Revista Brasileira de Física, Vol. 16, n? 4, 1986
Baryon Masses and Hyperfine Splittings
MARIAALINE B. DO VALE,ANTONIOS. DECASTRO and HELIO F. DE CARVALHO
Instituto de Física, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, Rio de Janeiro,
21944, RJ, 8 r d l
Recebido em 18 de julho de 1986
Assuming t h a t t h e t h r e e q u a r k i n t e r a c t i o n can be d e s c r i b e d i n
terms o f p a i r i n t e r a c t i o n s , and t h a t t h e quark- quark i n í e r a c t i o n i s r e l a t e d t o t h e q u a r k - a n t i q u a r k i n t e r a c t i o n (V(qq)=1/2 V(qq) ) ,
we
have
c a l c u l a t e d t h e baryon masses as t h r e e - q u a r k bound s t a t e s . We have a l s o
c a l c u l a t e d t h e r e l a t i v i s t i c c o r r e c t i o n s coming f r o m t h e s p i n - s p i n i n t e r a c t i o n . F i n a l l y , o u r r e s u l t s have been compared t o t h e a v a i l a b l e experimental data.
Abstract
1. INTRODUCTION
Using a phenomenological p o t e n t i a l we have
calculated
baryon masses and t h e h y p e r f i n e s p l i t t i n g s . T h i s model, w h i c h
t o reproduce t h e a s p e c t s o f confinement and a s y m p t o t i c
s t r o n g i n t e r a c t i o n t h e o r y (QCD) has been used
a u t h o r s i n hadron spectroscopy.
able
freedom o f
I n t h e p r e s e n t work, we
a ~ . " ~and z i c k e n d r a h t 3
is
w i t h success by
t h e baryon spectrum. For t h i s we have used a c o o r d i n a t e
made by FIÜgge e t
tlie
have
the
many
obtained
transformation
t h a t p e r m i t s us t o wr.ite
Schrodinger e q u a t i o n i n one v a r i a b l e o n l y . T h i s method, which
used f o r t h e ccc and bbb systems by d ' 0 l i v e i r a
e t aZ.'
the
has been
has now been ap-
p l i e d t o a1 1 t h e baryons formed by quark f l a v o r s u, d, s, c and b u s i n g
t h e parameters c a l c u l a t e d by C a s t r o e t
~ 2 . ~For
.
t h e ground s t a t e s ,
we
have c a l c u l a t e d t h e h y p e r f i n e s p l i t t i n g s coming f rom t h e s p i n - s p i n , t e m
o f t h e B r e i t - F e r m i Hami 1 t o n i a n .
2. NONRELATIVISTIC THREE-BODY PROBLEM
For t h e three-body systeni ( f i g . 1 ) t h e Schrodinger e q u a t i o n i n
t h e center-of-mass
system can be w r i t t e n as
Work p a r t i a 1 l y supported by CNPq
Agencies).
and
FINE? ( B r a z i 1 i a n
Government
Revista Brasileira de Física, Vol. 16. no 4, 1986
where
+
A, and A2 r e f e r t o t h e c o o r d i n a t e s xi and
+
+
x2
-+
where r l and r, a r e t h e p o s i t i o n v e c t o r s , and v i s t h e reduced mass.
Fol l o w i n g FlÜgge e t a ~ . " and
~ z i c k e n d r a h t 3 , we
can
perform
-+
a sequence o f coord i n q t e t r a n s f o r m a t i o n s and go f r o m t h e c o o r d i n a t e s s i
+
a, 6, C$, 0 , $.
and X, t o t h e c o o r d i n a t e y ,
I n t h i s system, t h e d i s t a n c e s a r e g i v e r ~by
where
6, = a r c cos
/
L
m l (m3-m2)-m, (m,+m j
2
(ml+m,)(m2+m3)
Revista Brasileira de Física, Vol. 16. n? 4, 1986
The ~ c h r o d i n ~ ee rq u a t i o n can be w r i t t e n as
H\Y = EY
+L
With
we have
T h i s l a s t e q u a t i o n can be s o l v e d n u m e r i c a l l y , g i v i n g t h e t h r e e
-quark bound s t a t e energy. The b o u n d - s t a t e rnass i s g i v e n by
I n o r d e r t o o b t a i n e q u a t i o n (15) we must know
we have t o p e r f o r m an average i n a,
6
B
of the p o t e n t i a l
v ( Y ) . For t h i s
v(Y,~,B).
l o w i n g Gerck and d l O l i v e i r a , we have t h a t , g i v e n a p o t e n t i a l
FOI-
Revista Brasileira de Física, Vol. 16, no 4, 1986
we o b t a i n
V(u)
=
BY
2V
where B can be w r i t t e n as
with
n a c o s ad&O
(20)
The t h r e e - q u a r k bound s t a t e e q u a t i o n s can be o b t a i n e d
as t h e
3. THREE-QUARK BOUND STATE EQUATION
q u a r k - a n t i q u a r k e q u a t i o n s . The g r a p h i c a l r e p r e s e n t a t i o n o f
the
three-
-quark B e t h e - S a l p e t e r e q u a t i o n i s shown i n f i g . 2 . Assurning two-body int e r a c t i o n s o n l y (G3=0),
(Flamm and Schroberl
we o b t a i n t h e t h r e e - q u a r k Bethe- Salpeter equation
)7
F i g . 2 - Graphical r e p r e s e n t a t i o n o f t h e Bethe-Salpeter equation.
r e p r e s e n t s t h e B - S amp l i t u d e ; G2 i s t h e t w o - p a r t i c l e a n d J t n e t h r e e
- p a r t i c l e i r r e d u c i b l e B - S kernel .
Revista Brasileira de Física, Vol. 16, n? 4, 1986
where
Hi,
H2
and
-mass system, A*+
H, a r e t h e quark H a m i l t o n i a n s i n t h e baryon c e n t e r - o f
+
+
i s t h e p o s i t i v e e n e r g y - p r o j e c t i o n o p e r a t o r , r and s
+
a r e the r e l a t i v e coordinates, x
+
jk
= x
i
-
+
xk, and U ( x . ) i s t h e g e n e r a l ~k
i z e d p o t e n t i a l t h a t c o n t a i n s a spin- dependent p a r t
dependent p a r t V S I . O m i t t i n g A*
Vs and
i n fhe l a s t equation
we
a
spin-in-
obtain
the
generalized B r e i t equation
+ -+
-f -f
H Y (r,s) = E Y ( r , s )
(24)
B
where
p i a r e t h e momenta o f t h e quarks.
Perforrning a g e n e r a l i z e d Foldy-Wouthuysen t r a n s f o r m a t i o n s , we
o b t a i n t h e B r e i t - F e r m i Hami l t o n i a n f o r t h r e e quarks (?i=e=l)
where
m . a r e t h e c o n s t i t u e n t quark masses and
vSI(r)
=
4as
-3r
I n o b t a i n i n g t h e B r e i t - F e r m i Harniltonian we have assumed t h a t
t h e QED equat ions can be a p p l i e d t o QCD. For t h i s we have t o r e p l a c e a ,
t h e QED c o u p l i n g c o n s t a n t , by
as,
t h e QED e f f e c t i v e c o u p l i n g c o n s t a n t .
Besides, we have t o t a k e t h e q u a r k c o l o r s i n t o account.The l o w e s t o r d e r
diagram f o r t h e quark- quark i n t e r a c t i o n due t o g l u o n exchange
i s shown
i n fig.3.
b2
gS'yv
a'
a2
-
t
\&r
Fig.3
Lowest o r d e r d i a grarn f o r t h e quark- q u a r k
i n t e r a c t i o n due t o glucin
exchange.
Revista Brasileira de Física, Vol. 16. no 4, 1986
Using t h e Feynman r u l e s , t h e i n t e r a c t i o n p o t e n t i a l i s
The sum o v e r t h e A m a t r i c e s can be r e p l a c e d by t h e s c a l a r p r o d u c t o f t h e
F s p i n s o f t h e two quarks. Hence, we o b t a i n
Since we can c o n s t r u c t t h e q u a d r a t i c Cas imi r o p e r a t o r o f S U ( ~ )
and t h e s p i n o f t h e two- quark system
5
i5
t h e suin o f t h e F s p i n s o f t h e
two quarks
and assuming t h a t i n d i v i d u a l quarks belongs t o t h e t r i p l e t
t a t i o n and have C'
represen-
= 4/3, we o b t a i n
'
Consequentl y, t h e bound- state energy i s p r o p o r t i c n a l t o C
For t h e q;
system, we have
and f o r t h e qq system
Looking a t t a b l e 1, we see t h a t
We conclude t h a t t h e q; c o l o r s i n g l e t i s lower i n mass than
the
other
Revista Brasileira de Física, Vol. 16, n'? 4, 1986
Table 1 - Eigenvalues o f t h e q u a d r a t i c Casimir o p e r a t o r f o r t h e l o w e s t -dimensional r e p r e s e n t a t i o n s o f SU(3). ( r n q i x e d ; s=symmetric).
l
Dimension
of
representat ion
Quark
indices
p
0 1
1 2
O 0
1 0
quark s t a t e s . T h i s way quark confinement can be made p l a u s i b l e .
argument can a l s o be a p p l i e d t o
This
n- quark s t a t e s , assuming o n l y two-bcbdy
interactions.
For t h r e e - q u a r k s t a t e s , we have
where
and f o r t h e qqq system
3 ~ 3 @ 3 = 1 @ 8 @ 8 @ 1 0
. we have
As can be seen f rorn t a b l e 1
E(qqq11 < ~ ( q q q ) s< ~ ( q q q ) 1 o
The c o l o r s i n g l e t s t a t e i s a g a i n t h e most f a v o r e d .
Revista Brasileira de Física, Vol. 16,,n? 4, 1986
Using eq. ( 3 3 ) , we o b t a i n t h a t
and
t hen
A l t h o u g h t h i s r e l a t i o n has been o b t a i n e d f o r t h e Coulomb p o t e n t i a l ,
it
i s b e l i e v e d t o be v a l i d t o a l i p o t e n t i a l s .
4. PHENOMENOLOGICAL POTENTIAL
The p o t e n t i a 1
has been appl i e d t o r n e s ~ n s and
~ ' ~ with
states.
as
= O t o the
bbb and ccc bound
I n t h i s p o t e n t i a l t h e f i r s t term i s t h e c o n f i n i n g p o t e n t i a 1 , t h e
second term i s t h e p o t e n t i a l due t o one- gluon exchange,andtheparameter
C, which depends on each q u a r k - a n t i q u a r k p a i r , r e f l e c t s t h e
fact
that
the
par-
we can o n l y c a l c u l a t e energy d i f f e r e n c e s .
C a l c u l a t i n g t h e average p o t e n t i a l
V($) and u s i n q
ameters f o r mesons5 w i t h t h e r e l a t i o n (43) we o b t a i n from e q u a t i o n (15)
t h e baryon masses shown i n t a b l e s 2,
3 , 4 and 5.
5. HYPERFINE SPLITTINGS
W i t h t h e H a m i l t o n i a n eq.(26)
we o b t a i n ,
using
perturbation
t n e o r y , t h e h y p e r f i n e s p l i t t i n g s between t h e J = 3 / 2 and
= 1/2 s t a t e s
i n t h e 1s leve1 o f t h e baryon spectrum.
The baryon wave f u n c t i o n s have been c o n s t r u c t e d
p l e t e l y antisymmetric.
Then, f o r t h e J = 3/2 baryon, we have
to
be
com-
Revista Brasileira de Física, Vol. 16, no 4, 1986
Table 2
uud
UUS
USS
SSS
UUC
USC
SSC
UCC
SCC
CCC
uub
usb
ssb
ucb
scb
ecb
ubb
sbb
cbb
bbb
-
Spectrum o f baryon masses ( r a d i a l e x c i t a t i o n s ) . Masses i n GeV.
Revista Brasileira de Física, Vol. 16, n? 4, 1986
Table 3
-
Spectrurn o f baryon masses ( o r b i t a l e x c i t a t i o n s ) . Masses i n GeV.
4, 42, 4,
uud
UKS
USS
SSS
UKC
use
S SC
UCC
SCC
CCC
uub
usb
ssb
ucb
scb
ccb
ubb
sbb
cbb
bbb
Revista Brasileira de Física, Vol. 16,
Table 4
uud
UUS
USS
SSS
UUC
USC
SSC
UCC
SCC
CCC
uub
usb
ssb
ucb
scb
ccb
ubb
sbb
cbb
bbb
-
no
4, 1986
Spectrurn o f baryon masses ( r a d i a l e x c i t a t i o n s ) . M a s s e s i n GeV.
Revista Brasileira de Física, Vol. 16, n'? 4, 1986
Table 5
uud
UUS
USS
SSS
UUC
use
ssc
UCC
SCC
CCC
uub
usb
ssb
ucb
scb
ccb
Ubb
sbb
cbb
bbb
-
Spectrum o f baryon masses ( o r b i t a l e x c i t a t ions). Masses i n GeV.
Revista Brasileira de Física, Vol. 16, no 4. 1986
'4 =
TxSdg
where
T
i s the c o l o r wave f u n c t i o n
xS
i s the symnetric s p i n wave f u n c t i o n
-$s i s t h e symmetric f l a v o r wave f u n c t i o n
$
i s the ground s t a t e space wave f u n c t i o n
with
For the baryons w i t h J = 1/2
where p and h a r e the mixed representations
x ~1 ( T1 ~) =,
i
I
1/&
x,(~,-~)
= I/&
Applying HSS
(+$-i+)+
(f+-t+)+
t o t h e wave f u n c t i o n s (451,
t a i n e d t h a t t h e s p i n - s p i n c o r r e c t i o n i s g i v e n by
(461, we
have
ob-
Revista Brasileira de Física, Vol. 16, n? 4, 1986
For t h e p o t e n t i a l
where
(44) w i t h as
= O , we have
B 1 , B 2 and B 3 r e p r e s e n t t h e average i n a, B and a r e shown i n t a b l e
6.
7
Using eq.(48) we have o b t a i n e d t h e r e s u l t s shown i n t a b l e s
and 8.
Assuming t h a t t h e p o t e n t i a 1 i s a m i x t u r e o f s c a l a r and vector
5
p o t e n t i a l s , as i n t h e case o f t h e mesons , o f t h e form
vs
=
fV
conf
.( r )
we have o b t a i n e d t h e r e s u l t s o f t h e t a b l e s
9 and 10. The s c a l a r term does
n o t c o n t r i b u t e t o t h e s p i n - s p i n c o r r e c t i o n . Then t h e
f o r many v a l u e s o f
f
results
obtained
can be compared t o t h e e x p e r i m e n t a l d a t a
to
give
t h e c o r r e c t L o r e n t z n a t u r e o f t h e conf i n i n g p o t e n t i a 1 .
Tables 11 and 12 show o u r r e s u l t s and those from l3jorkenl0,and
Samuel and b r i a r t y " .
Those a u t h o r s use d i f f e r e r i t procedures t o c a l c u -
l a t e t h e baryon masses.
We have c o n s i d e r e d
as = O
i n t h e r e s u l t s shown i n t a b l e s 7,8,
9 and 10. In t h i s case we have c a l c u l a t e d t h e
hx/perfine s p l i t t i h g s ,
t a k i n g i n t o account t h e c o n f i n i n g p o t e n t i a l . T h i s i s due
t h a t the hyperfine s p l i t t i n g s involve a
to
the
fact
6- function. This i s equivalent
t o h a v i n g t h e wave f u n c t i o n a t t h e o r i g i n . T h i s c a l c u l a t i o n t u r n s o u t t o
be d i f f i c u l t i f we use t h e Z i c k e n d r a h t v a r i a b l e s . Besides,
developed by Z i c k e n d r a h t showed t o be i n e x c e l l e n t agreement
the
method
with
the
Revista Brasileira de Física, Vol. 16, n? 4, 1986
T a b l e 6 - B i , B2 and B 3 values which represent a
and 8 averaged i n t h e L a p l a c i a n p o t e n t i a l .
4, qb
uud
bbtc
CCS
CCU
SSC
SSU
uub
UUC
UUS
CSU
bbc
bbs
ccb
ssb
bcs
bcu
bsu
qc
Revista Brasileira de Física, Vol. 16, n? 4. 1986
T a b l e 7 - Masses o f t h e g r o u n d s t a t e s o f b a r y o n s u s i n g s p i n - s p i n correct i o n s . Masses i n GeV.
:ls)
(J1
Theo r .
1.344
1 .344
1.344
1.344
1.447
1.447
1.447
1.583
1.583
1.701
2.572
2.572
2.572
2.683
2.683
2.812
3.640
3.640
Syrnbol
1
1
rheor.
1
I
Experimental
"1'
I
I
Revista Brasileira de Física. Vol. 16, n'? 4, 1986
-
Table 8
Masses o f t h e ground s t a t e s o f baryons using spin- spin correct i o n s . Masses i n GeV.
Symbol
SCC
R*+
CCC
R++
uub
'%+
udb
c*O
D
ddb
1;-
CC
CCC
udb
usb
dsb
=*o
-b
-*-b
ssb
ucb
=*+
-cb
dcb
=*o
scb
",*i
ccb
ubb
dbb
-cb
%b
=*o
-bb
,*-
-bb
sbb
%E-
cbb
*2b
bbb
Slbbb
Theo r .
Experimenta 1
Symbol
Theor.
Experimental
Revista Brasileira de Física, Vol. 16, n? 4, 1986
Table 9 - Masses o f baryons w i t h ?=I/z+
usinq the h y p e r f i n e spl i t t i n g s
o b t a i n e d f o r s e v e r a l v a l u e s o f J (K=0.855 G ~ v ~ ' ~Masses
) .
i n GeV.
+
Table 10 - Masses o f baryons w i t h 2 = 1 / 2 u s i n g t i i e h y p e r f i n e s p l i t t i n g s
o b t a i n e d f o r s e v e r a l values o f
(K=0.855 G ~ v ~ / " Masses
) .
i n GeV.
Revista Brasileira de Física. Vol. 16, no 4, 1986
The colurrins
Table 1 1 - Masses o f baryons w i t h ?=3/2+.
Masses i n MeV
correspond t o : ( 1 ) t h e p o t e n t i a l parameters: K=0.855 G~v"', as= O; ( 2 )
t h e p o t e n t ia1 parameters: K=0.767 G ~ v ~ / as=O.
',
187; ( 3 ) t h e v a l u e s f r o m
~ a m u e l and M o r i a r t y ; ( 4 ) t h e v a l u e s from ~ j õ r k e n
UUU
UUS
USS
sss
UUC
USC
SSC
UCC
SCC
CCC
uub
usb
ssb
usb
scb
ccb
ubb
sbb
cbb
bbb
Revista Brasileira de Física, Vol. 16, no 4, 1986
-
T a b l e 12
Masses o f baryons w i t h F = 1 / 2 + . Masses i n Me".
The columns
correspond t o : ( I ) t h e p o t e n t i a l parameters: K=0.855 G ~ v " ' and xs = O
f o r masses and s p i n - s p i n c o r r e c t i o n ; ( 2 ) %=0.767 G ~ v " ' and as = 0.187
f o r masses. K= 0.767 G e v 3 I 2 arid a, = O f o r s p i n - s p i n c o r r e c t i o n s ;
( 3 )
v a l u e s f r o m Samuel and M o r i a r t y ; ( 4 ) v a l u e s from B j o r k e n .
uud
UUS
uds
USS
UUC
udc
USC
SSC
UCC
SCC
energy-eigenva1ues;but
tions.
t h e same c o u l d n o t be s a i d about t h e wave f u n c -
I n the c a l c u l a t i o n s o f the hyperfine s p l i t t i n g s
using
c o n f i n i n g p o t e n t i a l , t h e accuracy o f t h e c a l c u l a t i o n s i s
t h a t o f energy l e v e l s .
only
a
equivalent t o
I n t h i s way we avoided, t o c o n s i d e r t h e v a l u e o f
t h e wave f u n c t i o n a t t h e o r i g i n . But i n t h e a n a l y s i s o f
the
s p l i t t i n g s we have c o n s i d e r e d O,<f<I. I n t h e case where f=l
hyperfine
theconfining
p o t e n t i a l does n o t c o n t r i b u t e t o t h e h y p e r f i n e s p l i t t i n g s . S u c h c o n f i n i n g
p o t e n t i a l would be a L o r e n t z s c a l a r .
I n t h a t case t h e c o n t r i b u t i o n s
t h e h y p e r f i n e s p l i t t i n g s can o n l y come f r o m t h e (.oulombic p a r t
to
of
the
p o t e n t i a l . We a r e s t u d y i n g t h i s q u e s t i o n and one p o s s i b l e s o l u t i o n
can
be found u s i n g t h e method proposed by H i l l e r e t c ~ í . ' ~ . ~ hmethod
is
shows
t h a t t h e v a l u e o f t h e wave f u n c t i o n a t t h e o r i g i n ,
for
two
b o d i e s , can be r e l a t e d t o t h e d e r i v a t i v e o f t h e p o t e n t i a l . We
p r e s e n t t h i s c a l c u l a t i o n as soon as p o s s i b l e .
or
hope
more
to
Rwista Brasileira de Física, Vol. 16, n? 4, 1986
6. CONCLUSION
When o u r r e s u l t s a r e compared w i t h those o b t a i n e d by Guimarães
K harmonics method, f o r t h e square r o o t
e t aZ.13, u s i n g t h e
potentiali,
we n o t e a d i f f e r e n c e . T h i s i s n o t s u r p r i s i n g because f o r t h e
harmonic-
- o s c i l l a t o r p o t e n t i a l o u r r e s u l t s , which agree w i t h t h e a n a l y t i c values,
d o n ' t agree w i t h those a u t h o r s r e s u l t s .
J = 3/2 baryons t h e comparison o f o u r
For t h e
results
t h e e x p e r i m e n t a l d a t a show a l i t t l e d i s c r e p a n c y . From t a b l e 1 1
t h a t i n a g e n e r a l way o u r r e s u l t s agree w i t h those o b t a i n e d
.
authors"'
"
The d i s c r e p a n c y i s g r e a t e r f o r t h e
we
by
uuu systems
and
see
other
and
for
t h e h e a v i e r baryons.
In table
7 t h e comparison o f o u r r e s u l t s and t h e e x p e r i m e n t a l
d a t a shows t h a t t h e masses o f J
= 1/2 baryons a r e below t h e experimeni-
t a l ones. T h i s f a c t leads u s t o t h e c o n c l u s i o n t h a t t h e
w-
confining
t e n t i a l i s n o t a p u r e v e c t o r b u t a m i x t u r e o f v e c t o r and s c a l a r ,
which
can be c o n f i r m e d by t a b l e 12 f r o m t h e comparison w i t h o t h e r a u t h o r s .
I n t a b l e 9 t h e e x p e r i m e n t a l v a l u e s show t h a t
f
must be below
0.5. For mesons, comparison o f t h e c a l c u l a t e d and e x p e r i m e n t a l
showed t h a t f
must be between
culated f o r these
f values
0.5
and 0.6.
values
I f t h e baryon masses c a l -
a r e compared w i t h t h o s e o b t a i n e d
by
other
I n t h e a n a l y s i s o f t h e J = 1/2 baryon r e s u l t s we must
recall
a u t h o r s we f i n d a good agreement.
t h a t we have n o t c a l c u l a t e d t h e s p i n - s p i n c o r r e c t i o n f o r
the
complete
p o t e n t i a l . The i n c l u s i o n o f t h e Coulombic t e r m would a f f e c t o u r r e s u l t s
p r o b a b l y l e a d i n g t o a b e t t e r agreement w i t h t h e f v a l u e
mesons
obtained
for
.
We would l i k e t o thank A . B . d l O l i v e i r a ,
A.C.B.Antunes
f o r use-
f u l d i s c u s s i o n s , A.Vaidya and C.Sigaud f o r r e a d i n g t h e e n g l i s h
script.
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Resumo
Considerando que a i n t e r a ç ã o e n t r e t r ê s quarks pode s e r desc r i t a em termos das i n t e r a ~ õ e se n t r e pares e que a i n t e r a ç ã o q w r k - q u a r k
e s t á r e l a c i o n a d a 5 i n t e r a ç a o q u a r k - a n t i q u a r k ('J(qq) = 1/2 V ( @ ) ) , c a l culamos as massas dos b ã r i o n s como estados l i g a d o s de t r ê s quarks. Calculamos também as c o r r e ç õ e s r e l a t i v í s t i c a s p r o v e n i e n t e s das i n t e r a ç ã o
s p i n - s p i n . Comparamos nossos r e s u l t a d o s com o s dados e x p e r i m e n t a i s d i s pon i v e i S.