Table Of ContentImplications of the HyperCP Data on B and τ decays
Chuan-Hung Chen1,2 and Chao-Qiang Geng3,4
∗ †
1Department of Physics, National Cheng-Kung University, Tainan 701, Taiwan
2National Center for Theoretical Sciences, Taiwan
3Department of Physics, National Tsing-Hua University, Hsinchu 300, Taiwan
4Theory Group, TRIUMF, 4004 Wesbrook Mall,
7
Vancouver, B.C V6T 2A3, Canada
0
0
(Dated: February 2, 2008)
2
n
Abstract
a
J
8 If the HyperCP three events for the decay of Σ+ pµ+µ− are explained by a new pseudoscalar
→
2 (axial-vector) boson X with a mass of 214.3 MeV, we study the constraints on the couplings
P(A)
v
2 between X and fermions from the experimental data in K and B processes. Some implications
P(A)
4
1 of the new particle on flavor changing B and τ decays are given. Explicitly, we show that the
2
1 decay branching ratios of B φX φµ+µ−, B K∗(1430)P(A) K∗(1430)µ+µ− and
6 s → P(A) → d → 0 → 0
0
τ µX µµ+µ− can be as large as 2.7 (2.8) 10−6, 7.4 (7.5) 10−7 and 1.7 (0.14) 10−7,
/ P(A)
h → → × × ×
p respectively.
-
p
e
h
:
v
i
X
r
a
∗ Email: [email protected]
† Email: [email protected]
1
The HyperCP collaboration [1] has presented the branching ratio of Σ+ pµ+µ− to be
→
(8.6+6.6 5.5) 10−8, which is hardly explained within the Standard Model [1, 2, 3], and
−5.4 ± ×
suggested a new boson X with a mass of 214.3 0.5 MeV to induce the flavor changing
±
transition of s dµ+µ−. It has been demonstrated [4, 5, 6] that to explain the data the new
→
particle cannot be a scalar (vector) but pseudoscalar (axial-vector) boson X based on
P(A)
the direct constraints from K+ π+µ+µ− and K µ+µ−. A possible candidate with a
L
→ →
lightsgoldstinoinspantaneously localsupersymmetry breaking theorieshasbeenextensively
discussed in the literature [7]. Recently, He, Tandean and Valencia [8] have also shown that
the light pseudoscalar Higgs boson in the next-to-minimal supersymmetric standard model
can be identified as X . In addition, searching for the new light boson at colliders has been
P
studied by Ref. [9].
In this paper, we will explore the implications of the HyperCP Data on flavor changing B
and τ decays. In particular, we will examine constraints on the effective interactions induced
by X from the experimental data in B processes and study the possibility of having large
P,A
effects in semileptonic B decays. For the tau decays, we will concentrate on the decays of
d,s
τ ℓµ+µ−.
→
We start by writing the effective interactions for the new pseudoscalar (X ) or axial-
P
vector (X ) particle coupling to quarks and leptons to be [4]
A
= igQ q¯γ q igL ℓ¯γ ℓ +H.c. X ,
−LP − Pij i 5 j − Pij i 5 j P
(cid:16) (cid:17)
= gQ q¯γ γ q +gL ℓ¯γ γ ℓ +H.c. Xµ, (1)
−LA Aij i µ 5 j Aij i µ 5 j A
(cid:16) (cid:17)
where gF (gF ) (F = Q,L) denote the couplings of X (Xµ) to quarks and leptons, respec-
Pij Aij P A
tively, and the indices i,j stand for the quark or lepton flavors. Although the exotic events
observed in the HyperCP are associated with the flavor changing neutral current (FCNC)
in the first two generations of quark flavors and lepton flavor (LF) conservation, to study
the effects of the new particle on B and τ decays, we will include all FCNCs in both quark
and lepton sectors. It has been studied and known that the constraint on the s d X
− −
coupling from the decay branching ratio (BR) of K µ+µ− is more strict than that from
L
→
the K K¯ mixing [4]. In order to search for the most strict bounds on X-mediated B-
−
meson decay processes, we will examine those measured well in experiments, such as the
B B¯ mixings and the decays of B µ+µ− and B K∗µ+µ−.
d(s) d(s) d,s
− → →
To study the B related processes, in terms of Eq. (1), we explicitly write the relevant
q
2
interactions to be
= igQ ¯bγ qX igL µ¯γ µX +gQ ¯bγ γ qXµ +gL µ¯γ γ µXµ +H.c.. (2)
−L − Pbq 5 P − Pµ 5 P Abq µ 5 A Aµ µ 5 A
We note that for simplicity, we have abbreviated gQ (gQ ) to denote the effective coupling
Pbq Abq
for b q X . ¿From Eq. (2), the effective Hamiltonian for ∆B = 2 could be obtained
P(A)
− − | |
by
g2
H|∆B|=2 = X ¯bΓγ q ¯bΓγ q , (3)
eff m2 m2 5 5
Bq − X
(cid:0) (cid:1)(cid:0) (cid:1)
where g = (gQ , gQ ), m = (m , m ) and Γ = (1, γ ) for X = (X , Xµ). The
X Pbq Abq X XP XA µ P A
¯
B B oscillations are dictated by the two physical mass differences, defined by ∆m =
q − q Bq
2 M∆B=2 = 2 B¯ H∆F=2 B . Explicitly, we have
| 12 | |h q| eff | qi|
4g2 m f2
∆m = X Bq Bq PSLL PLR ,
Bq XP 3(m2 m2 ) 1 − 1
Bq − XP
(cid:0)∆m (cid:1) = 4gX2 mBqfB2q (cid:12)(cid:12) (PVLL P(cid:12)(cid:12)LR)+ (mb +mq)2 PSLL PLR , (4)
Bq XA 3(m2 m2 ) − 1 − 2 m2 1 − 1
Bq − XA (cid:12) XA (cid:12)
(cid:0) (cid:1) (cid:12) (cid:0) (cid:1)(cid:12)
(cid:12) (cid:12)
where we have used the hadronic ma(cid:12)trix elements, defined by [10] (cid:12)
m f2
B¯ (¯bγµP q)(¯bγ P q) B = Bq BqPVLL,
h q| L(R) µ L(R) | qi 3 1
m f2
B¯ (¯bγµP q)(¯bγ P q) B = Bq BqPLR,
h q| L µ R | qi 3 1
m f2
B¯ (¯bP q)(¯bP q) B = Bq BqPSLL,
h q| L(R) L(R) | qi 3 1
m f2
B¯ (¯bP q)(¯bP q) B = Bq BqPLR, (5)
h q| L R | qi 3 2
with f (m ) being the B decay constant (mass). It is clear that by using s(d) instead
Bq Bq q
of b(q), Eq. (5) can be applied to the K system. In Table I [10], we give the values of
PVLL, PLR and PSLL for K and B systems. Note that PLR and PSLL in the B system are
1 1,2 1 1,2 1 q
much smaller than those in the K system as there is no enhancement from chiral symmetry
breaking in B .
q
We now examine P µ+µ− with P = (K , B ). Note that the decay of K µ+µ−
L q L
→ →
gives the strongest constraint on gQ. To estimate the decay BRs, we use
sd
f m2
0 q¯γ γ q P(p) = if p , 0 q¯γ q P(p) = i P P . (6)
i µ 5 j P µ i 5 j
h | | i h | | i − m +m
qi qj
3
TABLE I: The form factors for the P P¯ transitions with P being the pseudoscalar mesons.
−
P PVLL PLR PSLL PLR
1 1 1 2
K 0.48 36.1 18.1 59.3
− −
B 0.84 1.62 1.47 2.46
q
− −
By neglecting the mixing induced CP violation in the K system and using K K , the
L 2
≈
X-mediated decay amplitudes for P µ+µ− could be summarized as
→
gQ gL f m2
M(P µ+µ−) = ia Pij Pµ P P µ¯γ µ,
→ XP − P m2 m2 m +m 5
P − XP qi qj
gQ gL
M(P µ+µ−) = i2a Pij Pµm f µ¯γ µ, (7)
→ XA P m2 µ P 5
XA
where q = (s, b), q = (d, q) and a = (√2,1) for P = (K ,B ). The corresponding decay
i j P L q
rates are given by
Γ(P µ+µ−) = mP aPgPQij fPm2P 2 1−4m2µ/m2P Γ(X µ+µ−),
→ XP mXP (cid:12)(cid:12)m2P −m2XP mqi +mqj(cid:12)(cid:12) q1−4m2µ/m2XP P →
(cid:12) (cid:12)
(cid:12) (cid:12) q
Γ(P µ+µ−) = 3 mP(cid:12) 2aPgAQijm f 2 1(cid:12) −4m2µ/m2P Γ(X µ+µ−), (8)
→ XA 2m m2 µ P (1q 4m2/m2 )3/2 A →
XA (cid:12)(cid:12) XA (cid:12)(cid:12) − µ XA
(cid:12) (cid:12)
with (cid:12) (cid:12)
(cid:12) (cid:12)
gL 2m 4m2
Γ(X µ+µ−) = | Pµ| XP 1 µ ,
P → 8π s − m2
XP
gL 2m 4m2 3/2
Γ(X µ+µ−) = | Aµ| XA 1 µ . (9)
A → 12π − m2
(cid:18) XA(cid:19)
After introducing ∆F = 2 (F = S,B) and purely rare dileptonic decays of K and B ,
L q
| |
we investigate other X-mediated rare semileptonic processes, such as B MX Mµ+µ−
q
→ →
with M being a light meson. For the s-wave states in the processes, M has to be a vector-
meson (V) due to parity. On the other hand, for the production of p-wave states in the
decays, M can be either scalar (S) or axial-vector (A) mesons, such as f (980), a (980), κ,
0 0
f (1500), K∗(1430) and K . To study the decay rates, we illustrate the formulas in the case
0 0 1
with M = V and X = Xµ. We write the corresponding decay amplitude to be
A
gµν +qµqν/m2
M = Vℓ+ℓ− B = h − XA µ¯γ γ µ, (10)
h |Heff| qi µq2 m2 +iΓ m ν 5
− XA XA XA
4
where h = V ¯bγ γ q′ B denotes the hadronic transition matrix element and Γ is the
µ h | µ 5 | qi XA
total decay width of Xµ. In terms of the narrow width approximation, given by
A
2
1 π
δ(q2 m2 ), (11)
q2 m2 +iΓ m ≈ Γ m − XA
(cid:12) − XA XA XA(cid:12) XA XA
(cid:12) (cid:12)
the squared decay a(cid:12)mplitude can be writte(cid:12)n as
(cid:12) (cid:12)
2
π
M 2 = h εµ∗ (λ)εν (λ)µ¯γ γ µ δ(q2 m2 ),
| | (cid:12)(cid:12)λX=0,± µ XA XA ν 5 (cid:12)(cid:12) ΓXAmXA − XA
(cid:12) (cid:12)
(cid:12) (cid:12) π
= (cid:12) h εµ∗ (λ) 2 εν(cid:12) (λ)µ¯γ γ µ 2 δ(q2 m2 ), (12)
µ XA XA ν 5 Γ m − XA
"λ=0,± #"λ=0,± # XA XA
X (cid:12) (cid:12) X (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
and the differential decay rates for the semileptonic B decays are given by
q
1
dΓ(B Vµ+µ−) = h εµ∗ (λ) 2 d (B VX )
q → 2m µ XA 2 d → A
Bq "λ=0,± #
X (cid:12) (cid:12)
(cid:12) (cid:12)
1
εν (λ)µ¯γ γ µ 2 d (X µ+µ−)
×2m Γ XA ν 5 2 A →
XA XA "λ=0,± #
X (cid:12) (cid:12)
= 3dΓ(B VX )dB(cid:12)R(X µ+µ(cid:12)−), (13)
q A A
→ →
where the factor 3 is due to the spin-degree of freedom from X . The formula in Eq. (13)
A
could be applied to any decaying chain of P MX Mµ+µ−. As a result, we find that
→ →
Γ(B MX Mµ+µ−) = f Γ(B MX) BR(X µ+µ−), (14)
q X q
→ → → × →
where f = (1,3) for X = (X ,X ), representing the spin degree of freedom.
X P A
In our following analysis, we will focus on the decays of B MX. It is clear that X is
q
→
emitted and the meson in the final state owns the same spectator quark as B . The hadronic
q
effects for the decays are only related to B M transition form factors. The relevant form
q
→
fcators for various mesons are parametrized by [11]
ε∗ (λ) P
V(p,ε )¯bγµγ q′ B (p ) = i (m m ) εµ∗(λ) V · qµ ABq
h V | 5 | q B i Bq − V V − q2 1
(cid:20) (cid:18) (cid:19)
ε∗ P P q
V · Pµ · qµ ABqV(q2)
−m +m − q2 2
Bq V (cid:18) (cid:19)
ε∗ P
+2m · qµABqV(q2) ,
V q2 0
(cid:21)
ABqA(q2)
A(p,ε )¯bγ γ q′ B (p ) = ǫ εν∗(λ)Pρqσ,
h A | µ 5 | q B i −m m µνρσ A
Bq − A
P q P q
S(p)¯bγµγ q′ B (p ) = i P · q FBqS(q2)+ · q FBqS(q2) , (15)
h | 5 | q B i − µ − q2 µ 1 q2 µ 0
(cid:20)(cid:18) (cid:19) (cid:21)
5
where ε (λ) denotes the polarization of V(A) with the helicity state λ, P = (p + p)
V(A) µ B µ
and q = (p p) . Consequently, the decay amplitudes for the decays B (V, S)X can
µ B µ q P
− →
be written as
2m
A(B VX ) = gQ V ε∗ qABqV(m2 ),
q → P − Pbq′mb +mq′ V · 0 XP
m2 m2
A(B SX ) = gQ Bq − SFBqS(m2 ). (16)
q → P Pbq′ mb +mq′ 0 XP
Since m is as light as 0.214 GeV, it is a good approximation to adopt F(0) F(m2 ) for
X ≈ X
the various form factors. From Eq. (16), we obtain
2
Γ(B VX ) m3Bq A0BqV(0) 1 m2V 3 gQ 2 ,
q → P ≈ 16π((cid:16)mb +mq′)(cid:17)2 − m2Bq! (cid:12) Pbq′(cid:12)
2 (cid:12) (cid:12)
Γ(B SX ) m3Bq F0BqS(0) 1 m2S 3 (cid:12)gQ (cid:12)2 . (17)
q → P ≈ 16π((cid:16)mb +mq′(cid:17))2 − m2Bq! (cid:12) Pbq′(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Similarly, the decay amplitudes for B (V, A, S)X are given by
q A
→
A(B VX ) = igQ (m +m )ABV(0)ε∗(λ) ε∗ (λ)
q → A Abq′ Bq V 1 V · XA
2ABV(0)
(cid:2)2 ε∗ (λ) p ε∗ (λ) p ,
−m +m V · B XA · B
Bq V (cid:21)
2gQ ABqA(0)
A(B AX ) = Abq′ ǫ εµ∗ (λ)εν∗(λ)pρqσ,
q → A − m m µνρσ XA A B
Bq − A
A(B SX ) = 2igQ ε∗ (λ) p FBqS(0). (18)
q → A − Abq′ XA · B 1
To get the unified formulas for the decay rates with two spin-1 mesons in the final states,
we can write the general decay amplitude in terms of the helicity basis as
b c
A = ε∗ (λ)ε∗ (λ) agµν + pµpν +i ǫµναβp p (19)
λ 1µ 2ν m m B B m m 1α 2β
(cid:20) 1 2 1 2 (cid:21)
where m stand for the meson masses. Then, the longitudinal and transverse components
1,2
are given by H = ax b(x2 1) and H = a c√x2 1, respectively, with x = (m2
0 − − − ± ± − B −
m2 m2)/(2m m ). Since B is a spinless particle, both spin-1 mesons in the final states
1 − 2 1 2 q
should have the same helicity. Hence, the decay rates are given by [12, 13]
p~
Γ(B M M ) = | | H 2 + H 2 + H 2 (20)
q → 1 2 8πm2 | 0| | +| | −|
Bq
(cid:0) (cid:1)
6
with p~ = m m √x2 1/m2 . By comparing Eq. (18) with Eq. (19), we find that
| | 1 2 − Bq
2ABqV(0)
a = igQ (m +m )ABqV(0), b = igQ m m 2 , c = 0, (21)
Abq′ Bq V 1 − Abq′ V XAm +m
Bq V
for B VX and
q A
→
2ABqA(0)
a = 0 b = 0, c = igQ m m , (22)
Abq′ A XAm m
Bq − A
for B AX . The decay rates for B SX are given by
q A q A
→ →
2 3
m3 FBqS(0) m2 2
Γ(B SX ) Bq 1 1 S gQ . (23)
q → A ≈ 16π m − m2 Abq′
XA ! Bq! (cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Finally, we study the rare decays of B γX . Although γ is a vector boson, unlike an
q A
→
ordinary vector meson, it is massless and only has transverse degrees of freedom. With the
transition form factor for B γ, defined by [14]
q
→
F (q2)
γ(k)¯bγ γ q′ B (p ) = ie A εµ∗(λ)p k ε∗(λ) p k , (24)
h | µ 5 | q B i m γ B · − γ · B µ
B
(cid:2) (cid:3)
the decay amplitudes are given by
F (q2)
A(B γX ) = ie A p kε∗(λ) ε∗ (λ), (25)
q → A m B · γ · XA
Bq
leading to
α 2
Γ(B γX ) = emm F (0) 2 gQ (26)
q → A 8 Bq| A | Abq′
(cid:12) (cid:12)
(cid:12) (cid:12)
with α = e2/4π. (cid:12) (cid:12)
em
After the B processes, we now discuss the pure leptonic decays. As the new particle of
q
X or Xµ couples to the muon, it is natural to speculate that it will also couple to other
P A
leptons such as e and τ. Moreover, the new particle could also give rise to the LFV, like the
FCNCs in the quark sector, such as µ eγ, µ 3e and τ ℓµ+µ−. Recently, the LFV in
→ → →
τ decays has been improved up to O(10−7) [16, 17]. It should be interesting to explore the
LFV due to X . To illustrate the effects of the LFV, we will concentrate on the processes
P,A
related to τ. We write the relevant effective interactions as
= igL ℓ¯γ τX +gL ℓ¯γ γ τXµ +H.c.. (27)
−L − Pℓτ 5 P Aℓτ µ 5 A
7
The rates for τ ℓX ℓµ+µ− are described by Γ(τ ℓX ℓµ+µ−) = f Γ(τ
X
→ → → → →
ℓX)BR(X µ+µ−). Similar to the B decays, we only discuss τ ℓX. In terms of the
q
→ →
interactions in Eq. (27), the decay rates with X and X are given by
P A
m m3
Γ(τ ℓX ) τ gL 2 , Γ(τ ℓX ) τ gL 2 , (28)
→ P ≈ 16π Pℓτ → A ≈ 32πm2 Pℓτ
XA
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
respectively, where we have neglected the masses of ℓ and X due to m m > m .
P,A τ ≫ XP,A ℓ
TABLE II: The input values of parameters in units of GeV.
fK fBd fBs mX mK0 mBd mBs
0.16 0.20 0.22 0.214 0.497 5.28 5.37
m m m τ τ τ
s d b KL Bd Bs
0.15 0.01 4.4 7.87 1016 2.33 1012 2.22 1012
× × ×
In order to do the numerical estimations, the input values for the various parameters are
presented in Table II. To see the effects of the new particle on low energy physics, we first
consider its contributions to ∆F = 2 processes. ¿From the current experimental data, the
mass differences in the K and B systems are given by ∆m = (3.483 0.006) 10−15,
q K
± ×
∆m = (3.337 0.033) 10−13 and ∆m = (11.45+0.20) 10−12 GeV. By utilizing these
Bd ± × Bs −0.13 ×
values and those inputs in Tables I and II, from Eq. (4) the direct constraints on the
couplings are found to be
2 2 2
gQ < 2.3 10−15, gQ < 2.2 10−10, gQ < 6.4 10−9,
Psd × Pbd × Pbs ×
(cid:12) (cid:12)2 (cid:12) (cid:12) 2 (cid:12) (cid:12)2
(cid:12)gQ (cid:12) < 0.67 10−15, (cid:12) gQ(cid:12) < 5.1 10−13 (cid:12)gQ (cid:12) < 1.4 10−11. (29)
(cid:12) Asd(cid:12) × (cid:12) Ab(cid:12)d × (cid:12) Abs(cid:12) ×
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
Due to th(cid:12)e stro(cid:12)ng cancelation betwe(cid:12)en P(cid:12)SLL and PLR in B(cid:12) , th(cid:12)e constraints from ∆m
1 1 q Bq
are two orders of magnitude less than the naive expectation. Next, we discuss the decays of
P µ+µ−. It is well known that the long-distance effect dominates the process of K
L
→ →
µ+µ−, while the short-distance contribution usually is taken to be BR(K µ+µ−) <
L SD
→
3.6 10−10 [4]. As to the dileptonic decays in B decays, we also know the upper bounds of
q
×
BR(B µ+µ−) < 2.3 10−8 and BR(B µ+µ−) < 8 10−8 [15]. With these constraints
d s
→ × → ×
8
and Eq. (8), we have
2 2
gQ Γ(X µ+µ−) < 6.5 10−29, gQ Γ(X µ+µ−) < 3.8 10−20,
Psd P → × Pbd P → ×
(cid:12) (cid:12)2 (cid:12) (cid:12)2
(cid:12)gQ (cid:12) Γ(X µ+µ−) < 1.2 10−19, (cid:12)gQ (cid:12) Γ(X µ+µ−) < 1.0 10−29,
(cid:12) Pbs(cid:12) P → × (cid:12) Asd(cid:12) A → ×
(cid:12) (cid:12)2 (cid:12) 2(cid:12)
(cid:12)gQ (cid:12) Γ(X µ+µ−) < 2.3 10−24 (cid:12)gQ (cid:12)Γ(X µ+µ−) < 7.0 10−24. (30)
(cid:12) Abd(cid:12) A → × (cid:12)Abs (cid:12) A → ×
(cid:12) (cid:12) (cid:12) (cid:12)
It has(cid:12)been(cid:12)shown that the current strict bound(cid:12)s on(cid:12) gL 2 and gL 2 are from muon g 2,
(cid:12) (cid:12) (cid:12) (cid:12)| Pµ| | Aµ| −
given by gL 2 < 2.6 10−7 and gL 2 < 6.7 10−8 [4], respectively. ¿From Eq. (9), one
| Pµ| × | Aµ| ×
gets the upper bounds on the rates as Γ(X µ+µ−) < 4.3 10−10 GeV and Γ(X
P A
→ × →
µ+µ−) < 2.7 10−12 GeV. To illustrate the constraints on the couplings, we take Γ(X
P,A
× →
µ+µ−) (10−10,10−12) GeV and we obtain
∼
2 2 2
gQ < 6.5 10−19, gQ < 3.8 10−10, gQ < 1.2 10−9,
Psd × Pbd × Pbs ×
(cid:12) (cid:12)2 (cid:12) (cid:12)2 (cid:12) (cid:12)2
(cid:12)gQ (cid:12) < 1.0 10−17, (cid:12)gQ (cid:12) < 2.3 10−12 (cid:12)gQ (cid:12) < 7.0 10−12, (31)
(cid:12) Asd(cid:12) × (cid:12) Abd(cid:12) × (cid:12)Abs (cid:12) ×
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
respectively. It is clear that the constraints from ∆m are weaker than those from K
(cid:12) (cid:12) (cid:12) (cid:12) K (cid:12) (cid:12) L
→
µ+µ−, which are consistent with the HyperCP data in the decay of Σ+ p+µ+µ−, given
→
by [4]
2
gQ BR(X µ+µ−) = (8.4+6.5 4.1) 10−20,
Psd P → −5.1 ± ×
(cid:12) (cid:12)2
(cid:12)gQ (cid:12) BR(X µ+µ−) = (4.4+3.4 2.1) 10−20.
(cid:12) Asd(cid:12) A → −2.7 ± ×
(cid:12) (cid:12)
On the other hand,(cid:12)the b(cid:12)ounds from ∆m and BR(B µ+µ−) are similar.
(cid:12) (cid:12) Bq q →
To estimate the BRs of semileptonic B decays, we use the B M transition form
q q
→
factors in Eq. (15), calculated by the light-front quark model (LFQM) and summarized in
Table III [11]. In the table, the states of 1P and 3P will be used to consist of the physical
1 1
TABLE III: Values of form factors at q2 = 0 defined in Eq. (15) and calculated by the LFQM [11].
ABqK∗ ABqK∗ ABqK∗ ABqK3P1 ABqK1P1 FBqK0∗ FBqK0∗
0 1 2 1 0
0.31 0.26 0.24 0.26 0.11 0.26 0.26
states K (1270) and K (1400) and their relations are parametrized by [11, 18],
1 1
K1(1270) = K1P1 cosθ+K3P1sinθ,
K1(1400) = −K1P1sinθ+K3P1 cosθ, (32)
9
with θ = 58◦ being the mixing angle [13]. The decay of B K∗0µ+µ− has been measured
d
→
at the B-factories with the world average on the decay BR being (1.22+0.38) 10−6 [19].
−0.32 ×
¿From Eqs. (17), (18) and (20) and the values in Tables II and III, we obtain
2
BR(B K∗0X K∗0µ+µ−) = 3.1 1010 gQ BR(X µ+µ−),
d → P → × Pbs P →
(cid:12) (cid:12)2
BR(B K∗0X K∗0µ+µ−) = 3.9 1013(cid:12)gQ (cid:12) BR(X µ+µ−).
d → A → × (cid:12) Abs(cid:12) A →
(cid:12) (cid:12)
(cid:12) (cid:12)
If we regard BR(B K∗0µ+µ−) = 1.22 10−6 as the(cid:12)uppe(cid:12)r bound, we have
d
→ ×
2
gQ BR(X µ+µ−) 3.9 10−17,
Pbs P → ≤ ×
(cid:12) (cid:12)2
(cid:12)gQ (cid:12) BR(X µ+µ−) 3.1 10−20. (33)
(cid:12) Abs(cid:12) A → ≤ ×
(cid:12) (cid:12)
(cid:12) (cid:12)
For BR(X µ+µ−) (cid:12) 1,(cid:12)we find that the decay B K∗0µ+µ− gives the strongest
P,A d
→ ∼ →
limits on the couplings of gQ and gQ .
Pbs Abs
From Eq. (33), we can study the contributions of X to other B decays. The first
P,A q
direct application is B φX φµ+µ−. In terms of the formulas shown in Eqs. (17),
s P,A
→ →
(18) and (20), we get
BR(B φX φµ+µ−) 2.74 10−6,
s P
→ → ≤ ×
BR(B φX φµ+µ−) 2.81 10−6, (34)
s A
→ → ≤ ×
where we have used ABsφ(0) = 0.474, ABsφ(0) = 0.311 and ABsφ(0) = 0.234 for B φ
0 1 2 s →
transition form factors calculated by the light cone sum rules (LCSRs) [20]. Interestingly,
the bounds are just under the D0 upper limit of BR(B φµ+µ−) < 3.2 10−6 [21]. If
s
→ ×
the events observed by the HyperCp collaboration [1] are indeed from the new particle, the
decay of B φµ+µ− should be observed soon as the standard model prediction is around
s
→
1.6 10−6 [22].
×
Next, we discuss the productions of p-wave mesons in B decays. As mentioned before,
q
the p-wave mesons could be f (980), a (980), κ, K∗(1430) and K . However, since the
0 0 0 1
quark contents for the light p-wave mesons (<1 GeV) are not certain, we will only focus
on K∗(1430) and K . By using the values given in Tables II and III, we directly display
0 1
the predicted upper BRs in Table IV. From the table, we see clearly that only the produc-
tion of K∗(1430) is interesting, which is accessible to the current B-factories. As only the
0
transverse degrees of freedom are involved in the decays of B AX , where the effects
q A
→
10
