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 diagram for the e ＋ e －→P two-photon production process.

This figure was used to measure the π0 transition form factors. Here, this technique is applied to study the η,η’and ηc form factors.•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.

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+ e- γ*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+e-γ*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

0|, , ,

22 |

2211

2133

21

21

21

kdkbkz

kkPdkdkPQQ

qqQQ

, , kkk , , 21 kkk ,

22

k

kmk

where ΦQQ is the amplitude of the corresponding and k1

(2) is the on-mass shell light front momentum of the internal quark.

QQ

. )2(2 3

2

kddk

dk

Matrix elements and Form Factors

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

kzkvku

M

kkkz

QQQQ, , ,

2, 22

511

2

1

20

2121

and wave function can be expressed

Wei-Min Zhang, Chin. J. Phys,31,717(1994).

)2

exp(),( 2

2

QQ

zQQ

k

dx

dkNkx

Matrix elements and Form Factors

Decay constant ： The decay constant of pion is defined by

),(~2

)1(

020

2

kx

M

xx

Mp QQ

QQ

W

imp

mpi

impp

mppirpdNpQQA

QQc 221

15

221

153

4 T ][)(||0

pifpQQAQQ

)(||0

The amplitude can be written as ：

p3

p1Axial vector current

ΛQQ is the bound state vertex function. One could relate to the distribution functuion ψQQ by,

For the decay constant, the result are

223

2

),(16

2

km

mkx

kdxdNf

QQcQQ

Input parameters to fix the parameter ω.

pQQ

Matrix elements and Form Factors

Form factors are calculated in one-loop approximation.

The hadronic matrix elements which contribute to FPγ* are:

γ*(q1)

γ*(q2)

21221

1

222

2

223

353

42**

,

Tr ][21

qqimp

mpi

imp

mpi

imp

mpipdeeeQQA

Q

Q

Q

Q

Q

Q

QQqq

21

22

21

** ,* qqqqieFQQAP

The amplitude can be written as ：

p3 p2

p1

pQQ

Matrix elements and Form Factors

To numerical the meson P → γ*γ* (P ＝ η, η’and ηC) transition from factors within LFQM, we have to decompose into a Fock state for meson. The valence state of η, η’and ηC can be written as ：

,)1(1 pxp

The distribution functuion ψ is in term of the light-front relative monentun variable （ x, k⊥ ） .

,3 xpp ,)1(1

kpxp .3

kxpp

and ,1

222220 x

km

x

kmM QQ

.)( 2200 QQ mmMM

QQ

C

s

q

PCPC

PC

PC

C

|

|

|

1cossin

coscossin

sinsincos

|

'|

|

Matrix elements and Form Factors

where , and . 2

||

dduuq sss || ccC ||

Consequently, the transition from factors of P → γ γ h∗ave the forms

ssqqssqqFFFFFF sin cos, sin cos

CCssPCqqPC FFFF

C cossinan

d

ssqqssqqffffff sin cos, sin cos

the decay constant of η, η’and ηC to be

CCssPCqqPC ffff

C cossin

Numerical Results

The decay constants are

,140qqf 168ssf in MeV The branching ratios of η and η’mesons to 2γ are

%20.030.392 Br %14.012.22' Br

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⊥).

Numerical Results

which lead to |F(0,0)P→2γ|≡ |FPγ(0)| = 0.260 （ η ） and 0.341 （ η‘ ） in GeV−1

The factor FP (0,0) →2γ can be determined via

2

23

2

2 )0,0( 64

)4(

PPP

P FmBr

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

003.0069.0)0(2 CF

1.22.72 c

The decay widths of ηC are

5.03.52 c

We input the parameters ： mq=0.22 ～ 0.3, ms=0.40 ～ 0.45 GeV The mixing angle are , and

.

00 42~37 00 1.01 C02.21P

I.

II.

keV PDG data

which lead toI.

II. 011.0081.0)0(2 CF

keV Lattice QCD prediction

GeV-1

Numerical Results

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.

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

GeVin 7.4 , 2.2

)/()/(1

1

)0(

)(42

2

QQF

QF

c

c

Numerical Results

Summary of the decays constant of ηC

LFQM I 5.3 ± 0.5

(a) (b) (c)

LFQM II 7.2 ± 2.1

(a) (b) (c)

Lattice QCD

7.2 ± 2.1

394.7 ± 2.4

CLEO － 335 ± 52 ± 47 ± 12 ± 25

)(2 keVc )(MeVf

c

3.330.470.194

7.330.449.196

2.520.615.230

5.127

4.846.243

2.1434.840.2.249

2.1154.1166.303

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.

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

η and η’

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 , we have found that and MeV in the LFQM I and II based.

Conclusions

ηc

ccC ~

•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.

2.520.615.230

2.1154.1166.303

Acknowledgements

This work was partially supported by National Center for Theoretical Sciences, SZL-10004008, National Science Council (NSC-97-2112-M-471-002-MY3, NSC-98-2112-M-007-008-MY3, and NSC-101-2112-M-007-006-MY3) and National Tsing-Hua University (102N1087E1 and 102N2725E1) and SZMC-SZL10204006.