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The Heavy baryon masses and Spin-Isospin Dependence. Z. Ghalenovi, A.A. Rajabi, A. Tavakolinezhad. Shahrood University of Technology, Shahrood , Iran. Outline 1- The Models 2- The Heavy Baryons Masses 3- The Heavy Baryons Magnetic Moments. The First Model. - PowerPoint PPT Presentation
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The Heavy baryon masses and Spin-Isospin Dependence
Z. Ghalenovi ,A.A. Rajabi ,
A. Tavakolinezhad
Shahrood University of Technology, Shahrood, Iran
Outline
1- The Models
2- The Heavy Baryons Masses
3- The Heavy Baryons Magnetic Moments
The First Model
4
To describe the baryon containing three quarks , we use the Jacobi coordinates and as
(1)
Such that
(2)
the hyperradius and the hyperangle , defined by
(3)
),2(6
1),(
2
132121 rrrrr
)(2
)(3;
2
321
213
21
21
mmm
mmmm
mm
mmm
x
,22
x )(tan 1
Therefore
)4()(2
),(22
222
xVm
pV
m
p
m
pH
The remaining hyperradial part of the wave function is determined by
)5(
)()(2)()4(5
22
2
xxVEmxxdx
d
xdx
d
Where
) 6 (
and
)7(
)2(
mm
mmm
2
ll 2
5
Then the Schrodinger equation will be
)9(0)()]
8
15)([2
)(22
2
xx
xVEdx
xd
)()( 2
5
xxx
We use the transformation
)8(
Where
)10(xcbxaxxV /)( 2
6
We use a new variable , then the equation (9) becomes
)11(
xx
0)()]8
15)([2)(
2
xx
xVEx
We introduce a simple variational ansatz for
)12(
)(x
222/5322)( xpexpx
7
And the energy
)13(
)(min0 pEEp
Where
)14(pcpb
appHpE .2
.4
3..
2.
16
15.
2
33)( 122
8
The Second Model
)()(2)()4(5
22
2
xxVEmxxdx
d
xdx
d
By using the hyperspherical coordinates , The hypercentral Schrodinger equation will be
) 15(
)()( 2
5
xxx
)()(]4
)52)(32([)(
21
12
1 xxxx
cxbxax
xcbxaxxV /)( 2
Where)16(
10
reduces Eq. (15) to the form
)18(
The transformation
)17 (
mccmbbmaamE 2,2,2,2 111
let us assume for the wave function
)20 ()](exp[()()( xgxfx
Now for the function f(x) and g(x) we make use of the ansatz
)21 (
1
0),(
01
)(
ii ifax
if
xf
xxxxg ln2
1)( 2
and
11
Where
)19(
and
)22(
,1a ,2 1
1
a
b,21 c
2)21(, 2
5
2/1)2
(m
aThe frequence of oscillating is , Therefore
)23(
,2 mma .)52(
2
mcb
.)2
5(
mb
c
12
.)52(
2
2)62(
2
2
0
mcE
))52(
2
2exp( 2
002
5
00 xmc
xm
xNxN
The energy eigenvalue and the normalized eigenfunction of the system are obtained as
)24(
and
)25(
13
The Heavy Baryons Masses
The ground state heavy baryon masses are obtained by
)26( int0321 HEmmmM baryon
)()()()(int xHxHxHxH SIIS
where
and
)27(
).)(/exp()1
( 21223 ssxAH S
s
SS
).)(/exp()1
( 21223 ttxAH I
I
II
).)(.)(/exp()1
( 2121223 ttssxAH SI
SI
SISI
15
The Heavy Baryons Magnetic Moments
The effective quark mass is defined as
)28()1( int
i i
ieffi
m
HEmm
Such that the mass of the baryon is
)29(. i
effiB mM
The magnetic moment of baryon is as
)30( sfiii
sfB ||
where
effi
ii m
e
2
17
um
dm
sm
cm
2.37
quark massvalue
320MeV
325MeV
440MeV
1310MeV
4690MeV
S fm
sA
2fm
SI fm
SIA2fm
I fm
IA
2fm
3fm
2fm
54.7
2.15
90
1.12
85
0.23
parametersvalue
0.52
Table 1 The fitted values of the parameters of potentials,obtained with a global fit to the experimental masses.
bm
0.61
a
b
c
18
The first model
um
dm
sm
cm
2.87
quark massvalue
350MeV
350MeV
400MeV
1500MeV
4980MeV
S fm
sA
2fm
SI fm
SIA2fm
I fm
IA
2fm
1fm
67.4
2.31
-106.2
3.45
51.7
o.832
parametersvalue
0.62
Table2 The fitted values of the parameters of potentials,fited to the experimental masses.
bm
c
19
The second model
c
c
0c
0c
c
c
Table 3 Single charm baryon masses (masses are in GeV)
baryonOur model (1)Our model (2)Exp[9][11][12]
2.4502.4572.4512.4482.4602.461
2.5372.5862.5182.5052.5252.526
2.4552.461--2.4772.471
2.5422.591--2.5442.536
2.4682.4662.4672.4962.5302.485
2.5562.5962.6462.6332.6032.672
2.4732.4712.471-2.5482.494
2.5612.6012.646-2.6232.680
2.5882.4762.6992.7012.6202.696
2.6762.606-2.7592.7042.757
0c
0 c
0c
0c20
b
b
b
b
Table 5 Single beauty baryon masses (masses are in GeV)
baryonOur model (1)Our model (2)Exp[9][10][12]
5.8035.8075.807--5.801
5.9125.9365.829--5.823
5.8135.8185.815--5.821
5.9225.9465.836--5.834
5.8485.8215.7925.8795.9705.872
5.9365.956-5.9675.9805.936
5.968--6.0376.0816.005
6.0565.961-6.0906.1026.065
0b
0b
b
b
21
2/3pJ
N
c
0c
c
0c
0c
Table 6 Magnetic moments of single charm baryons with
in terms of nuclear magneton
.
baryonOur model (1)[4][8][11][12]
1.0971.1701.1301.1581.252
-1.115-1.230-1.146-1.101-0.848
1.3901.4301.2641.2421.513
-0.960-1.000-0.986-1.002-0.688
-0.770-0.770-0.833-0.904-0.865
22
2/3pJ
NTable 7 Magnetic moments of single beauty baryons with
in terms of nuclear magneton
b
b
0b
b
baryonOur model (1)[8][12]
3.4603.0823.234
-1.790-1.634-1.655
-1.423-1.477-1.095
-1.368-1.292-1.199
23
24
REFS:
1. H. HASSANABADI, A.A.RAJABI AND S. ZARRINKAMAR: SPECTRUM OF BARYONS AND SPIN-ISOSPIN DEPENDENCE. 2. S. BOFFI, C. CIOFI DEGLE ATTI AND M. M. GIANNINI (WORLD SCIENTIFIC, 19980, P. 363). 3. M. NARODETSKII, M. A. TRUSOV: THE HEAVY BARYONS IN THE NONPERTURBATIVE STRING APPROACH. ARXIV; HEP-PH/0104019V2 (2001) 72, 4, 679-688 (2009) 4. R. DHIR, R.C. VERMA: MAGNETIC MOMENTS OF () HEAVY BARYONS USING EFFECTIVE MASS AND SCREENED CHARGED SCHEME. EUR. PHYS. J. A 42, 243-249 (2009) 5. B. PATEL, A. MAJETHIYA AND P.C. VINODKUMAR: MASSES AND MAGNETIC MOMENTS OF TRIPLE HEAVY FLAVOUR 6. M.M GIANNINI, E.SANTOPINTO A. VASSALO: AN OVERVIEW OF THE HYPERCENTRAL CONSTITUENT QUARK MODEL. PROG. PART. NUCL. PHYS. 50, 263-272 (2003)7. D. EBERT, R.N. FAUSTOV, V.O. GALKIN, PHYS. REV. D 72 034026 (2005)8. A. MAJETHIYA, B. PATEL, P.C. VINODKUMAR: SINGLE HEAVY FLAVOUR BARYONS USING COULOMB PLUS A POWER LAW INTERQUARK POTENTIAL. EUR. PHYS. J. A 38, 307-315 (2008)9. H. GARCILAZO, J. VIJANDE, A. VALCARCE, J. PHYS. G: NUCL. PART. PHYS. 34, 961-76 (2007) 10. W. ROBERTS, MUSLEMA PERVIN: HEAVY BARYONS IN A QUARK MODEL. ARXIV:0711.2492V2 [NUCL-TH] (2008)11. B. PATEL, A. KUMAR RAI, P. C. VINODKUMAR: MASSES AND MAGNETIC MOMENTS OF CHARMED BARYONS USING HYPER CENTRAL MODEL. ARXIV: 0803.0221V1 [HEP-PH] (2008) BARYONS IN HYPERCENTRAL MODEL. PRAMANA J. PHYS. 12. B. PATEL, A. KUMAR RAI, P.C. VINODKUMAR: MASSES AND MAGNETIC MOMENTS OF HEAVY BARYONS IN THE HYPER CENTRAL MODEL. J. PHYS. G: NUCL. PART. PHYS. 35, 065001 (2008).13. K. NAKAMURA ET AL (PDG), JPG 37, 075021 (2010)
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