Study light scalar mesons from
heavy quark decays
Cai-Dian Lü (呂才典)
IHEP, Beijing
Wei Wang, CDLu, arXiv:0910.0613 [hep-ph]
CD Lu
1
Outline

Scalar meson study status
Four quark states or two-quark states:
semileptonic decays
Non-leptonic decays

Summary


CD Lu
2
Scalar Mesons (JP=0+)
K 0* (1430)
*
0
K (1430)
qq
2

0
a (1470)
q q
or
2
qq
CD Lu
0
0
a (1470)
f 0 (1500)
a0 (1470)
K 0* (1430)
K 0* (1430)
 (800)
 (800)

0
a (980)
 (800)
a00 (980)
f 0 (980)
a0 (980)
 (800)

f 0 (1370)

 (600)
f 0 (1710)
3
Scalar Mesons around
1GeV (JP=0+)
 exist or not for many years
Mass too light to be
qq
Because 0++ states are S=1, L=1, should be heavier
than 1– –(,  K*)
 (800)
2
q q
2
CD Lu

0
a (980)
 (800)
 (800)
a00 (980)
f 0 (980)
a0 (980)
 (800)

 (600)
4
the neutral scalar meson 0++ has the
same quantum number with vacuum




They can mix with vacuum, glueball, even
molecular states
So a lot of explanations on the market—
tetra quark states
Most study focus on the decay property of
the scalar mesons
The production of scalar mesons from
heavy quark decays are more interesting
CD Lu
5
Feynman diagrams of semileptonic decays of D to Scalar
Mesons
D+
D+  f0 +, D+   + have been measured
CD Lu
6
Isospin relation
Isospin 0 and isospin 1 contribution should
be 1:1, derived from the Clebsch-Gordan
coefficients
 Isospin conserved by strong interaction, no
matter perturbative or non-perturbative
--model independent


0 



similarly, B / D   e  e  B / D  e  e
already verified by exp.

CD Lu
7
2-quark picture of ordinary
light Scalar Mesons
 - f0 mixing:
f0
 cos 
 

  sin 
sin   ss

cos   nn
with
CD Lu
8
2-quark picture of ordinary
light Scalar Mesons
 - f0 mixing:
f0
 cos 
 

  sin 
sin   ss

cos   nn
with
CD Lu
9
Sum rule independent of mixing
only d d component contributes, in isospin
symmetry, we have
where
Br ~ |A|2,
CD Lu
We can get sum rule as
10
4-quark picture of ordinary
light Scalar Mesons
Group theory
assignment
 - f0 mixing:
 cos 
 

  sin 
f0
with
CD Lu
sin   nn ss

cos   uu d d
uu  dd
nn 
2
11
4-quark picture of ordinary
light Scalar Mesons
Group theory
assignment
 - f0 mixing:
 cos 
 

  sin 
f0
with
CD Lu
sin   nn ss

cos   uu d d
uu  dd
nn 
2
12
4-quark picture of Scalar
Mesons
where
We can get sum rule as
CD Lu
13
Define a ratio R
It is one for 2-quark picture, while 3 for 4-quark
picture
Similarly, for B meson decays, we have
CD Lu
14
These channels have large
enough BRs to be measurable
CD Lu
15
CD Lu
16
c
B J/ (c) f0



Semileptonic B decays B   f 0l  l are clean,
but the neutrino is identified as missing
energy, thus the efficiency is limited
The lepton pair can also be replaced by a
charmonium state such as J/, since J/
does not carry any light flavor either.
B J/ f0 decays may provide another ideal
probe to detect the internal structure of
the scalar mesons.
CD Lu
17
c
B J/ (c) f0



Semileptonic B decays B   f 0l  l are clean,
but the neutrino is identified as missing
energy, thus the efficiency is limited
The lepton pair can also be replaced by a
charmonium state such as J/, since J/
does not carry any light flavor either.
B J/ f0 decays may provide another ideal
probe to detect the internal structure of
the scalar mesons.
CD Lu
18
B J/(c) S
CD Lu
19
CD Lu
20
Uncertainties mainly from
SU(3) breaking




Form factor difference – no problem, since it
makes the R for 4-quark picture even larger than 3
Mostly by mass difference, only problem:
m = (0.4~1.2) GeV, but any way phase space
is easy to calculate
Relatively large uncertainty in D decays. But in B
decays, there is no problem, since m negligible
Governed by heavy quark effective theory, the
SU(3) breaking is suppressed by 1/mQ
CD Lu
21
Summary


The semi-leptonic decays of D and/or B
mesons to scalar mesons can provide a
theoretically clean way to distinguish the 2quark and 4-quark picture of light scalar
mesons
So as the non-leptonic decays of B J/(c) S
CD Lu
22
Thank you!
CD Lu
23
The
measurem
ent of
ratio R
can
determine
the
mixing
angle of f0
and 
CD Lu
24