Transcript Form factors of η, η’ and ηc in light front quark model

Form factors of η, η’ and ηc in light
front quark model
C. C. Lih and C. Q. Geng
The International Workshop on Particle Physics and Cosmology after Higgs and Planck,
Chongqing, China, 2013
Motivation
The motive mainly comes from the new experimental data.
This figure was used to
the π0 transition form
Here, this technique is
to study the η,η’and
form factors.
measure
factors.
applied
ηc
•CLEO experiment study the
ηand η’form factors cover
the Q2 region from 1.5 to about
20 GeV2.
•Babar experiment study the
ηand η’form factors in the
Q2 range from 4 to 40 GeV2.
This diagram for the e＋e－→P
two-photon production process.
Motivation
The transition form factors multiplied by Q2 for (a) η and (b) η’.
Motivation
The transition form factors multiplied by Q2 for (a) η and (b) η’. The solid line
shows the result of the fit to BABAR data. The dashed lines indicate the
average form factor values over the data points with Q2 > 14 GeV2.
Motivation
 The measurements on ηc→γ∗γ
form factor have been done
by both L3 and BaBar
Collaborations based on the
process of e+ e-→ e+ eγ*γ*→ e+ e- ηc in the
range of Q2 from 2 to 50 GeV2.
 Since ηc is composed of two
massive charm quarks, it is
important to know the
behavior of the ηc→e+eγ*γ* transition form
factor at a high Q2 momentum
transfer to compare with
those of the light
pesudoscalar ones.
Matrix elements and Form Factors
 Light-Front Quark Model
Within the light front formalism, the meson bound
state, which consists of a quark q and an anti-quark
with the total momentum P and spin s, can be written as
| QQP   
 1 2
3
3
P  k1  k 2 






dk
1
dk
2
2
2



 1 2
z, k   bq k1 , 1  d q k 2 , 2  | 0
 Q
Q
where ΦQQ is the amplitude of the corresponding QQ and
k1(2) is the on-mass shell light front momentum of the
internal quark.
k  k  , k ,
m 2  k 2
,
k  k1 , k2  , k 

k

dk  d 2 k
dk 
.
3
2(2 )
Matrix elements and Form Factors
and wave function can be expressed
12
QQ
k k
z, k   
 2M

1

2
2
0
1
2

5



u
k
,


vk2 , 2  Q Q z, k 
1
1


with ψ being the space part wave function that
depends on the dynamics. This distribution
functuion ψ is in term of the light-front
relative monentun variable （x, k⊥）. One wave
function that has often been used for the meson
is Gassian type
2
QQ ( x, k )  N
dkz
k
exp(
)
2
dx
2Q Q
Wei-Min Zhang, Chin. J.
Phys,31,717(1994).
Matrix elements and Form Factors
 Decay constant：
p3
The decay constant of pion is defined by
 0 | A  | QQ( p )  if Q Q p 
Axial vector current
QQ  p
W
p1
The amplitude can be written as：
 5 i p 1  p  m 
i p 1  m  

4
 5
0 | A | QQ( p)   N c  [d p3 ]  Q Q Tr 
  2

2
2
2


p

p

m

i

p

m

i

1 
1


ΛQQ is the bound state vertex function. One could relate to the
distribution functuion ψQQ by,
QQ
x(1  x)

~  ( x, k )
p 2  M 02
2M 0 Q Q
For the decay constant, the result are
fQQ
dxd 2 k 
m
 2 Nc 

(
x
,
k
)

2
16 3 Q Q
m2  k
Input parameters
to fix the parameter ω.
Matrix elements and Form Factors
 Form factors are calculated in
γ*(q1)
one-loop approximation.
p3
 The hadronic matrix
QQ  p
elements which contribute
to FPγ* are:



p2
p1
γ*(q2)

A QQ   * *  ieFP * q12 , q22   q1 q2
The amplitude can be written as：

A QQ   
*

*

 5 i p
 3  mQ 

 eq1 eq e  [ d p3 ]  Q Q T r

2
2
2
p3  mQ  i

2
i p
 2  mQ 
p  m  i
2
2
2
Q

4

i p
 1  mQ  
     , q1  q2 
2
2
p1  mQ  i 
Matrix elements and Form Factors
The distribution functuion ψ is in term of the light-front
relative monentun variable （x, k⊥）.
p1  (1  x) p  , p3  xp , p1  (1  x) p  k , p3  xp  k .

and
M 02 
mQ2  k2
1 x

mQ2  k2
x
2
2
, M 0  M 0  (mQ  mQ ) .
 To numerical the meson P → γ*γ* (P＝ η, η’and
ηC) transition from factors within LFQM, we have to
decompose into a QQ Fock state for meson. The
valence state of η, η’and ηC can be written as：
cos
 sin 
  C sin  P  |  q  
|  


 

sin 
cos
 C cos P  | s  
| '    
 |       sin       cos   
 |   
1
P
C
P
 C   C
 C 
Matrix elements and Form Factors
Consequently, the transition from factors of P → γ∗γ
have the forms
F  cos Fq q  sin  Fs s , F  cos Fq q  sin  Fs s
and F   sin  F  cos  F  F
C
C
P
C
P
qq
ss
CC
| uu  d d 
, | s  | s s and | C  | cc .
2
the decay constant of η, η’and ηC to be
where | q  
f  cos f q q  sin  f s s ,
f  cos f q q  sin  f s s
fC  C sin P  f q q C cos P  f s s  fCC
Numerical Results
We use the decay constant and the branching ratio of
P → 2γ to specify the quark masses of mu,d,s and the
meson scale parameter of ωQQ in ψQQ(x,k⊥).
 The decay constants are
f qq  140,
f ss  168
in MeV
 The branching ratios of η and η’mesons to 2γ are
Br 2  39.30  0.20%
Br '2  2.12  0.14%
Numerical Results
The factor FP (0,0) →2γ can be determined via
BrP2
2
(4 ) 2 3

mP FP2 (0,0)
64 P
which lead to |F(0,0)P→2γ|≡ |FPγ(0)| = 0.260（η） and
0.341（η‘） in GeV−1
Numerical Results
Note that the upper (lower) edges of the green bands in
figures correspond to mq=0.3(0.25) and ms=0.45(0.4) GeV,
while those of the yellow bands ψ=37°(42°).
Numerical Results
In these figures, we draw Q2Fη(η’)γ(Q2) as a function of Q2,
where the green and yellow bands represent the inputs of
mq=0.22～0.3, ms=0.40 ～ 0.45 GeV and ψ=40° and
mq=0.25,ms=0.45 GeV and ψ=37 ～ 42°, respectively.
Numerical Results
 The decay widths of ηC are
c 2  5.3  0.5
II.  2  7.2  2.1
I.
keV
PDG data
keV
Lattice QCD prediction
c
which lead to
FC 2 (0)  0.069 0.003
II. FC 2 (0)  0.081 0.011
I.
GeV-1
 We input the parameters：
mq=0.22～0.3, ms=0.40 ～ 0.45 GeV
0
0
0
0
 The mixing angle are   37 ~ 42 , C  1  0.1
and  P  21.20 .
Numerical Results
they are about 10 % smaller than the data
points for Q2 in the range of 7.5 ∼ 20
GeV2.
it is sensitive to the mixing angles as
well as the mass of the charm quark.
we can also fit the result of the
LFQM I by a double pole form
F c (Q 2 )
F c (0)

1
1  (Q /  ) 2  (Q /  ) 4
  2.2 ,   4.7 in GeV
The bands for LFQM I and II correspond
to the calculations based on Γ (ηc→2γ)
given by PDG and Lattice QCD
calculation, respectively.
In this figure, we show charm
quark mass dependence for ηc
decay constant.
Numerical Results
 Summary of the decays constant of ηC
c 2 (keV )
fc (MeV )
.3
194.033
47.0
(b)
.7
196.933
44.0
.2
(c) 230.552
61.0
LFQM I
5.3 ± 0.5
(a)
LFQM II
7.2 ± 2.1
.2
.5
115.2
(a) 243.6127
(b) 249.2.0143
(c) 303.6116
84.4
84.4
.4
Lattice QCD
7.2 ± 2.1
394.7 ± 2.4
CLEO
－
335 ± 52 ± 47 ± 12 ± 25
Both results are within the error of the CLEO data, but they are
somewhat smaller than Lattice QCD result, in which Γηc→γγ = 7.2
± 2.1 keV is used like the LFQM II. However, the Lattice QCD result
can easily be accounted when a larger value of the charm-quark mass
is used.
Conclusions
 η and η’
As shown in figures, our results for Q2Fη(η’)γ(Q2) are in good
agreement with the experimental data.
- We remark that the form factors Q2Fη(η’)γ(Q2) increase
(decrease) with quark masses mq(the mixing angle ψ.
- The effect from the uncertainty of ms is small due to the small
quark charge.
- It is interesting to point out that the form factors can be
better fitted for a larger mq with a fixed ψ or ψ = 40° with a
fixed mq in the lower Q2 region.
Conclusions
 ηc
We have illustrated the transition form factor of ηc →γ∗γ as a
function of the momentum transfer Q2. We have shown that although our
results are consistent with the experimental data by the BaBar
collaboration. We have also evaluated the decay constant of ηc. We
have shown that it is sensitive to the mixing angles as well as the
mass of the charm quark. Explicitly, for C ~ cc , we have found
that 230.552.2 and 303.6115.2 MeV in the LFQM I and II based.
61.0
116.4
•Future precision measurements on the decay width of ηc →γ∗γ
are clearly needed in order to determine the ηc decay constant in
the LFQM.
Acknowledgements

This work was partially supported by National Center for Theoretical Sciences,
SZL-10004008, National Science Council (NSC-97-2112-M-471-002-MY3, NSC98-2112-M-007-008-MY3, and NSC-101-2112-M-007-006-MY3) and National
Tsing-Hua University (102N1087E1 and 102N2725E1) and SZMC-SZL10204006.