Rare Decays of ψand Υ

18 Pages·0.12 MB·

by  K. K. Sharma

Υ
Rare Decays of ψ and
K. K. Sharma and R. C. Verma
∗
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9 Centre for Advanced Study in Physics, Department of Physics,
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Panjab University, Chandigarh -160014,India.

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a
J ∗ Department of Physics, Punjabi University, Patiala - 147 002, India.

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v Abstract
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2 We study two-body weak hadronic decays of ψ and Υ employing the factorization
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0 scheme. Branching ratios for ψ →PP/PV and Υ→PP/PV decays in the Cabibbo-
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angle-enhanced and Cabibbo-angle-suppressed modes are predicted.
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1 Introduction
A wealth of experimental data on masses, lifetimes and decay rates of charm
and bottom mesons has been collected [1]. Models based on factorization
scheme [2,3], successfully used to describe weak decays of naked flavor mesons,
can also be used to study weak decays of hidden flavor mesons. Low lying
states of quarkonia systems usually decay through intermediate photons or
¯
gluons produced by the parent cc¯and bb quark pair annihilation. For instance,
charmonium state ψ(1S) decays predominantly to hadronic states (87.7%),
whereas its leptonic modes are observed to be around 6%. The same trend
follows for the bottomonium states Υ(1S) also. These OZI violating but flavor
conserving decays lead to narrow widths to ψ and Υ states.
IntheStandardModelframework, theflavorchangingdecays ofthesestates
arealso possiblethoughtheseareexpected tohaverather lowbranching ratios.
At present, experiments have crossed over to many million ψ-events and the
prospectus are good for a 100-fold increase in these events in the high energy
electron-positron collider. It is expected that some of its rare decay modes
may also become detectable in future. With this possibility in mind, we study
two-body weak decays of ψ in Cabibbo-angle-enhanced and Cabibbo-angle-
suppressed modes. Employing the factorization scheme, we predict branching
ratios for ψ PP/PV decays ( where P and V represent pseudoscalar meson
→
and vector meson respectively ) and extend this analysis to Υ PP/PV
→
decays involving b c transitions.
→
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2 Two-body Weak Decays of ψ
The structure of the general weak current current Hamiltonian is
⊗
G
F
H = V V a (s¯c) (u¯d) +a (s¯d) (u¯c) +h.c., (1)
W √2 ud c∗s{ 1 H H 2 H H}
for Cabibbo-angle-enhanced mode (∆C = ∆S = 1), and
−
G
F ¯
H = V V [a (u¯d) (dc) (u¯s) (s¯c)
W √2 ud c∗d 1{ H H − H H}
¯
+ a (u¯c) (dd) (u¯c) (s¯s) ]+h.c., (2)
2 H H H H
{ − }
forCabibbo-angle-suppressed mode(∆C = 1,∆S = 0). q¯q q¯γ (1 γ )q
1 2 1 µ 5 2
− ≡ −
represents the color singlet V-A current and V denote Cabibbo-Kobayashi-
ij
Maskawa (CKM) mixing matrix elements. The subscript H implies that (
q¯q ) is to be treated as a hadron field operator. as are the two undetermined
1 2 ′
coefficients assignedtotheeffective chargecurrent, a , andtheeffective neutral
1
current, a , parts of the weak Hamiltonian. These parameters can be related
2
to the QCD coefficients c as
1,2
a = c +ζc , (3)
1,2 1,2 2,1
where ζ is usually treated as a free parameter, to be fixed by the experiment.
We take
a = 1.26, a = 0.51, (4)
1 2
−
on the basis of D Kπ decays [4].
→
3
2.1 ψ PP Decays
→
The decay rate formula for ψ PP decays is given by
→
p3
Γ(ψ PP) = c A(ψ PP) 2, (5)
→ 24πm2 | → |
ψ
where p is the magnitude of the three momentum of final state meson in
c
the rest frame of ψ meson and m denote its mass. Following the procedure
ψ
adopted by one of us (RCV) with Kamal and Czarnecki [5] in determination
of the weak decay amplitudes of ψ decays, we express the decay amplitude of
ψ PP as (upto the scale GF CKMfactor QCD coefficient),
→ √2 × ×
A(ψ PP) = P Jµ 0 P J ψ , (6)
µ
→ h | | ih | | i
where Jµ is the weak V-A current. Matrix elements of the weak current are
given by
P(k) A 0 = ι f k , (7)
µ P µ
h | | i −
and
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P J ψ = ǫ ǫν(P +P )ρqσV(q2) ι (m +m )ǫµA (q2)
h | µ| i m +m µνρσ ψ ψ P − ψ P ψ 1
ψ P
ǫ q ǫ q
ι ψ · (P +P )µA (q2) + ι ψ · (2m )qµA (q2)
− m +m ψ P 2 q2 ψ 3
ψ P
ǫ q
ι ψ · (2m )qµA (q2), (8)
− q2 ψ 0
where f is the meson decay constant, ǫ is the polarisation vector of ψ, P
P ψ ψ
and P are the four-momenta of ψ and pseudoscalar meson respectively, and
P
qµ = ( P P )µ. A (q2) is related to A (q2) and A (q2) as
ψ P 3 1 2
−
( m + m ) ( m m )
2 ψ P 2 ψ P 2
A (q ) = A (q ) + − A (q ). (9)
3 1 2
2m 2m
ψ ψ
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In the Cabibbo-angle-enhanced mode, ψ can decay to D+π orD0K0. To illus-
s −
trate the procedure, we consider the color enhanced decay ψ D+π whose
→ s −
decay amplitude can be expressed as
G
A(ψ D+π ) = FV V a π Jµ 0 D+ J ψ , (10)
→ s − √2 ud c∗s 1 h −| | i h s | µ| i
which gets simplified to
G
A(ψ D+π ) = FV V a (2m )f (ǫ q)AψDs(m2). (11)
→ s − √2 ud c∗s 1 ψ π ψ · 0 π
Similarly, the factorization amplitude of the decay ψ D0K0 and Cabibbo-
→
angle-suppressed decays of ψ are obtained. We take the following values for
the decay constants (in GeV) [6]:
f = 0.132, f = 0.161,
π K
fη = 0.131, fη′ = 0.118. (12)
For the form factors at q2 = 0, we use
AψD(0) = 0.61, AψDs(0) = 0.66, (13)
0 0
obtained in the earlier work [5] using the BSW model wavefunctions [7] at
ω = 0.5 GeV. The oscillator parameter ω is the measure of the average
transverse momentum of the quark in the meson [7]. We use the following
basis for η η mixing:
′
−
1
¯
η = (uu¯+dd)sinφ (ss¯)cosφ , (14)
p p
√2 −
1
¯
η = (uu¯+dd)cosφ +(ss¯)sinφ , (15)
′ p p
√2
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where φ = θ θ . Using the decay rate formula (5), we compute
p ideal physical
−
branching ratiosofvariousψ PP decayswhich arelistedinTable-1. Among
→
the Cabibbo-angle-enhanced decays, we find that the dominant decay is ψ
→
D+π , having the branching ratio
s −
+ 7
B(ψ D π ) = (0.87 10 )%, (16)
→ s − × −
and the next in order is ψ D0K0, whose branching ratio is
→
0 0 7
B(ψ D K ) = (0.28 10 )%. (17)
−
→ ×
We hope that these branching ratios would lie in the detectable range.
2.2 ψ PV Decays
→
Similar to ψ PP decays, the weak decay amplitude for ψ PV decays can
→ →
be expressed as product of the matrix elements of the weak currents,
A(ψ PV) = V Jµ 0 P J ψ , (18)
µ
→ h | | ih | | i
where the vector meson is generated out of the vacuum, and the corresponding
matrix element is given by
V(k) V 0 = ǫ m f . (19)
h | µ| i ∗µ V V
Using the matrix elements given in (8), and (9), we obtain
2m f
A(ψ → PV) = [m +V mV ǫµνρσǫ∗1µǫ∗2νPψρPPρV(q2)
ψ P
2
+ι(m f ) ǫ .ǫ (m +m )A (q )
V V ∗1 ∗2 ψ P 1
{
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P +P
ψ P 2
ǫ .(P P )ǫ . A (q ) ], (20)
∗1 ψ P ∗2 2
− − m +m }
ψ P
which yields the following decay rate formula:
p
2 c 2 2
Γ(ψ PV) = (nonkinematic factor) ( m f ) ( m + m )
→ 24π m2 V V ψ P
ψ
2 2 2 2 2 2 2 2
α V(q ) + β A (q ) + γ A (q ) + δ Re[A (q ) A (q )] , (21)
1 2 ∗1 2
×{ | | | | | | × }
where
8 m2 p2
α = P c , (22)
( m + m )4
P ψ
m2 m2 m2
β = 2 + ψ − P − V 2, (23)
{ 2 m m }
ψ V
4 m4 p4
γ = P c , (24)
m2 m2 ( m + m )4
ψ V P ψ
and
2 (m2 m2 m2 ) m2 p2
δ = ψ − P − V P c , (25)
( m + m )2 m2 m2
P ψ ψ V
Nonkinematic factor is the product of scale factor (GFV Vcs ) and the ap-
√2 ud ∗
propriate QCD coefficient a or a . The terms corresponding to A (q2), V(q2),
1 2 1
and A (q2) represent S, P, and D partial waves in the final state. We have
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computed the numerical coefficients ( α, β, γ, δ ) for various decays which
are given in Table-2. We observe that the numerical coefficient of A (q2) is
1
the largest, thus the retention of S-wave only would appear to be an excellent
approximation. For the sake of simplicity, we have retained only this term in
our analysis. Following values of the vector meson decay constants (in GeV)
[6] are used in our analysis:
fρ = 0.216, fK∗ = 0.221, fω = 0.195, fφ = 0.237,fD∗ = 0.250, (26)
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and the form factors [5]
AψD(0) = 0.68, AψDs(0) = 0.78, (27)
1 1
at ω = 0.5 GeV. Using the decay rate formula (21), we obtain the branch-
ing ratios of ψ PV decays in Cabibbo-angle-enhanced and Cabibbo-angle-
→
suppressed modes which are given in Table-3. For the color enhanced decay of
the Cabibbo-angle-enhanced mode, we calculate
+ 6
B(ψ D ρ ) = (0.36 10 ),% (28)
→ s − × −
which is higher than the branching ratio of ψ D+π . Our analysis yields
→ s −
B(ψ D+ρ )
→ s − = 4.2, (29)
B(ψ D+π )
→ s −
and therefore ψ D+ρ can be expected to be measured soon.
→ s −
3 Two-body Weak Decays of Υ
In this section, we extend our analysis to Υ PP/PV decays.
→
3.1 Υ PP Decays
→
The effective weak Hamiltonian generating the dominant b quark decays in-
volving b c transition is given by
→
G
H∆b=1 = F V V [a (c¯b)(d¯u)+a (d¯b)(c¯u)]
W √2{ cb u∗d 1 2
+V V [a (c¯b)(s¯c)+a (s¯b)(c¯c)] +h.c., (30)
cb c∗s 1 2 }
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for the CKM favored mode and
G
H∆b=1 = F V V [a (c¯b)(s¯u)+a (s¯b)(c¯u)]
W √2{ cb u∗s 1 2
¯ ¯
+V V [a (c¯b)(dc)+a (db)(c¯c)] +h.c., (31)
cb c∗d 1 2 }
for the CKM singly suppressed mode. In our analysis we use
a = 1.03, a = 0.23, (32)
1 2
as guided by B PP/PV data [8]. Similar to ψ PP decays, the factoriza-
→ →
tion scheme expresses weak decay amplitudes as a product of matrix elements
of the weak currents ( upto the scale GF CKMfactor QCDfactor) as :
√2 × ×
A(Υ PP) = < P Jµ 0 >< P J Υ > . (33)
µ
→ | | | |
For instance the decay amplitude for the color enhanced mode Υ B+π of
→ c −
the CKM-favored decays is given by
G
A(Υ B+π ) = F V V a f (2m )AΥ Bc(m2). (34)
→ c − √2 × cb u∗d × 1 π Υ 0→ π
We take (in GeV)
f = 0.240, f = 0.280, f = 0.393, (35)
D DS ηc
values [6] in our calculations. Generally the form factors for the weak tran-
sitions between heavy mesons are calculated in the quark model framework
using meson wave functions. However, in the past few years the discovery of
new flavor and spin symmetries has simplified the heavy flavor physics [9], and
it has now become possible to calculate these form factors from certain mass
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factors involving the Isgur-Wise function. In the framework of heavy quark
effective theory (HQET), these can be expressed as
m +m
AΥ Bc(q2) = Υ Bc ξ(ω), (36)
0→
2 (m m )
q Bc Υ
with
m2 + m2 m2
ω = v v = Υ Bc − π, (37)
Υ · Bc 2 m m
Υ Bc
where the Isgur-Wise function ξ is normalized to unity at kinetic point v
Υ
·
v = 1. As ω 1.08 for Υ B+π decay, we have ignored the ω
Bc ≈ → c −
dependence of the Isgur-Wise function ξ and calculate
AΥ Bc(q2) 0.98. (38)
0→
≈
This in turn yields
+ 8
B(Υ B π ) = 0.33 10 %. (39)
→ c − × −
The decay amplitudes for other CKM-favored and CKM-suppressed decay
modes are obtained similarly. Branching ratios for various Υ PP decays
→
are given in Table 4. We find that the dominant decays are Υ B+D and
→ c s−
Υ B+π .
→ c −
3.2 Υ PV Decays
→
The decay rate formula for such decays has been discussed in the section
3. Here also for comparison of the contributions of the various form factors
involved, wehave calculatedthenumerical coefficients (α, β, γ, δ )forvarious
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