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Interactions of - mesons with heavy quarks. Avakyan E. Z. , Avakyan S. L. GSTU, Gomel. MIXING SCHEMES (only u,d,s -quarks). 1. Mixing: Where The flavor structure of and :. MIXING SCHEMES only u,d,s -quarks. - PowerPoint PPT Presentation
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Interactions of - mesons with heavy quarks
Avakyan E. Z., Avakyan S. L.
GSTU, Gomel
Avakyan E. Z., Avakyan S. L.
GSTU, Gomel
,
MIXING SCHEMES (only u,d,s-quarks)
1. Mixing:
WhereThe flavor structure of and :
ssqq
s
qU
)(
cossin
sincos)(U
q s
ss
dduu
s
q
)(
2
1
MIXING SCHEMES only u,d,s-quarks
2. MIXING IN THE OCTET – SINGLET BASISThe pseudoscalar nonet can be displayed as:
So
36
2362
362
080
0080
080
KK
K
K
P
cossin
sincos
08
08
HEAVY QUARK ADMIXTURES TO LIGHT PSEUDOSCALAR MESONS
Problem: ClEO mesuarment of branching fraction unexpetedly large. [CLEO coll ab. B.H.Behrens et.al. Phis.Rev.Lett.80,3710(1998)]
Mixing :
KB
ccssqq
01)cos()sin(
coscossin
sinsincos
c
s
q
ycyc
yc
yc
c
WAYS TO DEFINE MIXING PARAMETERS
• The radiative decays of η – mesons[T.Feldman, P.Kroll and B. Stech, Phys.Rew. D58,114006 (1998)
• Radiative Ј/Ψ – decays• [K.Chao, Nucl.Phys B317,597(1989)
• Rations of weak decays constants
QUARK MODEL
• The hadronic interactions will be described in the Relativistic Constituent Quark Model (RCQM).
[M.A.Ivanov,M.P.Locher and V.E.Lyubovitskij, Few-Body Syst. 21,131(1996),
M.A.Ivanov, V.E.Lyubovitskij, Phys.Lett., B408,435(1997)]
The coupling of the meson M into its constituents and is given by Lagrangian
where are Dirac and Gell-Mann matrices.
1q 2q
)()(),,()( 212121int xqxqxxxdxdxxMgL MMMM
MM ,
The function characterizes the finite size of meson and is given by:
The coupling constant is determined by compositeness condition:
where - the derivative of the mass function of meson.
QUARK MODEL
M
))(()(),,( 221
21
221121 xxf
mm
xmxmxxxxM
Mg
0)(~
4
31
2
2
MMM
M mg
Z
)(~ 2pM
QUARK MODEL
MM k )( 2 MM k )( 2
pkm ˆˆ,2
km ˆ,1
Example of evaluation of one – loop diagram:
After calculating Tr and performing α-parameterization one finds:
where Next we use Cauchy integral representation for
QUARK MODEL
)( 22 kp
1
0))()((
)(2
22
2
4
2221
)(
)(
2)(~
pkD
pkkmmp
ki
d
i
kddp
)1()1()( 222
21 pmmD
We use α-parameterization once more and find
The contour integral can be done again by Cauchy’s theorem.On substitution of and shift , after rotation ,one receives:
QUARK MODEL
t
t
1
pt
tkt
K
11
1
ukkikk E 2240 ,
)
11
1())((
1)(
~ 221
0 0
22
1
0
2
t
tpu
tmmzuduu
t
dtdp Pp
t
tptDz
1)(
22
MESON-QUARK INTERACTION CONSTANTS
Pseudoscalar meson constant
THE MIXING:
)(~
1
4
32
2
MM
MM
m
gh
ccssqq
),(sinsin),(cos),(
2)(
22222
cPPycsppnpp cmFmFmF
h
),(cossin),(sin),(
2)(
222222
cPPycsppnpp cmFmFmF
h
),()(cos),()(sin),(
2)(
222222
cppycsppycnpp c
c mFmFmFh
DECAY
Decay amplitude:
Decay width:
Experimental values (PDG)
P
2121
2 )()()( qqqqgePA P
223
4)( PP gmPW
keVW
keVW
)19.027.4()(
,)04.046.0()(
exp
exp
p̂
2q̂
1q̂
)( 25 kP
DECAYP
THE MIXING:
DECAYP
ccssqq
),(sin9
5sin),(
9
1
cos),(9
5
2
1)(3
)(22
2
mcmFmsmF
mumFhg
PycP
P
),(cos9
5cos),(
9
1
sin),(9
5
2
1)(3
)(22
2
mcmFmmF
mmFhg
PycsP
uP
To define the experimental values
were used.
For mixing:For mixing:So we find the quark content of physical mesons
1exp1exp 341.0,259.0
GeVgGeVg
yc cos,
ssqq 3,39ccssqq 012,0cos,1,33 yc
RADIATIVE TRANSITIONS BETWEEN LIGHT PSEUDOSCALAR AND VECTOR MESONS
Decay amplitude:
Decays width:
VVVP pqpqegPVA )()()(
VVPV pqpqegVPA )()()(
223)( PVV gmVPW 223
3
1)( VPP gmPVW
• NUMERICAL RESULTS FOR RADIATIVE DECAYS
RADIATIVE TRANSITIONS BETWEEN LIGHT PSEUDOSCALAR AND VECTOR MESONS
EXPERIMENT[PDG] scheme
1,51
0,51
0,72
V
)GeV(g 1V
g
g
g
28,0
25,047,1
04,053,0
02,069,0
ccssqq
012,0cos,1,33 yc
• NUMERICAL RESULTS FOR RADIATIVE DECAYS
RADIATIVE TRANSITIONS BETWEEN LIGHT PSEUDOSCALAR AND VECTOR MESONS
VEXPERIMENT
[PDG] scheme
1,24
0,37
0,79
)( 1
GeV
g Vccssqq
012,0cos,1,33 yc
g
g
g
06,031,1
03,045,0
21,0
28,000,1