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PHENOMENOLOGICAL DESCRIPTION OF
SEMI-LEPTONIC DECAYS OF MESONS
CHIA SWEE PING
INTI College Malaysia
71800 Bandar Baru Nilai, Negeri Sembilan, Malaysia
and
EPHRANCE ABU UJUM
Institute of Mathematical Science, University of
Malaya
50603 Kuala Lumpur, Malaysia
CONTENTS
1.
Introduction
2.
Bound quarks in hadronic states
3.
Semi-leptonic decays of mesons
4.
CKM mixing matrix elements
5.
Comparison with experimental values
6.
Quality of the comparison
7.
Comparison with the muonic modes
8.
Unmeasured decay modes
9.
Conclusion
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Introduction
 Standard Model (SM) is successful.
 Electroweak processes are simple at the
quark level.
 Quarks are tightly bound inside hadrons.
 Interactions of bound quarks in hadrons are
not well understood.
 Calculations are complicated for electroweak
processes involving hadronic states.
 I will present here a phenomenological
model for the description of the electroweak
interactions of bound quarks in hadronic
states.
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Bound Quarks in Hadronic States
 Consider M1 changes into meson M2
electroweak process.
 M1 and M2 are bound states of q1q and
q 2q .
 This process is to be thought of as bound q1
in M1 changes into bound q2 in M2.
 Propose here a simple phenomenological
model.
 Bound quarks in hadrons behave like free
particles.
 Both q1 and q2 are to be treated as free
quarks.
 All QCD effects affecting a bound quark q1 in
the bound system q1q is considered
adequately taken care of by assigning an
effective mass to q1.
 In the present approach, the effective quark
mass is to be determined empirically.
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Semi-Leptonic Decays of Mesons
 Semi-leptonic decays of mesons
M1  M2
,
M1= q1q , M2= q 2q ,  = e, 
 The process is described by the quark level
process:
q1  q2
 = e, 
 Calculation at the tree level is
straightforward.
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 Decay rate
q1  q2   
GF2
2
m15 V21 R( y , x )
192 3
V21 = the CKM mixing matrix element
m1 and m2 = masses of q1 and q2
E1
dE [(1  y  x )(22  12 )  4(23  13 ) / 3]
E2
y  m22 / m12 , x  m2 / m12
R( y , x )  48 
E1  (1  y  x ) / 2 , E2  x
2  (E1  E ) /(1  E  q )
1  (E1  E ) /(1  E  q )
q  E2  x
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CKM Mixing Matrix Elements
 Quark flavour transitions and the
corresponding CKM mixing matrix elements:
u d
Vud
0.9738  0.0005
s u
Vus
0.2200  0.0026
c d
Vcd
0.224  0.012
c s
Vcs
0.996  0.013
b u
Vub
0.00367  0.00047
b c
Vcb
0.0413  0.0016
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 The
following
decay
transitions
considered:
 
ud  uu , dd
K 
us  uu
K  
ds  du
D   ,  
cd  dd
D   K  , K 
cd  sd
D    ,  
cu  du
D   K  ,K 
cu  su
B   ,  
ub  uu
B   D  , D 
ub  uc
B   ,  
db  dd
B   D  ,D
db  dc
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Comparison with experimental
values
 Altogether there are 37 decay modes,
counting both electronic and muonic modes.
 Of these, 24 modes have their branching
ratios measured experimentally.
 Out of the above, 18 of the decay modes are
the electronic modes.
 The experimental partial decay rates are
shown in Table 1.
 14 parameters: effective masses of quarks
in mesons +, o, K+, Ko, D+, Do, B+, Bo , +,
o , K*+, K*o, D*+, and D*o.
 Assumption 1: For q1q in M1 effective mass
of q1 and that of q are the same.
 Assumption 2: For o and o, (superposition
of uu and dd ) effective mass of u and that
of d are the same.
 An effective mass is hence assigned to each
hadronic state (effective mass of bound
quark in hadron).
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Quality of the comparison
 Reasonable fit with 2 = 81.38 for N=18.
 Calculated values do not compare well with
experimental values for:
B    e  e , B   e  e .
 The decays are associated with the ill-
measuredVub.
 Ignoring the two B to  modes:
B    e  e and B   e  e
yields 2 = 6.50 for N=16.
 The computed decay rates for these 18
decay modes are also displayed in Table 1.
 The effective masses of the hadronic states
obtained from the fit are shown in Table 2.
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Table 1. Semi-Leptonic Decays of Mesons
(electronic modes)
Decay Mode
    e  e
Γexp (MeV)
Γcomp (MeV)
(2.5920.086)E-22 2.597E-22
K    e  e
(2.5880.032)E-15
2.586E-15
K L    e  e
(4.9320.051)E-15
4.933E-15
D    e  e
(1.960.95)E-12
2.069E-12
D    e  e
(1.580.63)E-12
9.275E-13
D   K e  e
(4.240.57)E-11
4.231E-11
D   K e  e
(3.480.44)E-11
3.482E-11
D    e  e
(5.780.96)E-12
5.276E-12
D  K  e  e
(5.740.29)E-11
5.751E-11
D  K e  e
(3.450.56)E-11
3.454E-11
B    e  e
(3.51.1)E-14
9.543E-14
B    e  e
(5.31.3)E-14
8.507E-14
B   D e  e
(8.470.87)E-12
8.480E-12
B   D e  e
(2.560.20)E-11
2.346E-11
B    e  e
(5.700.94)E-14
1.883E-13
B    e  e
(1.110.30)E-13
1.669E-13
B  D  e  e
(9.170.86)E-12
9.169E-12
B  De  e
(2.330.10)E-11
2.328E-11
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Table 2: Effective Quark Masses
Meson Effective Mass (MeV)
200.51
+
197.00
o
K+
450.8
Ko
451.0
D+
1336
Do
1400
B+
3635
Bo
3626
500
+
489
o
K*+
585
K*o
502
D*+
285
D*o
300
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Comparison with the muonic modes
 There are 6 muonic modes that are
experimentally measured.
 The partial decay rates are computed with
the values of the effective masses
determined from the electronic modes.
 Comparison with experimental values is
shown in Table 3.
Table 3. Semi-Leptonic Decays of Mesons
(muonic modes)
Γexp (MeV)
(1.7380.032)E-15
Γcomp (MeV)
1.075E-15
K L      
(3.4550.041)E-15
2.006E-15
D      e
(2.150.51)E-12
8.536E-13
D   K   
(4.430.95)E-11
3.917E-11
D   K    
(3.480.25)E-11
3.198E-11
D  K    
(4.970.27)E-11
5.374E-11
Decay Mode
K      
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Unmeasured decay modes
 The predictions for the 13 unmeasured
decay modes are shown in Table 4.
Table 4. Unmeasured Decay Modes
comp (MeV)
1.960E-12
Decay Mode
D      
D      
5.025E-12
D    e  e
2.454E-12
D      
2.279E-12
D  K    
3.163E-11
B      
9.479E-14
B      
8.446E-14
B   D   
8.378E-12
B   D    
2.330E-11
B      
1.871E-13
B      
1.657E-13
B  D    
9.063E-12
B  D   
2.312E-11
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Conclusion
 Formulated a simple model to describe the
electroweak interactions of bound quarks
inside hadrons.
 Quark tightly bound inside a hadron behaves
as a free particle with an effective mass,
which can be taken as the result of all the
QCD effects on this quark.
 Model allows calculations of the electroweak
processes at the quark level to be applied
directly to the corresponding hadronic
processes.
 Model applied to semi-leptonic decays of
mesons yielded reasonable fit, and effective
masses of quarks in mesons are empirically
estimated.
 For decays that involve Vub, the fit is
rather out, it can be improved if the value of
Vub is reduced.
 The K to πμ modes do not fit well.
Improvement is needed.
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 We intend to extend the model to fit the
semi-leptonic decays of baryons.
 Ultimately, we want to apply the model to
rare decays of mesons with the emissions of
neutrino pair or charged lepton pair. A
preliminary run earlier yielded reasonable
agreement with experimental data.
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