running quark masses are collected in Appendix B. As for the parameter σNR in Eq.
(2.18), in principle we can set its phase φσ to zero and use the measured KSKSKS rate, namely,
B(B0 → KSKSKS) = (6.2±0.9)×10−6 [50], to fix the parameter σNR and then use the data obtained
from the Dalitz plot analysis to determine the strong phases φr for resonant amplitudes. However,
in doing so one needs the data of invariant mass spectra. In the absence of such information, instead
we will treat φσ as a free parameter and do not assign any other strong phases to the resonant
amplitudes except for those arising from the Breit-Wigner formalism. It turns out that if φσ is
small, the K+K− mass spectrum in B
0 → K+K−KS will have a prominent hump at the invariant
mass mK+K− = 3 GeV, which is not seen experimentally (see Fig. 2(c)). We found that φσ ≈ π/4
will yield K+K− mass spectrum consistent with the data
= eiπ/4
3.36+1.12−0.96
GeV. (2.26)
Note that the phase of σ
is consistent with the BaBar measurement shown in Table II, namely,
φBaBarσ = 1.19± 0.37.
The calculated branching ratios of resonant and nonresonant contributions to B
0 → K+K−K0
are summarized in Table III. The theoretical errors shown there are from the uncertainties in (i)
TABLE III: Branching ratios (in units of 10−6) of resonant and nonresonant (NR) contributions to
0 → K+K−K0. Theoretical errors correspond to the uncertainties in (i) α
, (ii) ms, F
0 and
, and (iii) γ = (59 ± 7)◦. We do not have 1/mb power corrections within this model. However,
systematic errors due to model dependent assumptions may be sizable and are not included in the
error estimates that we give. Experimental results are taken from Table II.
Decay mode BaBar [16] Theory
2.98 ± 0.45 2.6+0.0+0.5+0.0−0.0−0.4−0.0
f0(980)K
9.57 ± 2.51 5.8+0.0+0.1+0.0−0.0−0.5−0.0
X0(1550)K
− 0.98 ± 0.33 0.93+0.00+0.16+0.00−0.00−0.15−0.00
NR 26.7 ± 4.6 18.1+0.6+5.1+0.2−0.7−3.8−0.2
total 23.8 ± 2.0± 1.6 19.8+0.4+0.5+0.1−0.4−0.4−0.2
the parameter αNR which governs the momentum dependence of the nonresonant amplitude, (ii)
the strange quark mass ms, the form factor F
0 and the nonresonant parameter σNR , and (iii) the
unitarity angle γ.
In QCD calculations based on a heavy quark expansion, one faces uncertainties arising from
power corrections such as annihilation and hard-scattering contributions. For example, in QCD
factorization, there are large theoretical uncertainties related to the modelling of power corrections
corresponding to weak annihilation effects and the chirally-enhanced power corrections to hard
spectator scattering. Even for two-body B decays, power corrections are of order (10-20)% for
tree-dominated modes, but they are usually bigger than the central values for penguin-dominated
decays. Needless to say, 1/mb power corrections for three-body decays may well be larger. However,
as stressed in Introduction, in this exploratory work we use the phenomenological factorization
model rather than in the established theories based on a heavy quark expansion. Consequently,
uncertainties due to power corrections, at this stage, are not included in our calculations, by
assumption. In view of such shortcomings we must emphasize that the additional errors due to
such model dependent assumptions may be sizable.
From Table III we see that the predicted rates for resonant and nonresonant components are
consistent with experiment within errors. The nonresonant contribution arises dominantly from
the transition process (88%) via the scalar-density-induced vacuum to KK̄ transition, namely,
〈K+K−|s̄s|0〉, and slightly from the current-induced process (3%). Therefore, it is natural to
conjecture that nonresonant decays could also play a prominent role in other penguin dominated
3-body B decays.
The K+K−KS mode is an admixture of CP -even and CP -odd components. By excluding the
major CP -odd contribution from φKS , the 3-bodyK
+K−KS final state is primarily CP -even. The
K+K− mass spectra of the B0 → K+K−KS decay from CP -even and CP -odd contributions are
shown in Fig. 2. For the CP -even spectrum, there are peaks at the threshold and mK+K− = 1.5
GeV region. The threshold enhancement arises from the f0(980)KS and the nonresonant f
1 1.5 2 2.5 3 3.5 4 4.5
m K+ K- HGeVL
1 1.5 2 2.5 3
m K+ K- HGeVL
1 1.02 1.04
   (GeV)K+K-m
1 1.5 2 2.5 3 3.5 4 4.5
   (GeV)K+K-m
1 1.5 2 2.5 3 3.5 4 4.5
BABAR
preliminary
(a) (b) (c)
FIG. 2: The K+K− mass spectra for B0 → K+K−KS decay from (a) CP -even and (b) CP -odd
contributions. The insert in (b) is for the φ region. The full K+K−KS spectrum, which is the sum
of CP -even and CP -odd parts, measured by BaBar [16] is depicted in (c).
contributions [see Eq. (2.18)]. 2 For the CP -odd spectrum, the peak on the lower end corresponds
to the φKS contribution, which is also shown in the insert. The b → u transition is governed by
the current-induced process 〈B0 → K+K0〉 × 〈0 → K−〉 [see Eq. (A4)]. From Eq. (2.8) it is clear
that the b→ u amplitude prefers a small invariant mass of K+ and K0 and hence a large invariant
mass of K+ and K−. In contrast, the b → c amplitude prefers a small s23. Consequently, their
interference is largely suppressed. The full K+K−KS spectrum, which is the sum of the CP -even
and the CP -odd parts, has been measured by BaBar [Fig. 2(c)]. It clearly shows the phenomenon
of threshold enhancement and the scalar resonances X0(1550) and χc0.
The decay B
0 → KSKSKS is a pure penguin-induced mode [cf. Eq. (A7)] and it receives
intermediate pole contributions only from the iso-singlet scalar mesons such as f0(980). Just like
other KKK modes, this decay is governed by the nonresonant background dominated by the σ
term defined in Eq. (2.18). Hence, this mode is ideal for determining the unknown parameter σ
which is given in Eq. (2.26). Time-dependent CP violation in neutral 3-body decay modes with
fixed CP parity was first discussed by Gershon and Hazumi [51].
Results for the decay rates and CP asymmetries in B0 → K+K−KS(L), KSKSKS(L) are dis-
played in Table IV and Table V, respectively. (For the decay amplitudes of B0 → KSKSKS(L), see
[40] for details.) The mixing-induced CP violations are defined by
SKKK,CP± =
Im(e−2iβACP±Ā
CP±)ds12ds23
|ACP±|2ds12ds23 +
|ĀCP±|2ds12ds23
SKKK =
Im(e−2iβAĀ∗)ds12ds23
|A|2ds12ds23 +
|Ā|2ds12ds23
= f+ SKKK,CP+ + (1− f+)SKKK,CP−, (2.27)
2 In our previous work [40] we have argued that the spectrum should have a peak at the large mK+K−
end. This is because we have introduced an additional nonresonant contribution to the ω− parameter
parametrized as ωNR− = κ
2pB ·p2
and employed the B− → D0K0K− data and applied isospin symmetry
to the B → KK matrix elements to determine the unknown parameter κ. Since this nonresonant term
favors a small mK+KS region, a peak of the spectrum at large mK+K− is thus expected. However, such a
bump is not seen experimentally [16]. In this work we will no longer consider this term.
TABLE IV: Branching ratios for B0 → K+K−KS , KSKSKS , KSKSKL decays and the fraction
of CP-even contribution to B
0 → K+K−KS , f+. The branching ratio of CP-odd K+K−KS with
φKS excluded is shown in parentheses. Results for (K
+K−KL)CP± are identical to those for
(K+K−KS)CP∓. For theoretical errors, see Table III. Experimental results are taken from [50].
Final State B(10−6)theory B(10−6)expt
K+K−KS 9.89
+0.19+2.28+0.07
−0.21−1.81−0.08 12.4± 1.2
(K+K−KS)CP+ 8.33
+0.10+1.82+0.05
−0.12−1.49−0.06
(K+K−KS)CP− 1.57
+0.09+0.46+0.02
−0.10−0.32−0.02
(0.14+0.06+0.14+0.01−0.06−0.06−0.01)
KSKSKS input 6.2± 0.9
KSKSKL 7.63
+0.01+1.37+0.03
−0.01−1.19−0.03 < 14
theory
K+K−KS 0.98
+0.01+0.01+0.00
−0.01−0.02−0.00 0.91 ± 0.07
theory
K+K−KL 0.98
+0.01+0.01+0.00
−0.01−0.02−0.00
where A is the decay amplitude of B
0 → K+K−KS(L) or KSKSKS(L) and Ā is the conjugated B0
decay amplitude, and f+ is the CP even fraction defined by
ΓCP+ + ΓCP+
Γ + Γ
φKS excluded.
(2.28)
Generally, it is more convenient to define an effective sin 2β via Sf ≡ −ηf sin 2βeff with ηf = 2f+−1
for K+K−KS . The predicted value of f+ is consistent with the data but it is on the higher end of
the experimental measurement because the CP -odd contributions from the vector mesons ρ, ω, · · · ,
are OZI suppressed and the CP -odd nonresonant contribution is constrained by the π+π−π− rate.
The deviation of the mixing-induced CP asymmetry in B0 → K+K−KS and KSKSKS from
that measured in B → φcc̄KS , i.e. sin 2βφcc̄KS = 0.681± 0.025 [50], namely, ∆ sin 2βeff ≡ sin 2βeff −
sin 2βφcc̄KS , is calculated from Table V to be
∆ sin 2βK+K−KS = 0.047
+0.028
−0.033 ,
∆sin 2βKSKSKS = 0.038
+0.027
−0.032 . (2.29)
The corresponding experimental values are 0.049 ± 0.10 and −0.101 ± 0.20, respectively. Due to
the presence of color-allowed tree contributions in B
0 → K+K−KS , it is naively expected that this
penguin-dominated mode is subject to a potentially significant tree pollution and hence ∆ sin 2βeff
can be as large as O(10%). However, our calculation indicates the deviation of the mixing-induced
CP asymmetry in B
0 → K+K−KS from that measured in B
0 → φcc̄KS is very similar to that of
the KSKSKS mode as the tree pollution effect in the former is somewhat washed out. Nevertheless,
TABLE V: Mixing-induced and direct CP asymmetries sin 2βeff (top) and Af (in %, bottom),
respectively, in B
0 → K+K−KS and KSKSKS decays. Experimental results for K+K−KS and
K+K−KL modes are obtained from the data of B
0 → K+K−K0. Results for (K+K−KL)CP± are
identical to those for (K+K−KS)CP∓. For theoretical errors, see Table III. Experimental results
are taken from [50].
Final state sin 2βeff Expt.
(K+K−KS)φKS excluded 0.728
+0.001+0.002+0.009
−0.002−0.001−0.020 0.73 ± 0.10
(K+K−KS)CP+ 0.732
+0.003+0.006+0.009
−0.004−0.004−0.020
(K+K−KL)φKL excluded 0.728
+0.001+0.002+0.009
−0.002−0.001−0.020 0.73 ± 0.10
KSKSKS 0.719
+0.000+0.000+0.008
−0.000−0.000−0.019 0.58 ± 0.20
KSKSKL 0.718
+0.000+0.000+0.008
−0.000−0.000−0.019
Af (%) Expt.
(K+K−KS)φKS excluded −4.63
+1.35+0.53+0.40
−1.01−0.54−0.34 −7± 8
(K+K−KS)CP+ −4.86+1.43+0.52+0.42−1.09−0.55−0.35
(K+K−KL)φKL excluded −4.63
+1.35+0.53+0.40
−1.01−0.54−0.34 −7± 8
KSKSKS 0.69
+0.01+0.01+0.05
−0.01−0.01−0.06 14 ± 15
KSKSKL 0.77
+0.01+0.01+0.05
−0.01−0.03−0.07
direct CP asymmetry of the former, being of order −4%, is more prominent than the latter.3
B. B− → KKK decays
The B− → K+K−K− decay amplitude has a similar expression as Eq. (A4) except that one
also needs to add the contributions from the interchange s23 → s12 and put a factor of 1/2 in the
decay rate to account for the identical particle effect.
Branching ratios of resonant and nonresonant contributions to B− → K+K−K− are shown in
Table VI. It is clear that the predicted rates of resonant and nonresonant components are consistent
with the data except for the broad scalar resonance X0(1550). Both BaBar and Belle have seen
a large fraction from X0(1550), (121 ± 19 ± 6)% by BaBar [8] and (63.4 ± 6.9)% by Belle [9], 4
while our prediction is similar to that in B
0 → K+K−K0. It is not clear why there is a huge
3 In our previous work [40], ∆ sin 2βeff is found to be
∆ sin 2βK+K−KS = 0.06
+0.09
−0.04 , ∆sin 2βKSKSKS = 0.06
+0.03
−0.04 ,
for sin 2βJ/ψKS = 0.687 ± 0.032, while direct CP asymmetry is less than 1% in both modes. Note that
due to an oversight the experimental error bars were not included in our previous paper for the theoretical
calculation of ∆ sin 2βeff .
4 Belle [9] actually found two solutions for the fraction of X0(1550)K
−: (63.4± 6.9)% and (8.21± 1.94)%.
The first solution is preferred by Belle.
disparity between B− → K+K−K− and B0 → K+K−K0 as far as the X0(1550) contribution is
concerned. Obviously, a refined measurement of the X0(1550) contribution to the K
+K−K− mode
is urgently needed in order to clarify this issue. Our result for the nonresonant contribution is in
good agreement with Belle, but disagrees with BaBar. Notice that Belle did not see the scalar
resonance f0(980) as Belle employed the E791 result [53] for g
f0→KK̄ which is smaller than gf0→ππ.
In contrast to E791, the ratio gf0→KK̄/gf0→ππ is measured to be larger than 4 in the existing e+e−
experiments [45, 54]
TABLE VI: Branching ratios (in units of 10−6) of resonant and nonresonant (NR) contributions to
B− → K+K−K−. For theoretical errors, see Table III.
Decay mode BaBar [8] Belle [9] Theory
φK− 4.14 ± 0.32± 0.33 4.72 ± 0.45 ± 0.35+0.39−0.22 2.9
+0.0+0.5+0.0
−0.0−0.5−0.0
f0(980)K
− 6.5± 2.5± 1.6 < 2.9 7.0+0.0+0.4+0.1−0.0−0.7−0.1
X0(1550)K
− 43± 6± 3 1.1+0.0+0.2+0.0−0.0−0.2−0.0
f0(1710)K
− 1.7± 1.0± 0.3
NR 50± 6± 4 24.0 ± 1.5± 1.8+1.9−5.7 25.3
+0.9+4.8+0.3
−1.0−4.4−0.3
Total 35.2 ± 0.9± 1.6 32.1 ± 1.3± 2.4 25.5+0.5+4.4+0.2−0.6−4.1−0.2
We next turn to the decay B− → K−KSKS . Following [55], let us consider the symmetric state
of K0K
|K0K0〉sym ≡
|K0(p1)K
(p2)〉+ |K
(p1)K
0(p2)〉
= [|KS(p1)KS(p2)〉 − |KL(p1)KL(p2)〉] /
2. (2.30)
Hence,
B(B− → K−KSKS) =
[B(B− → K−KSKS) + B(B− → K−KLKL)]
B(B− → K−(K0K0)sym). (2.31)
The factorizable amplitude of B− → K−K0K0 is given by Eq. (A8). Just as other KKK modes,
this decay is also expected to be dominated by the nonresonant contribution (see Table VII).
The calculated total rate is in good agreement with experiment. Just as the pure penguin mode
KSKSKS , the decay B
− → K−KSKS also can be used to constrain the nonresonant parameter
As pointed out in [55], isospin symmetry implies the relation
A(B− → K−K0K0) = −A(B0 → K0K+K−). (2.32)
This leads to
B(B− → K−(K0K0)sym) =
τ(B−)
τ(B0)
B(B0 → K+K−K0)φK excluded. (2.33)
Experimentally, this relation is well satisfied: LHS=(23.0±2.6)×10−6 and RHS=(22.1±2.1)×10−6 .
Hence, the isospin relation Eq. (2.32) is well respected.
TABLE VII: Branching ratios (in units of 10−6) of resonant and nonresonant (NR) contributions
to B− → K−KSKS . For theoretical errors, see Table III.
Decay mode f0(980)K
− X0(1550)K
− NR total
Theory 5.2+0.0+0.3+0.1−0.0−0.5−0.1 0.92
+0.00+0.16+0.00
−0.00−0.15−0.00 12.4
+0.2+2.1+0.1
−0.3−2.0−0.1 12.2
+0.0+1.5+0.0
−0.0−1.7−0.0
Expt. 11.5 ± 1.3
III. B → Kππ DECAYS
In this section we shall consider five B → Kππ decays, namely, B− → K−π+π−, K0π−π0,
0 → K−π+π0, K0π+π− and K0π0π0. They are dominated by b → s penguin transition and
consist of three decay processes: (i) the current-induced process, 〈B → ππ〉 × 〈0 → K〉, (ii) the
transition processes, 〈B → π〉 × 〈0 → πK〉, and 〈B → K〉 × 〈0 → ππ〉, and (iii) the annihilation
process 〈B → 0〉 × 〈0 → Kππ〉.
The factorizable amplitudes for B− → K−π+π−, K0π−π0, B0 → K−π+π0, K0π+π− and
π0π0 are given in Eqs. (A10-A14), respectively. All five channels have the three-body matrix
element 〈ππ|(q̄b)V−A|B〉 which has the similar expression as Eqs. (2.3) and (2.4) except that the
pole B∗s is replaced by B
∗ and the kaon is replaced by the pion. However, there are additional
resonant contributions to this three-body matrix element due to the intermediate vector ρ and
scalar f0 mesons
〈π+(p2)π−(p3)|(ūb)V −A |B
−〉R =
→π+π−
m2ρi − s23 − imρiΓρi
ε∗ · (p2 − p3)〈ρ0i |(ūb)V −A |B
gf0i→π
m2f0i
− s23 − imf0iΓf0i
〈f0i|(ūb)V −A |B
−〉, (3.1)
where ρi denote generic ρ-type vector mesons, e.g. ρ = ρ(770), ρ(1450), ρ(1700), · · ·. Applying Eqs.
(B1) and (B6) we are led to
〈π+(p2)π−(p3)|(ūb)V−A|B−〉R 〈K−(p1)|(s̄u)V −A |0〉
→π+π−
− s23 − imρiΓρi
(s12 − s13)
(mB +mρi)A
mB +mρi
(s12 + s13 − 3m2π)− 2mρi [A
2)−ABρi0 (q
f0i→π
m2f0i
− s23 − imf0iΓf0i
(m2B −m2f0i)F
2). (3.2)
Likewise, the 3-body matrix element 〈K−π+|(s̄b)
|B0〉 appearing in B0 → K−π+π0 also receives
the following resonant contributions
〈K−(p1)π+(p2)|(s̄b)V −A |B
0〉R =
→K−π+
− s12 − imK∗
ε∗ · (p1 − p2)〈K
i |(s̄b)V −A |B
(3.3)
with K∗i = K
∗(892),K∗(1410),K∗(1680), · · ·.
For the two-body matrix elements 〈π+K−|(s̄d)V−A|0〉, 〈π+π−|(ūu)V−A|0〉 and 〈π+π−|s̄s|0〉, we
note that
〈K−(p1)π+(p2)|(s̄d)V −A |0〉 = 〈π
+(p2)|(s̄d)V −A |K
+(−p1)〉 = (p1 − p2)µFKπ1 (s12)
m2K −m2π
(p1 + p2)µ
−FKπ1 (s12) + FKπ0 (s12)
, (3.4)
where we have taken into account the sign flip arising from interchanging the operators s ↔ d.
Hence,
〈K−(p1)π+(p2)|(s̄d)V−A|0〉〈π−(p3)|(d̄b)V−A|B−〉
= FBπ1 (s12)F
1 (s12)
s23 − s13 −
(m2B −m2π)(m2K −m2π)
+ FBπ0 (s12)F
0 (s12)
(m2B −m2π)(m2K −m2π)
. (3.5)
However, the form factor F1 also receives resonant contributions
− s12 − imK∗
− s12 − imK∗
(p1 − p2)µ
, (3.6)
K∗πK = 〈K
−(p1)π
+(p2)|K∗〉 = gK
∗→πK ε∗ · (p1 − p2), (3.7)
where K∗0 i = K
0 (1430), · · ·. Hence, the resonant contributions to the form factor FKπ1 are
FKπ1,R (s) =
− s− imK∗
− s− imK∗
. (3.8)
In principle, the weak vector form factor F π
+π− defined by
〈π+(pπ+)π−(pπ−)|ūγµu|0〉 = (pπ+ − pπ−)µF π
+π− , (3.9)
can be related to the time-like pion electromagnetic form factors. However, unlike the kaon case,
the time-like e.m. form factors of the pions are not well measured enough allowing us to determine
the resonant and nonresonant parts. Therefore, we shall only consider the resonant part which has
the expression
F ππR (s) =
mρifρig
ρi→ππ
m2ρi − s− imρiΓρi
. (3.10)
Following Eq. (2.18), the relevant matrix elements of scalar densities read
〈π+(p2)π−(p3)|s̄s|0〉 =
mf0i f̄
gf0i→π
m2f0i
− s23 − imf0iΓf0i
+ 〈π+(p2)π−(p3)|s̄s|0〉NR, (3.11)
〈K−(p1)π+(p2)|s̄d|0〉 =
→K−π+
− s12 − imK∗
+ 〈K−(p1)π+(p2)|s̄d|0〉NR. (3.12)
Note that for the scalar meson, the decay constants fS and f̄S are defined in Eq. (B1) and they
are related via Eq. (B2). The nonresonant contribution 〈π+(p2)π−(p3)|s̄s|0〉NR vanishes under the
OZI rule, while under SU(3) symmetry5
〈K−(p1)π+(p2)|s̄d|0〉NR = 〈K+(p1)K−(p2)|s̄s|0〉NR = fNRs (s12), (3.13)
with the expression of fNRs given in Eq. (2.18).
It is known that in the narrow width approximation, the 3-body decay rate obeys the factoriza-
tion relation
Γ(B → RP → P1P2P ) = Γ(B → RP )B(R→ P1P2), (3.14)
with R being a resonance. This means that the amplitudes A(B → RP → P1P2P ) and A(B → RP )
should have the same expressions apart from some factors. Hence, using the known results for
quasi-two-body decay amplitude A(B → RP ), one can have a cross check on the three-body decay
amplitude of B → RP → P1P2P . For example, from Eq. (A12) we obtain the factorizable
amplitude A(B
0 → K∗00 (1430)π0;K∗00 (1430) → K−π+) as
〈K−(p1)π+(p2)π0(p3)|Tp|B
0〉K∗0
(1430) =
(1430)→K−π+
− s12 − imK∗
−ap4 + r
10 − r
FBπ0 (m
)(m2B −m2π)
a2δpu +
(a9 − a7)
B −m2K∗
, (3.15)
where
χ (µ) =
2m2K∗
mb(µ)(ms(µ)−mq(µ))
. (3.16)
The expression inside {· · ·} is indeed the amplitude of B0 → K∗00 (1430)π0 given in Eq. (A6) of
[48].
The strong coupling constants such as gρ→π
+π− and gf0(980)→π
+π− are determined from the
measured partial widths through the relations
8πm2S
g2S→P1P2 , ΓV =
4πm2V
g2V→P1P2 , (3.17)
for scalar and vector mesons, respectively, where pc is the c.m. momentum. The numerical results
+π− = 6.0, gK
∗→K+π− = 4.59,
gf0(980)→π
+π− = 1.33+0.29−0.26 GeV, g
→K+π− = 3.84GeV. (3.18)
In determining the coupling of f0 → π+π−, we have used the partial width
Γ(f0(980) → π+π−) = (34.2+13.9+8.8−11.8−2.5)MeV (3.19)
5 The matrix elements of scalar densities can be generally decomposed into D-, F - and S(singlet)-type
components. Assuming that the singlet component is OZI suppressed, SU(3) symmetry leads to, for
example, the relation 〈Kπ|s̄q|0〉NR = 〈KK̄|s̄s|0〉NR.
measured by Belle [56]. The momentum dependence of the weak form factor FKπ(q2) is
parametrized as
FKπ(q2) =
FKπ(0)
1− q2/Λχ2 + iΓR/Λχ
, (3.20)
where Λχ ≈ 830 MeV is the chiral-symmetry breaking scale [57] and ΓR is the width of the relevant
resonance, which is taken to be 200 MeV [38].
The results of the calculation are summarized in Tables VIII-XII. We see that except for
f0(980)K, the predicted rates for K
∗π, K∗0 (1430)π and ρK are smaller than the data. Indeed, the
predictions based on QCD factorization for these decays are also generally smaller than experiment
by a factor of 2∼5. This will be discussed in more details in Sec. VI.
TABLE VIII: Branching ratios (in units of 10−6) of resonant and nonresonant (NR) contributions
to B− → K−π+π−. For theoretical errors, see Table III.
Decay mode BaBar [6] Belle [7] Theory
π− 9.04± 0.77 ± 0.53+0.21−0.37 6.45 ± 0.43 ± 0.48
+0.25
−0.35 3.0
+0.0+0.8+0.0
−0.0−0.7−0.0
0 (1430)π
− 34.4± 1.7 ± 1.8+0.1−1.4 32.0 ± 1.0± 2.4
−1.9 10.5
+0.0+3.2+0.0
−0.0−2.7−0.1
ρ0K− 5.08± 0.78 ± 0.39+0.22−0.66 3.89 ± 0.47 ± 0.29
+0.32
−0.29 1.3
+0.0+1.9+0.1
−0.0−0.7−0.1
f0(980)K
− 9.30± 0.98 ± 0.51+0.27−0.72 8.78 ± 0.82 ± 0.65
+0.55
−1.64 7.7
+0.0+0.4+0.1
−0.0−0.8−0.1
NR 2.87± 0.65 ± 0.43+0.63−0.25 16.9 ± 1.3± 1.3
−0.9 18.7
+0.5+11.0+0.2
−0.6− 6.3−0.2
Total 64.4± 2.5 ± 4.6 48.8 ± 1.1± 3.6 45.0+0.3+16.4+0.1−0.4−10.5−0.1
TABLE IX: Same as Table VIII except for the decay B− → K0π−π0.
Decay mode Theory Decay mode Theory
K∗−π0 1.5+0.0+0.3+0.2−0.0−0.3−0.2 K
π− 1.5+0.0+0.4+0.0−0.0−0.3−0.0
K∗−0 (1430)π
0 5.5+0.0+1.6+0.1−0.0−1.4−0.1 K
0 (1430)π
− 5.2+0.0+1.6+0.0−0.0−1.4−0.0
1.3+0.0+3.0+0.0−0.0−0.9−0.0 NR 10.0
+0.2+7.1+0.0
−0.2−3.7−0.0
Total 27.0+0.3+15.4+0.2−0.2− 8.8−0.2
While Belle has found a sizable fraction of order (35 ∼ 40)% for the nonresonant signal in
K−π+π− and K
π+π− modes (see Table I), BaBar reported a small fraction of order 4.5% in
K−π+π−. The huge disparity between BaBar and Belle is ascribed to the different parameteri-
zations adopted by both groups. BaBar [6] used the LASS parametrization to describe the Kπ
S-wave and the nonresonant component by a single amplitude suggested by the LASS collaboration
to describe the scalar amplitude in elastic Kπ scattering. As commented in [7], while this approach
is experimentally motivated, the use of the LASS parametrization is limited to the elastic region
of M(Kπ) <∼ 2.0 GeV, and an additional amplitude is still required for a satisfactory description
of the data. In our calculations we have taken into account the nonresonant contributions to the
TABLE X: Same as Table VIII except for the decay B
0 → K0π+π−.
Decay mode Belle [13] Theory
K∗−π+ 5.6± 0.7± 0.5+0.4−0.3 2.1
+0.0+0.5+0.3
−0.0−0.5−0.3
K∗−0 (1430)π
+ 30.8± 2.4 ± 2.4+0.8−3.0 10.1
+0.0+2.9+0.1
−0.0−2.5−0.2
6.1± 1.0± 0.5+1.0−1.1 2.0
+0.0+1.9+0.1
−0.0−0.9−0.1
f0(980)K
7.6± 1.7± 0.7+0.5−0.7 7.7
+0.0+0.4+0.0
−0.0−0.7−0.0
NR 19.9± 2.5 ± 1.6+0.7−1.2 15.6
+0.1+8.3+0.0
−0.1−4.9−0.0
Total 47.5± 2.4 ± 3.7 42.0+0.3+15.7+0.0−0.2−10.8−0.0
TABLE XI: Branching ratios (in units of 10−6) of resonant and nonresonant (NR) contributions
0 → K−π+π0. Note that the branching ratios for K∗−π+ and K∗0π0 given in [14] and
[15] are their absolute ones. We have converted them into the product branching ratios, namely,
B(B → Rh)× B(R→ hh). For theoretical errors, see Table III.
Decay mode BaBar [14] Belle [15] Theory
K∗−π+ 3.6± 0.8± 0.5 4.9+1.5+0.5+0.8−1.5−0.3−0.3 1.0
+0.0+0.3+0.1
−0.0−0.3−0.1
π0 2.0± 0.6± 0.3 < 2.3 1.0+0.0+0.3+0.2−0.0−0.2−0.1
K∗−0 (1430)π
+ 11.2± 1.5 ± 3.5 5.1± 1.5+0.6−0.7 5.0
+0.0+1.5+0.1
−0.0−1.3−0.1
0 (1430)π
0 7.9± 1.5± 2.7 6.1+1.6+0.5−1.5−0.6 4.2
+0.0+1.4+0.0
−0.0−1.2−0.0
ρ+K− 8.6± 1.4± 1.0 15.1+3.4+1.4+2.0−3.3−1.5−2.1 2.5
+0.0+3.6+0.2
−0.0−1.4−0.2
NR < 4.6 5.7+2.7+0.5−2.5−0.4 < 9.4 9.6
+0.3+6.6+0.0
−0.2−3.5−0.0
Total 34.9± 2.1 ± 3.9 36.6+4.2−4.1 ± 3.0 28.9
+0.2+16.1+0.2
−0.2− 9.4−0.2
two-body matrix elements of scalar densities, 〈Kπ|s̄q|0〉. Recall that a large nonresonant contribu-
tion from 〈KK|s̄s|0〉 is needed in order to explain the observed decay rates of B0 → KSKSKS and
B− → K−KSKS . From Tables VIII-XII we see that our predicted nonresonant rates are in agree-
ment with the Belle measurements. The reason why the nonresonant fraction is as large as 90% in
KKK decays, but becomes only (35 ∼ 40)% in Kππ channels (see Table I) can be explained as
follows. Under SU(3) flavor symmetry, we have the relation 〈Kπ|s̄q|0〉NR = 〈KK̄|s̄s|0〉NR. Hence,
the nonresonant rates in the K−π+π− and K
π+π− modes should be similar to that in K+K−K
or K+K−K−. Since theKKK channel receives resonant contributions only from φ and f0i mesons,
while K∗i ,K
0i, ρi, f0i resonances contribute to Kππ modes, this explains why the nonresonant frac-
tion is of order 90% in the former and becomes of order 40% in the latter. Note that the predicted
nonresonant contribution in the K−π+π0 mode is larger than the BaBar’s upper bound and barely
consistent with the Belle limit. It is conceivable that the SU(3) breaking effect in 〈Kπ|s̄q|0〉NR
may lead to a result consistent with the Belle limit.
It is interesting to notice that, based on a simple fragmentation model and SU(3) symmetry,
Gronau and Rosner [55] found the relations
Γ(B− → K+K−K−)NR = 2Γ(B
0 → K+K−K0)NR = 2Γ(B− → K−π+π−)NR
TABLE XII: Same as Table VIII except for the decay B
0 → K0π0π0.
Decay mode f0(980)K
0 (1430)π
0 NR Total
Theory 3.8+0.0+2.0+0.0−0.0−0.4−0.0 0.55
+0.00+0.16+0.00
−0.00−0.13−0.00 2.3
+0.0+0.8+0.0
−0.0−0.6−0.0 5.3
+0.0+1.8+0.0
−0.0−1.1−0.0 12.9
+0.0+4.0+0.1
−0.0−3.0−0.1
TABLE XIII: Branching ratios, mixing-induced and direct CP asymmetries for B
0 → KSπ+π−
decays. Results for (KLππ)CP± are identical to those for (KSππ)CP∓. For theoretical errors, see
Table III.
Final state Branching ratio
+π−)CP+ 13.52
+0.02+4.03+0.01
−0.03−3.06−0.01
+π−)CP− 7.45
+0.10+3.79+0.02
−0.08−2.32−0.02
f+ 0.65
+0.00+0.03+0.00
−0.00−0.04−0.00
Final state sin 2βeff
+π−)CP+ 0.693
+0.000+0.003+0.003
−0.000−0.002−0.014
+π−)full 0.718
+0.001+0.017+0.008
−0.001−0.007−0.018
Final state Af (%)
+π−)CP+ 4.27
+0.00+0.19+0.28
−0.00−0.12−0.35
+π−)full 4.94
+0.03+0.03+0.32
−0.02−0.05−0.40
= 2Γ(B
0 → K0π+π−)NR = 4Γ(B
0 → K−π+π0)NR. (3.21)
Again, a large nonresonant background in K−π+π− and K
π+π− is favored by this model.
Although the B
0 → KSπ0π0 rate has not been measured, its time-dependent CP asymmetries
have been studied by BaBar [58] with the results
sin 2βeff = −0.72± 0.71 ± 0.08, ACP = −0.23 ± 0.52 ± 0.13 . (3.22)
Note that this mode is a CP-even eigenstate. We found that its branching ratio is not so small,
of order 6 × 10−6, in spite of the presence of two neutral pions in the final state (see Table XII).
Theoretically, we obtain
sin 2βeff = 0.729
+0.000+0.001+0.009
−0.000−0.001−0.020 , ACP =
0.28+0.09+0.07+0.02−0.06−0.06−0.02
%. (3.23)
Finally, we consider the mode KSπ
+π− which is an admixture of CP-even and CP-odd compo-
nents. Results for the decay rates and CP asymmetries are displayed in Table XIII. We see that
the effective sin 2β is of order 0.718 and direct CP asymmetry of order 4.9% for KSπ
IV. B → KKπ DECAYS
We now turn to the three-body decay modes dominated by b → u tree and b → d penguin
transitions, namely, KKπ and πππ. We first consider the decay B− → K+K−π− whose factorizable
TABLE XIV: Same as Table VIII except for the decay B− → K+K−π−.
Decay mode f0(980)π
− K∗0K− K∗00 (1430)K
− NR Total
Theory 0.50+0.00+0.06+0.02−0.00−0.04−0.02 0.23
+0.00+0.04+0.02
−0.00−0.04−0.02 0.82
+0.00+0.18+0.09
−0.00−0.16−0.08 1.8
+0.5+0.4+0.2
−0.5−0.2−0.2 4.0
+0.5+0.7+0.3
−0.6−0.5−0.3
Expt. < 6.3 (BaBar)[59]
< 13 (Belle) [11]
TABLE XV: Same as Table VIII except for B− → π+π−π−. The nonresonant background is used
as an input to fix the parameter α
defined in Eq. (2.8).
Decay mode BaBar [5] Theory
ρ0π− 8.8 ± 1.0± 0.6+0.1−0.7 7.7
+0.0+1.7+0.3
−0.0−1.6−0.2
f0(980)π
− 1.2 ± 0.6± 0.1± 0.4 < 3.0 0.39+0.00+0.01+0.03−0.00−0.01−0.02
NR 2.3 ± 0.9± 0.3± 0.4 < 4.6 input
Total 16.2 ± 1.2± 0.9 12.0+1.1+2.0+0.4−1.2−1.8−0.3
amplitude is given by Eq. (A9). Note that we have included the matrix element 〈K+K−|d̄d|0〉.
Although its nonresonant contribution vanishes as K+ and K− do not contain the valence d or d̄
quark, this matrix element does receive a contribution from the scalar f0 pole
〈K+(p2)K−(p3)|d̄d|0〉R =
mf0i f̄
gf0i→π
m2f0i
− s23 − imf0iΓf0i
, (4.1)
where 〈f0|d̄d|0〉 = mf0 f̄df0 . In the 2-quark model for f0(980), f̄
f0(980)
= f̄f0(980) sin θ/
2. Also note
that the matrix element 〈K−(p3)|(s̄b)V −A|B−〉〈π−(p1)K+(p2)|(d̄s)V−A|0〉 has a similar expression
as Eq. (3.5) except for a sign difference
〈K−(p3)|(s̄b)V−A|B−〉〈π−(p1)K+(p2)|(d̄s)V−A|0〉
= −FBK1 (s12)FKπ1 (s12)
s23 − s13 −
(m2B −m2K)(m2K −m2π)
−FBK0 (s12)FKπ0 (s12)
(m2B −m2K)(m2K −m2π)
. (4.2)
As in Eq. (3.8), the form factor FKπ1 receives a resonant contribution for the K
∗ pole.
The nonresonant and various resonant contributions to B− → K+K−π− are shown in Table
XVI. The predicted total rate is consistent with upper limits set by BaBar and Belle.
V. B → πππ DECAYS
The factorizable amplitudes of the tree-dominated decay B− → π+π−π− and B0 → π+π−π0
are given by Eqs. (A15) and (A16), respectively. We see that the former is dominated by the ρ0
TABLE XVI: Same as Table VIII except for the decay B
0 → π+π−π0.
Decay mode ρ+π− ρ−π+ ρ0π0 f0(980)π
0 NR Total
Theory 8.5
+0.0+1.1+0.2
−0.0−1.0−0.1
+0.0+4.0+0.3
−0.0−3.5−0.3
+0.0+0.3+0.0
−0.0−0.2−0.0
0.010
+0.000+0.003+0.000
−0.000−0.002−0.000
+0.02+0.01+0.00
−0.02−0.01−0.00
+0.0+5.6+0.2
−0.0−5.0−0.2
pole, while the latter receives ρ± and ρ0 contributions. As a consequence, the π+π−π0 mode has a
rate larger than π+π−π− even though the former involves a π0 in the final state.
The π+π−π− mode receives nonresonant contributions mostly from the b → u transition as
the nonresonant contribution in the matrix element 〈π+π−|d̄d|0〉 is suppressed by the smallness of
penguin Wilson coefficients a6 and a8. Therefore, the measurement of the nonresonant contribution
in this decay can be used to constrain the nonresonant parameter α
in Eq. (2.8)
VI. DIRECT CP ASYMMETRIES
Direct CP asymmetries for various charmless three-body B decays are collected in Table XVII.
Mixing-induced and direct CP asymmetries in B0 → K+K−KS,L and KSKSKS,L decays are al-
ready shown in Table V. It appears that direct CP violation is sizable in K+K−K− and K+K−π−
modes.
The major uncertainty with direct CP violation comes from the strong phases which are needed
to induce partial rate CP asymmetries. In this work, the strong phases arise from the effective
Wilson coefficients a
i listed in (A3) and from the Breit-Wigner formalism for resonances. Since
direct CP violation in charmless two-body B decays can be significantly affected by final-state
rescattering [60], it is natural to extend the study of final-state rescattering effects to the case of
three-body B decays. We will leave this to a future investigation.
TABLE XVII: Direct CP asymmetries (in %) for various charmless three-body B decays. For
theoretical errors, see Table III. Experimental results are taken from [50].
Final state BaBar Belle Theory
K+K−K− −2± 3± 2 −10.4+1.7+0.9+0.9−1.3−1.0−0.8
K−KSKS −4± 11± 2 −3.9+0.0+0.6+0.3−0.0−0.8−0.3
K+K−π− 0± 10± 3 17.5+1.9+2.2+0.0−3.8−3.4−0.2
K−π+π− −1.3± 3.7± 1.1 4.9± 2.6± 2.0 −3.3+0.7+0.4+0.3−0.5−0.4−0.2
K−π+π0 7± 11± 1 6.3+0.6+1.4+0.5−0.7−1.4−0.5
π+π− 4.9+0.0+0.0+0.3−0.0−0.1−0.4
π0π0 −23± 52± 13 −17± 24± 6 0.28+0.09+0.07+0.02−0.06−0.06−0.02
π−π0 0.4+0.0+0.4+0.0−0.0−0.4−0.0
π+π−π− −1± 8± 3 4.4+0.8+1.2+0.0−0.6−0.9−0.2
π+π−π0 −3.0+0.1+0.2+0.3−0.1−0.3−0.2
VII. TWO-BODY B → V P AND B → SP DECAYS
Thus far we have considered the branching ratio products B(B → Rh1)B(R → h2h3) with
the resonance R being a vector meson or a scalar meson. Using the experimental information on
B(R→ h2h3) [4]
B(K∗0 → K+π−) = B(K∗+ → K0π+) = 2B(K∗+ → K+π0) =
B(K∗00 (1430) → K+π−) = 2B(K∗+0 (1430) → K
+π0) =
(0.93 ± 0.10),
B(φ→ K+K−) = 0.492 ± 0.006 . (7.1)
one can extract the branching ratios of B → V P and B → SP . The results are summarized in
Table XVIII.
Two remarks about the experimental branching ratios are in order: (i) The BaBar results for the
branching ratios of B
0 → K∗−π+, K∗0π0, K∗−0 (1430)π+ are inferred from the three-body decays
0 → K0π+π− (see Table XI) and Belle results are taken from B0 → K−π+π0 (see Table X). (ii)
Branching ratios of B
0 → φK0 shown in Table XVIII are not inferred from the Dalitz plot analysis
of B → KKK decays.
For comparison, the predictions of the QCD factorization approach for B → V P [61] and
B → SP [48] are also exhibited in Table XVIII. In order to compare theory with experiment for
B → f0(980)K → π+π−K, we need an input for B(f0(980) → π+π−). To do this, we shall use the
BES measurement [45]
Γ(f0(980) → ππ)
Γ(f0(980) → ππ) + Γ(f0(980) → KK)
= 0.75+0.11−0.13 . (7.2)
Assuming that the dominance of the f0(980) width by ππ and KK and applying isospin relation,
we obtain
B(f0(980) → π+π−) = 0.50+0.07−0.09 , B(f0(980) → K
+K−) = 0.125+0.018−0.022 . (7.3)
At first sight, it appears that the ratio defined by
R ≡ B(f0(980) → K
B(f0(980) → π+π−)
= 0.25± 0.06 (7.4)
is not consistent with the value of 0.69 ± 0.32 inferred from the BaBar data (see Tables VI and
VIII)
Γ(B− → f0(980)K−; f0(980) → K+K−)
Γ(B− → f0(980)K−; f0(980) → π+π−)
6.5± 2.5 ± 1.6
9.3± 1.0+0.6−0.9
, (7.5)
where we have applied the narrow width approximation Eq. (3.14).
The above-mentioned discrepancy can be resolved by noting that the factorization relation Eq.
(3.14) for the resonant three-body decay is applicable only when the two-body decays B → RP
and R → P1P2 are kinematically allowed and the resonance is narrow, the so-called narrow width
approximation. However, as the decay f0(980) → K+K− is kinematically barely or even not
allowed, the off resonance peak effect of the intermediate resonant state will become important.
TABLE XVIII: Branching ratios of quasi-two-body decays B → V P and B → SP obtained from the
studies of three-body decays based on the factorization approach. Unless specified, the experimental
results are obtained from the 3-body Dalitz plot analyses given in previous Tables. Theoretical
uncertainties have been added in quadrature. QCD factorization (QCDF) predictions taken from
[61] for V P modes and from [48] for SP channels are shown here for comparison.
Decay mode BaBar Belle QCDF This work
φK0 8.4+1.5−1.3 ± 0.5 a 9.0
−1.8 ± 0.7 b 4.1
+0.4+1.7+1.8+10.6
−0.4−1.6−1.9− 3.0 5.3
φK− 8.4± 0.7 ± 0.7 9.60 ± 0.92+1.05−0.84 4.5
+0.5+1.8+1.9+11.8
−0.4−1.7−2.1− 3.3 5.9
π− 13.5 ± 1.2+0.8−0.9 9.8± 0.9
−1.2 3.6
+0.4+1.5+1.2+7.7
−0.3−1.4−1.2−2.3 4.4
π0 3.0± 0.9 ± 0.5 < 3.5 0.7+0.1+0.5+0.3+2.6−0.1−0.4−0.3−0.5 1.5
K∗−π+ 11.0 ± 1.5± 0.7 8.4± 1.1+0.9−0.8 3.3
+1.4+1.3+0.8+6.2
−1.2−1.2−0.8−1.6 3.1
K∗−π0 6.9 ± 2.0± 1.3 b 3.3+1.1+1.0+0.6+4.4−1.0−0.9−0.6−1.4 2.2
K∗0K− 0.30+0.11+0.12+0.09+0.57−0.09−0.10−0.09−0.19 0.35
+0.06
−0.06
ρ0K− 5.1± 0.8+0.6−0.9 3.89 ± 0.47
+0.43
−0.41 2.6
+0.9+3.1+0.8+4.3
−0.9−1.4−0.6−1.2 1.3
4.9± 0.8 ± 0.9 6.1± 1.0 ± 1.1 4.6+0.5+4.0+0.7+6.1−0.5−2.1−0.7−2.1 2.0
ρ+K− 8.6± 1.4 ± 1.0 15.1+3.4+2.4−3.3−2.6 7.4
+1.8+7.1+1.2+10.7
−1.9−3.6−1.1− 3.5 2.5
8.0+1.4−1.3 ± 0.5 b 5.8
+0.6+7.0+1.5+10.3
−0.6−3.3−1.3− 3.2 1.3
ρ0π− 8.8± 1.0+0.6−0.9 8.0
−2.0 ± 0.7 b 11.9
+6.3+3.6+2.5+1.3
−5.0−3.1−1.2−1.1 7.7
ρ−π+ 21.2+10.3+8.7+1.3+2.0− 8.4−7.2−2.3−1.6 15.5
ρ+π− 15.4+8.0+5.5+0.7+1.9−6.4−4.7−1.3−1.3 8.5
ρ0π0 1.4± 0.6 ± 0.3 3.1+0.9+0.6−0.8−0.8 0.4
+0.2+0.2+0.9+0.5
−0.2−0.1−0.3−0.3 1.0
f0(980)K
0; f0 → π+π− 5.5± 0.7 ± 0.6 7.6± 1.7+0.8−0.9 6.7
+0.1+2.1+2.3
−0.1−1.5−1.1
c 7.7+0.4−0.7
f0(980)K
−; f0 → π+π− 9.3± 1.0+0.6−0.9 8.8± 0.8
−1.8 7.8
+0.2+2.3+2.7
−0.2−1.6−1.2
c 7.7+0.4−0.8
f0(980)K
0; f0 → K+K− 5.3± 2.2 5.8+0.1−0.5
f0(980)K
−; f0 → K+K− 6.5± 2.5 ± 1.6 < 2.9 7.0+0.4−0.7
f0(980)π
−; f0 → π+π− < 3.0 0.5+0.0+0.2+0.1−0.0−0.1−0.0 c 0.39
+0.03
−0.02
f0(980)π
−; f0 → K+K− 0.50+0.06−0.04
f0(980)π
0; f0 → π+π− 0.02+0.01+0.02+0.04−0.01−0.00−0.01 c 0.010
+0.003
−0.002
0 (1430)π
− 36.6 ± 1.8± 4.7 51.6 ± 1.7+7.0−7.4 11.0
+10.3+7.5+49.9
− 6.0−3.5−10.1 16.9
0 (1430)π
0 12.7 ± 2.4± 4.4 9.8± 2.5 ± 0.9 6.4+5.4+2.2+26.1−3.3−2.1− 5.7 6.8
K∗−0 (1430)π
+ 36.1 ± 4.8 ± 11.3 49.7 ± 3.8+4.0−6.1 11.3
+9.4+3.7+45.8
−5.8−3.7− 9.9 16.2
K∗−0 (1430)π
0 5.3+4.7+1.6+22.3−2.8−1.7− 4.7 8.9
K∗00 (1430)K
− < 2.2 b 1.3+0.3−0.3
aFrom the Dalitz plot analysis of B0 → K+K−K0 decay measured by BaBar (see Table III), we obtain
B(B0 → φK0) = (6.2± 0.9)× 10−6. The experimental value of BaBar cited in the Table is obtained from a
direct measurement of B0 → φK0.
bnot determined directly from the Dalitz plot analysis of three-body decays.
cWe have assumed B(f0(980) → π+π−) = 0.50 for the QCDF calculation.
Therefore, it is necessary to take into account the finite width effect of the f0(980) which has a
width of order 40-100 MeV [4]. In short, one cannot determine the ratio R by applying the narrow
width approximation to the three-body decays. That is, one should employ the decays B → Kππ
rather than B → KKK to extract the experimental branching ratio for B → f0(980)K provided
B(f0(980) → ππ) is available.
We now compare the present work for B → V P and B → SP with the approach of QCD
factorization [34, 48]. In this work, our calculation of 3-body B decays is similar to the simple
generalized factorization approach [62, 63] by assuming a set of universal and process independent
effective Wilson coefficients a
i with p = u, c in Eq. (A3). In QCDF, the calculation of a
rather sophisticated. They are basically the Wilson coefficients in conjunction with short-distance
nonfactorizable corrections such as vertex corrections and hard spectator interactions. In general,
they have the expressions [34, 61]
i (M1M2) =
Ni(M2) +
Vi(M2) +
Hi(M1M2)
i (M2), (7.6)
where i = 1, · · · , 10, the upper (lower) signs apply when i is odd (even), ci are the Wilson coefficients,
CF = (N
c − 1)/(2Nc) with Nc = 3, M2 is the emitted meson and M1 shares the same spectator
quark with the B meson. The quantities Vi(M2) account for vertex corrections, Hi(M1M2) for hard
spectator interactions with a hard gluon exchange between the emitted meson and the spectator
quark of the B meson and Pi(M2) for penguin contractions. Hence, the effective Wilson coefficients
i (M1M2) depend on the nature ofM1 andM2; that is, they are process dependent. Moreover, they
depend on the order of the argument, namely, a
i (M2M1) 6= a
i (M1M2) in general. In the above
equation, Ni(M2) vanishes for i = 6, 8 and M2 = V , and equals to unity otherwise. For three-
body decays, in principle one should also compute the vertex, gluon and hard spectator-interaction
corrections. Of course, these corrections for the three-body case will be more complicated than the
two-body decay one. One possible improvement of the present work is to utilize the QCDF results
for the effective parameters a
i (M1M2) in the vicinity of the resonance region.
We next proceed to the comparison of numerical results. For φK, K∗π and K∗K modes, the
QCDF and the present work have similar predictions. For the ρ meson in the final states, QCDF
predicts slightly small ρK and too large ρπ compared to experiment. 6 In contrast, in the present
work we obtain reasonable ρπ but too small ρK. This is ascribed to the form factor A
0 (0) =
0.37 ± 0.06 employed in [61] that is too large compared to ours ABρ0 (0) = 0.28 ± 0.03 (see Table
XIX). Recall that the recent QCD sum rule calculation also yields a smaller one A
0 (0) = 0.30
+0.07
−0.03
[64].
For B → f0(980)K and B → f0(980)π, QCDF [48] and this work are in agreement with
experiment. The large rate of the f0(980)K mode is ascribed to the large f0(980) decay constant,
f̄f0(980) ≈ 460 MeV at the renormalization scale µ = 2.1 GeV [48]. In contrast, the predicted
0 (1430)π
− and K∗−0 (1430)π
+ are too small compared to the data. The fact that QCDF leads to
too small rates for φK, K∗π, ρK and K∗0 (1430)π may imply the importance of power corrections
6 Recall that the world average of the branching ratio of B0 → ρ±π∓ is (24.0±2.5)×10−6 [50], while QCDF
predicts it to be ∼ 36.6× 10−6 [61].
due to the non-vanishing ρA and ρH parameters arising from weak annihilation and hard spectator
interactions, respectively, which are used to parametrize the endpoint divergences, or due to possible
final-state rescattering effects from charm intermediate states [60]. However, this is beyond the
scope of the present work.
VIII. CONCLUSIONS
In this work, an exploratory study of charmless 3-body decays of B mesons is presented using
a simple model based on the framework of the factorization approach. The 3-body decay process
consists of resonant contributions and the nonresonant signal. Since factorization has not been
proved for three-body B decays, we shall work in the phenomenological factorization model rather
than in the established theories such as QCD factorization. That is, we start with the simple idea
of factorization and see if it works for three-body decays. Our main results are as follows:
• If heavy meson chiral perturbation theory (HMChPT) is applied to the three-body matrix
elements for B → P1P2 transitions and assumed to be valid over the whole kinematic region,
then the predicted decay rates for nonresonant 3-body B decays will be too large and even
exceed the measured total rate. This can be understood because chiral symmetry has been
applied beyond its region of validity. We assume the momentum dependence of nonresonant
amplitudes in the exponential form e−αNRpB ·(pi+pj) so that the HMChPT results are recovered
in the soft meson limit pi, pj → 0. The parameter αNR can be fixed from the tree-dominated
decay B− → π+π−π−.
• Besides the nonresonant contributions arising from B → P1P2 transitions, we have identified
another large source of the nonresonant background in the matrix elements of scalar densi-
ties, e.g. 〈KK|s̄s|0〉 which can be constrained from the KSKSKS (or K−KSKS) mode in
conjunction with the mass spectrum in the decay B
0 → K+K−K0 .
• All KKK modes are dominated by the nonresonant background. The predicted branching
ratios of K+K−KS(L), K
+K−K− and K−KSKS modes are consistent with the data within
the theoretical and experimental errors.
• Although the penguin-dominated B0 → K+K−KS decay is subject to a potentially signifi-
cant tree pollution, its effective sin 2β is very similar to that of the KSKSKS mode. However,
direct CP asymmetry of the former, being of order −4%, is more prominent than the latter,
• The role played by the unknown scalar resonance X0(1550) in the decay B− → K+K−K−
should be clarified in order to see if it behaves in the same way as in the K+K−K
mode.
• Applying SU(3) symmetry to relate the nonresonant component in the matrix element
〈Kπ|s̄q|0〉 to that in 〈KK|s̄s|0〉, we found sizable nonresonant contributions in K−π+π−
and K
π+π− modes, in agreement with the Belle measurements but larger than the BaBar
results. In particular, the predicted nonresonant contribution in the K−π+π0 mode is consis-
tent with the Belle limit and larger than the BaBar’s upper bound. It will be interesting to
have a refined measurement of the nonresonant contribution to this mode to test our model.
• The π+π−π0 mode is predicted to have a rate larger than π+π−π− even though the former
involves a π0 in the final state. This is because the latter is dominated by the ρ0 pole, while
the former receives ρ± and ρ0 resonant contributions.
• Among the 3-body decays we have studied, the decay B− → K+K−π− dominated by b→ u
tree transition and b→ d penguin transition has the smallest branching ratio of order 4×10−6.
It is consistent with the current bound set by BaBar and Belle.
• Decay rates and time-dependent CP asymmetries in the decays KSπ0π0, a purely CP -even
state, and KSπ
+π−, an admixture of CP -even and CP -odd components, are studied. The
corresponding mixing-induced CP violation is found to be of order 0.729 and 0.718, respec-
tively.
• Since the decay f0(980) → K+K− is kinematically barely or even not allowed, it is crucial
to take into account the finite width effect of the f0(980) when computing the decay B →
f0(980)K → KKK. Consequently, one should employ the Dalitz plot analysis of Kππ mode
to extract the experimental branching ratio for B → f0(980)K provided B(f0(980) → ππ) is
available. The large rate of B → f0(980)K is ascribed to the large f0(980) decay constant,
f̄f0(980) ≈ 460 MeV.
• The intermediate vector meson contributions to 3-body decays e.g. ρ, φ, K∗
are identified through the vector current, while the scalar meson resonances e.g.
f0(980), X0(1550), K
0 (1430) are mainly associated with the scalar density. Their effects
are described in terms of the Breit-Wigner formalism.
• Based on the factorization approach, we have computed the resonant contributions to 3-
body decays and determined the rates for the quasi-two-body decays B → V P and B → SP .
The predicted ρπ, f0(980)K and f0(980)π rates are consistent with experiment, while the
calculated φK, K∗π, ρK and K∗0 (1430)π are too small compared to the data.
• Direct CP asymmetries have been computed for the charmless 3-body B decays. We found
sizable direct CP violation in K+K−K− and K+K−π− modes.
• In this exploratory work we use the phenomenological factorization model rather than in
the established theories based on a heavy quark expansion. Consequently, we don’t have
1/mb power corrections within this model. However, systematic errors due to such model
dependent assumptions may be sizable and are not included in the error estimates that we
give.
Note added: After the paper was submitted for publication, BaBar (arXiv:0708.0367 [hep-ex]) has
reported the observation of the decay B+ → K+K−π+ with the branching ratio (5.0± 0.5± 0.5)×
10−6. Our prediction for this mode (see Table XIV) is consistent with experiment.
Acknowledgments
This research was supported in part by the National Science Council of R.O.C. under Grant
Nos. NSC95-2112-M-001-013, NSC95-2112-M-033-013, and by the U.S. DOE contract No. DE-
AC02-98CH10886(BNL).
http://arxiv.org/abs/0708.0367
APPENDIX A: DECAY AMPLITUDES OF THREE-BODY B DECAYS
In this appendix we list the factorizable amplitudes of the 3-body decays B →
KKK,KKπ,Kππ, πππ. Under the factorization hypothesis, the decay amplitudes are given by
〈P1P2P3|Heff |B〉 =
p=u,c
λ(r)p 〈P1P2P3|Tp|B〉, (A1)
where λ
p ≡ VpbV ∗pr with r = d, s. For KKK and Kππ modes, r = s and for KKπ and πππ
channles, r = d. The Hamiltonian Tp has the expression [34]
Tp = a1δpu(ūb)V−A ⊗ (s̄u)V−A + a2δpu(s̄b)V−A ⊗ (ūu)V−A + a3(s̄b)V−A ⊗
(q̄q)V−A
(q̄b)V−A ⊗ (s̄q)V−A + a5(s̄b)V −A ⊗
(q̄q)V+A
−2ap6
(q̄b)S−P ⊗ (s̄q)S+P + a7(s̄b)V−A ⊗
eq(q̄q)V+A
−2ap8
(q̄b)S−P ⊗
eq(s̄q)S+P + a9(s̄b)V−A ⊗
eq(q̄q)V−A
(q̄b)V−A ⊗
eq(s̄q)V−A, (A2)
with (q̄q′)V±A ≡ q̄γµ(1 ± γ5)q′, (q̄q′)S±P ≡ q̄(1 ± γ5)q′ and a summation over q = u, d, s being
implied. For the effective Wilson coefficients, we use
a1 ≈ 0.99 ± 0.037i, a2 ≈ 0.19 − 0.11i, a3 ≈ −0.002 + 0.004i, a5 ≈ 0.0054 − 0.005i,
au4 ≈ −0.03− 0.02i, ac4 ≈ −0.04 − 0.008i, au6 ≈ −0.06− 0.02i, ac6 ≈ −0.06− 0.006i,
a7 ≈ 0.54 × 10−4i, au8 ≈ (4.5 − 0.5i) × 10−4, ac8 ≈ (4.4 − 0.3i) × 10−4, (A3)
a9 ≈ −0.010 − 0.0002i, au10 ≈ (−58.3 + 86.1i) × 10−5, ac10 ≈ (−60.3 + 88.8i) × 10−5,
for typical ai at the renormalization scale µ = mb/2 = 2.1 GeV which we are working on.
Various three-body B decay amplitudes are collected below.
B → KKK
〈K0K+K−|Tp|B
0〉 = 〈K+K0|(ūb)V−A|B0〉〈K−|(s̄u)V−A|0〉
a1δpu + a
4 + a
10 − (a
6 + a
+〈K+K−|(d̄b)V−A|B0〉〈K
0|(s̄d)V−A|0〉
+〈K0|(s̄b)V−A|B0〉〈K+K−|(ūu)V−A|0〉(a2δpu + a3 + a5 + a7 + a9)
+〈K0|(s̄b)V−A|B0〉〈K+K−|(d̄d)V−A|0〉
a3 + a5 −
(a7 + a9)
+〈K0|(s̄b)V−A|B0〉〈K+K−|(s̄s)V−A|0〉
a3 + a
4 + a5 −
(a7 + a9 + a
+〈K0|s̄b|B0〉〈K+K−|s̄s|0〉(−2ap6 + a
+〈K+K−|d̄(1− γ5)b|B0〉〈K
0|s̄(1 + γ5)d|0〉 (−2ap6 + a
+〈K+K−K0|(s̄d)V −A|0〉〈0|(d̄b)V−A|B0〉
+〈K+K−K0|s̄γ5d|0〉〈0|d̄γ5b|B0〉(−2ap6 + a
8), (A4)
with rPχ =
mb(µ)(m2+m1)(µ)
〈K+K−K−|Tp|B−〉 = 〈K+K−|(ūb)V−A|B−〉〈K−|(s̄u)V−A|0〉
a1δpu + a
4 + a
10 − (a
6 + a
+〈K−|(s̄b)V−A|B−〉〈K+K−|(ūu)V−A|0〉(a2δpu + a3 + a5 + a7 + a9)
+〈K−|(s̄b)V−A|B−〉〈K+K−|(d̄d)V −A|0〉
a3 + a5 −
(a7 + a9)
+〈K−|(s̄b)V−A|B−〉〈K+K−|(s̄s)V−A|0〉
a3 + a
4 + a5 −
(a7 + a9 + a
+〈K−|s̄b|B−〉〈K+K−|s̄s|0〉(−2ap6 + a
+〈K+K−|ū(1− γ5)b|B0〉〈K−|s̄(1 + γ5)u|0〉 (−2ap6 + a
+〈K+K−K−|(s̄u)V−A|0〉〈0|(ūb)V−A|B−〉
+〈K+K−K−|s̄γ5u|0〉〈0|ūγ5b|B−〉(−2ap6 + a
8). (A5)
Since there are two identical K− mesons in this decay, one should take into account the identical
particle effects. For example,
〈K+K−|(ūb)V−A|B−〉〈K−|(s̄u)V−A|0〉 = 〈K+(p1)K−(p2)|(ūb)V −A|B−〉〈K−(p3)|(s̄u)V−A|0〉
+ 〈K+(p1)K−(p3)|(ūb)V −A|B−〉〈K−(p2)|(s̄u)V−A|0〉,
and a factor of 1
should be put in the decay rate.
〈K0K0K0|Tp|B
0〉 = 〈K0K0|(d̄b)V−A|B0〉〈K0|(s̄d)V−A|0〉
10 − (a
+〈K0|(s̄b)V −A|B0〉〈K0K0|(d̄d)V−A|0〉
a3 + a5 −
(a7 + a9)
+〈K0|(s̄b)V −A|B0〉〈K0K0|(s̄s)V−A|0〉
a3 + a
4 + a5 −
(a7 + a9 + a
+〈K0|s̄b|B0〉〈K0K0|s̄s|0〉(−2ap6 + a
+〈K0K0K0|(s̄d)V−A|0〉〈0|(d̄b)V−A|B0〉
(a7 + a9 + a
+〈K0K0K0|s̄γ5d|0〉〈0|d̄γ5b|B0〉(−2ap6 + a
8). (A7)
The second and third terms do not contribute to the purely CP -even decay B
0 → KSKSKS .
〈K−K0K0|Tp|B−〉 = 〈K0K
0|(ūb)V−A|B−〉〈K−|(s̄u)V−A|0〉
a1δpu + a
4 + a
10 − (a
6 + a
+〈K0K−|(d̄b)V−A|B−〉〈K0|(s̄d)V−A|0〉
10 − (a
+〈K−|(s̄b)V−A|B−〉〈K0K0|(d̄d)V−A|0〉
a3 + a5 −
(a7 + a9)
+〈K−|(s̄b)V−A|B−〉〈K0K0|(s̄s)V−A|0〉
a3 + a
4 + a5 −
(a7 + a9 + a
+〈K−|s̄b|B−〉〈K0K0|s̄s|0〉(−2ap6 + a
+〈K−K0K0|(s̄u)V −A|0〉〈0|(ūb)V−A|B0〉(a1δpu + ap4 + a
+〈K−K0K0|s̄γ5u|0〉〈0|ū(1− γ5)b|B−〉(2ap6 + 2a
8). (A8)
The third and fourth terms do not contribute to the decay B− → K−KSKS .
B → KKπ
〈π−K+K−|Tp|B−〉 = 〈K+K−|(ūb)V−A|B−〉〈π−|(d̄u)V −A|0〉
a1δpu + a
4 + a
10 − (a
6 + a
+〈π−|(d̄b)V−A|B−〉〈K+K−|(ūu)V−A|0〉(a2δpu + a3 + a5 + a7 + a9)
+〈π−|d̄b|B−〉〈K+K−|d̄d|0〉(−2ap6 + a
+〈π−|(d̄b)V−A|B−〉〈K+K−|(s̄s)V−A|0〉
a3 + a5 −
(a7 + a9)
+〈K−|(s̄b)V−A|B−〉〈K+π−|(d̄s)V−A|0〉(ap4 −
+〈K−|s̄b|B−〉〈K+π−|d̄s|0〉(−2ap6 + a
+〈K+K−π−|(d̄u)V−A|0〉〈0|(ūb)V −A|B−〉
a1δpu + a
4 + a
+〈K+K−π−|d̄γ5u|0〉〈0|ūγ5b|B−〉(2ap6 − a
8). (A9)
B → Kππ
〈K−π+π−|Tp|B−〉 = 〈π+π−|(ūb)V−A|B−〉〈K−|(s̄u)V−A|0〉
a1δpu + a
4 + a
10 − (a
6 + a
+〈K−|(s̄b)V−A|B−〉〈π+π−|(ūu)V −A|0〉
a2δpu +
(a7 + a9)
+〈K−|s̄b|B−〉〈π+π−|s̄s|0〉(−2ap6 + a
+〈π−|(d̄b)V−A|B−〉〈K−π+|(s̄d)V−A|0〉(ap4 −
+〈π−|d̄b|B−〉〈K−π+|s̄d|0〉(−2ap6 + a
+〈K−π+π−|(s̄u)V−A|0〉〈0|(ūb)V−A|B−〉(a1δpu + ap4 + a
+〈K−π+π−|s̄γ5u|0〉〈0|ūγ5b|B−〉(2ap6 + 2a
8). (A10)
〈K0π+π−|Tp|B
0〉 = 〈π+π−|(d̄b)V−A|B
0〉〈K0|(s̄d)V −A|0〉
10 − (a
+〈K0|(s̄b)V−A|B
0〉〈π+π−|(ūu)V−A|0〉
a2δpu +
(a7 + a9)
+〈K0|s̄b|B0〉〈π+π−|s̄s|0〉(−2ap6 + a
+〈π+|(ūb)V−A|B
0〉〈K0π−|(s̄u)V−A|0〉(a1 + ap4 + a
+〈π+|ūb|B0〉〈K0π−|s̄u|0〉(−2ap6 − 2a
+〈K0π+π−|(s̄d)V −A|0〉〈0|(d̄b)V−A|B
0〉(a1δpu + ap4 + a
+〈K0π+π−|s̄(1 + γ5)d|0〉〈0|d̄γ5b|B
0〉(2ap6 − a
8). (A11)
〈K−π+π0|Tp|B
0〉 = 〈π+π0|(ūb)V−A|B
0〉〈K−|(s̄u)V −A|0〉
a1δpu + a
4 + a
10 − (a
6 + a
+〈K−π+|(s̄b)V−A|B
0〉〈π0|(ūu)V−A|0〉
a2δpu +
(−a7 + a9)
+〈π+|(ūb)V−A|B
0〉〈K−π0|(s̄u)V−A|0〉 [a1δpu + ap4 + a
+〈π0|(d̄b)V −A|B
0〉〈K−π+|(s̄d)V −A|0〉(ap4 −
+〈π+|ūb|B0〉〈K−π0|s̄u|0〉(−2ap6 − 2a
+〈π0|d̄b|B0〉〈K−π+|s̄d|0〉(−2ap6 + a
+〈K−π+π0|(s̄d)V−A|0〉〈0|(d̄b)V−A|B
0〉(ap4 −
+〈K−π+π0|s̄(1 + γ5)d|0〉〈0|d̄γ5b|B
0〉(2ap6 − a
8). (A12)
〈K0π−π0|Tp|B−〉 = 〈π0π−|(d̄b)V−A|B−〉〈K
0|(s̄d)V −A|0〉
10 − (a
+〈K0π−|(s̄b)V−A|B−〉〈π0|(ūu)V−A|0〉
a2δpu +
(−a7 + a9)
+〈π0|(ūb)V−A|B−〉〈K
π−|(s̄u)V−A|0〉 [a1δpu + ap4 + a
+〈π−|(d̄b)V−A|B−〉〈K
π0|(s̄d)V−A|0〉
+〈π0|ūb|B−〉〈K0π−|s̄u|0〉(−2ap6 − 2a
+〈π−|d̄b|B−〉〈K0π0|s̄d|0〉(−2ap6 + a
+〈K0π−π0|(s̄u)V −A|0〉〈0|(ūb)V−A|B−〉(a1δpu + ap4 + a
+〈K0π−π0|s̄(1 + γ5)u|0〉〈0|ūγ5b|B−〉(2ap6 + 2a
8). (A13)
〈K0π0π0|Tp|B
0〉 = 〈π0π0|(d̄b)V−A|B
0〉〈K0|(s̄d)V −A|0〉
10 − (a
+〈K0π0|(s̄b)V−A|B
0〉〈π0|(ūu)V−A|0〉
a2δpu +
(−a7 + a9)
+〈π0|(d̄b)V−A|B
0〉〈K0π0|(s̄d)V−A|0〉
+〈π0π0|(ūu)
|0〉〈K0|(s̄b)
a2δpu + 2a3 + 2a5 +
(a7 + a9)
+〈K0π0|s̄d|0〉〈π0|d̄b|B0〉(−2ap6 + a
+〈π0π0|s̄s|0〉〈K0|s̄b|B0〉(−2ap6 + a
+〈K0π0π0|(s̄d)V−A|0〉〈0|(d̄b)V−A|B
0〉(ap4 −
+〈K0π0π0|s̄(1 + γ5)d|0〉〈0|d̄γ5b|B
0〉(2ap6 − a
8). (A14)
B → πππ
〈π−π+π−|Tp|B−〉 = 〈π+π−|(ūb)V −A|B−〉〈π−|(d̄u)V−A|0〉
a1δpu + a
4 + a
10 − (a
6 + a
+〈π−|(d̄b)V−A|B−〉〈π+π−|(ūu)V−A|0〉
a2δpu − ap4 +
(a7 + a9) +
+〈π−|d̄b|B−〉〈π+π−|d̄d|0〉(−2ap6 + a
+〈π−π+π−|(d̄u)V−A|0〉〈0|(ūb)V−A|B−〉(a1δpu + ap4 + a
+〈π−π+π−|d̄(1 + γ5)u|0〉〈0|ūγ5b|B−〉(2ap6 + 2a
8). (A15)
〈π0π+π−|Tp|B
0〉 = 〈π+π0|(ūb)V−A|B
0〉〈π−|(d̄u)V−A|0〉
a1δpu + a
4 + a
10 − (a
6 + a
+〈π+π−|(d̄b)V −A|B
0〉〈π0|(ūu)V−A|0〉
a2δpu − ap4 + (a
(a7 + a9) +
+〈π+|(ūb)V−A|B
0〉〈π−π0|(d̄u)V−A|0〉 [a1δpu + ap4 + a
+〈π0|(d̄b)V−A|B
0〉〈π+π−|(ūu)V−A|0〉
a2δpu − ap4 +
(a7 + a9) +
+〈π0|d̄b|B0〉〈π+π−|d̄d|0〉(−2ap6 + a
(A16)
APPENDIX B: DECAY CONSTANTS, FORM FACTORS AND OTHERS
In this appendix we collect the numerical values of the decay constants, form factors, CKM
matrix elements and quark masses needed for the calculations. We first discuss the decay constants
of the pseudoscalar meson P and the scalar meson S defined by
〈P (p)|q̄2γµγ5q1|0〉 = −ifPpµ, 〈S(p)|q̄2γµq1|0〉 = fSpµ, 〈S|q̄2q1|0〉 = mS f̄S , (B1)
and 〈V (p, ε)|Vµ|0〉 = fVmV ε∗µ for the vector meson. For the scalar mesons, the vector decay
constant fS and the scale-dependent scalar decay constant f̄S are related by equations of motion
µSfS = f̄S, with µS =
m2(µ)−m1(µ)
, (B2)
where m2 and m1 are the running current quark masses. The neutral scalar mesons σ, f0 and a
cannot be produced via the vector current owing to charge conjugation invariance or conservation
of vector current:
fσ = ff0 = fa0
= 0. (B3)
However, the decay constant f̄S is non-vanishing. In [48] we have applied the QCD sum rules to
estimate this quantity. In this work we folow [48] to use
f̄f0(980) = 460MeV, f̄K∗0 (1430)
= 550MeV, (B4)
at µ = 2.1 GeV. As for the decay constants of vector mesons, we use (in units of MeV).
fρ = 216, fK∗ = 218, f̄f0(980) = 460, f̄K∗0 = 550. (B5)
Form factors for B → P, S transitions are defined by [41]
〈P (p′)|Vµ|B(p)〉 =
(p+ p′)µ −
m2B −m2P
FBP1 (q
m2B −m2P
〈S(p′)|Aµ|B(p)〉 = −i
(p + p′)µ −
m2B −m2S
FBS1 (q
m2B −m2S
〈V (p′, ε)|Vµ|B(p)〉 =
mB +mV
ǫµναβε
∗νpαp′βV (q2),
〈V (p′, ε)|Aµ|B(p)〉 = i
(mB +mV )ε
ε∗ · p
mB +mV
(p+ p′)µA
ε∗ · p
2)−ABV0 (q2)]
, (B6)
where q = p− p′, F1(0) = F0(0), A3(0) = A0(0), and
mP +mV
2)− mP −mV
2), (B7)
where Pµ = (p + p
′)µ, qµ = (p − p′)µ. As shown in [65], a factor of (−i) is needed in B → S
transition in order for the B → S form factors to be positive. This also can be checked from heavy
quark symmetry [65].
Various form factors for B → S transitions have been evaluated in the relativistic covariant
light-front quark model [65]. In this model form factors are first calculated in the spacelike region
and their momentum dependence is fitted to a 3-parameter form
F (q2) =
F (0)
1− a(q2/m2B) + b(q2/m2B)2
. (B8)
The parameters a, b and F (0) are first determined in the spacelike region. This parametrization is
then analytically continued to the timelike region to determine the physical form factors at q2 ≥ 0.
The results relevant for our purposes are summarized in Table XIX. In practical calculations, we
shall assign the form factor error to be 0.03. For example, FBK0,1 (0) = 0.35 ± 0.03.
The form factor for B to f0(980) is of order 0.25 at q
2 = 0 [48]. In the qq̄ model for the f0(980),
0 = FBf0 sin θ/
For the heavy-flavor independent strong coupling g in HMChPT, we use |g| = 0.59± 0.01± 0.07
as extracted from the CLEO measurement of the D∗+ decay width [39]. The sign is fixed to be
negative in the quark model [23].
For the CKM matrix elements, we use the Wolfenstein parameters A = 0.806, λ = 0.22717,
ρ̄ = 0.195 and η̄ = 0.326 [52]. The corresponding CKM angles are (sin 2β)CKM = 0.695
+0.018
−0.016 and
γ = (59 ± 7)◦ [52]. For the running quark masses we shall use
mb(mb) = 4.2GeV, mb(2.1GeV) = 4.95GeV, mb(1GeV) = 6.89GeV,
mc(mb) = 1.3GeV, mc(2.1GeV) = 1.51GeV,
ms(2.1GeV) = 90MeV, ms(1GeV) = 119MeV,
md(1GeV) = 6.3MeV, mu(1GeV) = 3.5MeV. (B9)
The uncertainty of the strange quark mass is specified as ms(2.1GeV) = 90± 20 MeV.
TABLE XIX: Form factors of B → π,K,K∗0 (1430), ρ transitions obtained in the covariant light-
front model [65].
F F (0) F (q2max) a b F F (0) F (q
max) a b
FBπ1 0.25 1.16 1.73 0.95 F
0 0.25 0.86 0.84 0.10
FBK1 0.35 2.17 1.58 0.68 F
0 0.35 0.80 0.71 0.04
1 0.26 0.70 1.52 0.64 F
0 0.26 0.33 0.44 0.05
V Bρ 0.27 0.79 1.84 1.28 A
0 0.28 0.76 1.73 1.20
1 0.22 0.53 0.95 0.21 A
2 0.20 0.57 1.65 1.05
0.31 0.96 1.79 1.18 ABK
0 0.31 0.87 1.68 1.08
1 0.26 0.58 0.93 0.19 A
2 0.24 0.70 1.63 0.98
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	Introduction
	BKKK decays
	B0KKK decays
	B-KKK decays
	BK decays
	BKK decays
	B decays
	Direct CP asymmetries
	Two-body BVP and BSP decays
	Conclusions
	Acknowledgments
	Decay amplitudes of three-body B decays
	Decay constants, form factors and others
	References
