Relativistic Effects in Two Photon Decay of Quarkonium

H.Q.Zhou

Institute of High Energy Physics, CAS, Beijing100039,P.R.China

with B.S.Zou

1:Introduction

2:Basic formalism

3:Methods in literatures and Our work

4:Application to and our result

5:Further discussion

2c

1: Introduction

The is the most simple and pure process related to QCD bound state, it is an ideal place to test various description of QCD bound state from theoretical point of view.

2Meson

Up to now there are two main methods to study this process: NRQCD and Salpeter equation+Mandelstam form . NRQCD can study the corrections order by order in and while Salpeter equation can study the whole relativistic effects

s

v

2

221

||642

1

M

M

dd cm

),(),(]),([)2(

3 22114

4

kkPpTrpd

M vuuv

2:Basic formalism

Decay width in Mandelstam form for is

P

1k 2k

pmomentum of meson

momentum of photons

relative momentum

),( 11 k ),( 22 k polarization of photons

20

When we discuss relativistic effects we chooseleading order in pQCD for

u

c

vv

c

uq

uv

mkpmkpee ^

2

^

1

^

1

^

1

22 11

The relativistic effects are mainly contained in BS wave function ),( Pp

3 Methods in literature and Our work

Methods in most of the literatures to discuss relativistic effects are in fact equivalent to assuming the following form for BS function:

We assume

zzmsss

z JJmLssssssNPpz

,|,;,,|,2

1;,

2

1),(

..,21

21

)(|)(|)()()( 02

'

1'

21 pfqRYpvpuqLmss

zzmsss

z JJmLssssssNPpz

,|,;,,|,2

1;,

2

1),(

..,21

21

)(|)(|)()()( 021 21pqRYpvpu

qLmss

In this form all the relativistic effects just come from momentum distribution

This gives two new kinds of corrections which relate with

|)(| qR )( 0qf'' ,vuwhen considering the quark

confinement and complicated

property of quark propagator

and spinor form

energy distribution

(in meson rest frame)

21, ss vu

pE

MiN

22

free spinor form

M

m

E

Mi

p

222or

LmYqR |)(| obey a Schrodinger -type equation

21

'' , ss vu not free spinor form

)( 0q )( 0qf

heavy quark limit

quark confinement: pole structure of quark propagator

heavy quark limit

(in [1,2,3,4]) (in [5])

Why there are such two new kinds of corrections ?

complicated quark propagator

These two new kinds of effects are not known clearly (while definitely exist) and are different from the situation in QED because of confinement. For simplicity we do the following parameterization based on heavy quark limit and some physical consideration :

20

2

2/10 )( qaea

qf

s

s

s

mc

pmcpu

2

2)(1

1'

)2

exp()(

)2(4)(

2

4/3

2/3

q

qR

2/32 rwith 2r is mean-square-radius of meson

In heavy quark limit we have

0)(, mEca p

In this limit the above BS wave function will give the same result with the literatures

4: Application to and our result 2c

With the formalism, we analysis the following situations and give their comparison.

1: Heavy quark limit

0)(, mEca p

)0,0,0,()0,0,0,2

(21 mM

pp

2: Momentum distribution with spinor form 1

mmpmEca p 22

,

3.Momentum distribution with spinor form 2

consca ,

5.Momentum and energy distribution with spinor form 2

4.Momentum and energy distribution with spinor form 1

consc

mmpmEc p 22

Result ： 2rb

Dependence of decay width on NR: non-relativistic static limit Re-1: for relativistic case with spinor assuming Re-2: for relativistic case with spinor assuming

cmcba ,,,

cmE c 22cmqE

Conclusion for the relativistic effects of

1: relative momentum distribution : ------ give about -50% correction comparing with static NR approximation 2: various treatments for the bound quark spinor : ------ cause about 6% uncertainty 3: relative energy distribution ------ gives little correction of -2% level.

)2( c

5:Further discussion

We also extend the formalism directly to situation and get the following result:

ss

Result of correction: momentum distribution: 75% spinor form: 10% energy distribution: 25%

1:What will really happen in the situation of light quarks

2: The comparison of the BS wave function form in heavy quark limit and chiral limit

Finally I think the following two problems are interesting

THE END

THANKS

cc

Appendix:

1: discussion of the parameter

: 1.2Gev-1.6Gevcm

c : )2

( cc mM

mVE c

b : fmfm 4.03.0

a : 2// 2

cMbvb q

2: literatures

[1]W-Y Keung, I.J. Muzinich Phys.Rev.D27:1518,1983[2]Z.P. Li, F.E. Close, T Barnes Phys.Rev.D43:2161-2170,1991[3]E.S. Ackleh ,T Barnes Phys.Rev.D45:232-240,1992 [4]S.N.Gupta, J M. Johnson, W. W. Repko Phys.Rev.D54:2075-2080,1996[5]D. Ebert, R.N. Faustov, V.O. Galkin Mod.Phys.Lett.A18:601-608,2003

22220 4

1|| mMqqA

mE

qq

mE

qq

mEmE

qq

mE

mq

mE

mqB

1

||

2

||

)2)(1(

||||

2

||

1

|| 003

])2)(1(

||1[

2

2

mEmE

qMC

)2

,0,0,2

(1

MMk u

)cos||,sinsin||,cossin||,(),( 00 qqqqqqqu

22

2002

2

|21])(

[)()(| mEmEqm

AB

qm

C

qmA

qmALogqqfqRdqdq

M

3:decay width