relativistic effects in two photon decay of quarkonium
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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
mE
mEmE
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