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Universal confining potential and symmetry breaking in baryons
Volume 79B, number 4,5
UNIVERSAL
PItYSICS LETTERS
CONFINING
POTENTIAL
AND
SYMMETRY
4 December 1978
BREAKING
IN BARYONS
T. KOBAYASHI and Y. T A K A I W A i h~stitute for Physics, University of Tsukuba, Ibaraki, Japan 300-3I Received 10 July 1978
It is pointed out that in a universal confining potential model quark mass differences induce symmetry breaking in baryons. We show why SU8 ® 03 symmetry can be a very good symmetry in low-lying baryons and also that high-mass baryon resonances split into quite different types of baryon trajectories. The breaking effects in masses and form factors ot octet (JP = l/2 +) and decuplet (3/2 +) baryons are evaluated.
It is easily seen that SU2N ® 0 3 (the generalization of SU 6 ® 0 3 to N-flavour s y m m e t r y ) symmetric wave functions of baryons are n o t the eigenfunctions of the Schr6dinger equation with a universal (quark-mass-independent) potential because of the mass differences between the quark masses. In this article, as a simple example, we will consider this problem for b a r y o n s which are constructed from two identical quarks (a) and one nonidentical quark (~) with different nrasses (m a 4: rn~). The h a m i l t o n i a n for this system has a s y m m e t r y under the exchange of the two identical quarks only. Then the eigenfunctions of the hamiltonian have either of the following forms + ~, ~(1) (O~loQ j33 ;rl) = -0Xs(bs( (oe1 oQ)/33 ;t/),
( 1)
~(2) (oq o~2133;n ) = ~Xa ~a( [eq oQ]33;n),
(2)
where ~, X and ~b, respectively, stand for the color singlet, the flavour-and-spin and the space wave functions and the suffixes s and a represent symmetric and anti-symlnetric properties under the exchange of the two quarks in the parentheses, respectively , 2 hr general, the energy eigenvalues for qb(1) and q5(2) are different from each other, which was pointed out by Isgur and Karl [ 1 ] for excited b a r y o n states. The SU2N ® 0 3 symmetry, however, appears to be badly b r o k e n in these wave t'unctions. In order to clarify this situation we specify the problem to the SU 6 X 0 3 case, of which the generalization to the N-flavour case is straightforward. We can write the X'S in eqs. (1) and (2) in terlns of SU 6 irreducible representations: 1
Xs
_ 3-1/2
[X56(8) + X70s(8)
+
X70s(10)],
1 = 3-1/2
Xa
[X20(8) + X70a(8)
+
XT0a(10)]'
(3a)
for total quark spin l / 2 , and 3 = 3-1/2 Xs [X56(10) + ~
×70s(8)1
(3b)
for total quark spin 3/2. The suffixes A and D in XA(D) denote the dimensions of the SU 6 and SU 3 representations, respectively. Such a large SU 6 breaking as shown in eqs. (3) c a n n o t be accepted for low-lying baryons. That i Postdoctoral Fellow of the Japan Society for the Promotion of Science. +1 The suffixes of ee and t3 have no physical meaning at this stage. That is, they can be written as ~0) (al~2ee3 ;n) = r~Xs¢,s ({cqc~3}~z;n) and q~0)(~tee2a3;n) = ~Xses((a2a3}t~ l ;n) for ,l,0)(eeia2~3;n) and also ~l,(2)(alB2a3;n) and q~(2)(Blee2a3; n) for eo(2) (ce1c~2B3 ;n). (See fig. 1 .) • . *2 The exphc~t forms ot- the x , s are written down as x 1s , a = eeee~!o 1.a~, ) x~3 = c~al3~(3 ) , where the spin wave functions ~ for sz = 1/2 are 1 defined by ~ ) = (2~t; - l'~t - ~"* t)/,,/-6, ~l) = (tl~" - *t tI/x/2and ~(3) = (tt~ + ' t , t + *t t)/x/5. 411
Volume 79B, number 4,5
PItYSICS LETTERS
4 December 1978
is, we know that the classification of the ground states (1/2 +, 3/2 +) in terms of the {56}-dimensional representation and also the lp-wave states (l/2 , 3/2-, 5/2 -) in terms of {70} is quite well established. In order to reproduce the SU2N ® 0 3 symmetry a symmetrization of the wave function Xga is needed. The symmetrized wave function, however, is no more an eigenfunction of the original hamiltonian, because Xsgas ({al/32 }a3) and/or Xa 4~a(Ill/32 ] a3), which must be included in the symmetrized wave function, are not eigenfunctions. On the other hand, we know that the flavour exchange interactions, which are called non-planar or Xtype interactions in dual quark models, induce transitions between the eigenstates and non-eigenstates. On the assmnption that transitions between states with different radial and/or angular momentum quantum numbers (n) by the X-type interactions (Hx) are negligible, i.e., [(n'lHxln)/(E n, - E n)l "~ 1 for n'4, n, the eigenstate wave functions for the total hamiltonian including H x are well approximated by the following wave functions ,s :
vP(i) = 3 - 1/2 [~(i)
(Ctl a2~3 ;n) + ~(i) (o~1/32ct3 ;n) + c[~(i) (/31&2ct3 ;n)],
(4)
where i = 1 or 2. It is also easily seen that in the limit mc~ = m# they reduce to the right SU2N ® 0 3 wave functions. (We will explicitly show this later on.) In the present case (m~ 4- rn~) q,(i) still includes the breaking of SU2N ® 0 3. In terms of SU 6 symmetry we can manifestly see this by rewriting eq. (4) as
+ ~ ~{XT0s(a)l~s(12) - ½@23) - ½~s(31)l + ½x/5 XT0a(8)I~s(23) - @31)1} + ½r/{X70s(10 ) [gas(12) - ½~s(23) - ½gas(31)] + ½N/~ XT0a(10) [gas(23) - ~bs(31)] },
(S)
where gas(i/) - ~bs((~ict/)s~k ;n ) (i :¢:j :¢: k 4= i). The function xp(2) is given by tire replacement of X56 with X20 and the suffixes s and a with a and s, respectively. It is clear that in the limit mc~ = m# the {56} for the ground states w h e r e ~s(12) = 4~s(23) = ~s(31) and the {70} for the lp states where q~s(a)(12) + gas(a)(23) + gas(a)(31) = 0 are derived. The mixing parameters of the {56}, {20} and (70} in eq. (5) are calculated by
Cs6=l fd3xd3ylgas(12)+(Js(23)+~s(31)12,
C20=l fd3xd3y[(~a(12)+qSa(23)+gaa(31)12, (6)
GO(A)
= l f d3xd3y
{{gas(a)(12)
½gas(a)(23) - ½gas(a)(31)[2 + ~lgas(a)(23)
-
~)s(a)(31)12}'
In order to estimate the symmetry breaking in the ground state baryons, we use the harmonic oscillator approximation for the universal confining potential, that is ~a, 3 p2 3
"t i H h o = . =i~=l 2m
+ -~k.= ' i~1 [ri -
r0[2+V0,
(7)
where k and V0 are, respectively, a universal spring constant and a universal constant, and ro, r 1 , r 2 and r 3 with the constraint Z3q (ri - r0) = 0 are defined in fig. I in the Y-shaped baryon. The mixing parameters (7) for the
ground states are obtained as
C56 =~(1
+2X),
C70(A ) =}(1 - X ) ,
(8)
where
,3 The X-type interaction hamiltonian is effectively described by a sum of all permutations of the quarks. Besides the above assumption in the text one can see that H,,¢ qb(i)(al o~2~/3;n) = ep(i) (al c~2¢/3 ;n) + q~(i)(cqt32a3 ;n) + qs(i) (t31ee2ce3 ;n). Therefore HXq~(i) is again written by ,I'(i) itself. "~ #4 A different choice of potential like (1/2)k' Zi>ilr i - rfr z" which corresponds to A-shaped baryons also gives the same results derived from eq. ( 7 ) b y the replacement o f k ' w'ith k/3.
412
Volume 79B, number 4,5
PHYSICS LETTERS
4 December 1978
13
r~,
Fig, 1, Configuration of the ",'-shaped baryon, where x = - r a andy = $I ( r c q + r 3 should be taken as interrrc~}variagles, rc~2) -
~,
1613(2x + 1)] 1/2
)3/2,
X = 3(2x + 1) + 1013(2x + 1)] 1/2 +9 with x = nl~/rn~. One has the following values for the choice o f m u = 336, m d = 336 + 5, m s = 540 and nl c = 1650 in MeV, where the values o f m s and m e are taken from De Rujula et al. [2], uud C56 = ~ 1 C70(A) -- 2.3 X 10 -6
uus
uuc
uuQ (mQ = o~)
0.9961 1.97 X 10 -3
0.9737 1.31 X 10 - 2
0.9459 2.70 X 10 -2.
(9)
It is surprising that the mixing of {70} in the ground states is less than 6% even in the limit of the infinite mass difference (x = 0) and that the SU2N synrmetry is a remarkably good sylmnetry in the ground states. It is clear that the above values are quite insensitive to variation of the quark masses. This situation is also realized by evaluating the difference of the energy expectation values between the eigenstate (~s((C~l c~2)s33)) and the non-eigenstate (0s((%32)sC~3)), which is estimated to be 8 E = (Hho)non.eigenstat e -- 32 w c~ {1 + [(2x + 1)/3] 1/2} - V0 = ~(1 - X ) { [ 3 / ( l + 2x)] 1/2 -- 1}coa,
(10)
where c% = ~ ~ 4 0 0 - 5 0 0 MeV is expected from the mass difference between the ground states and the lpstate (Os((alaZ)s33)) and the non-eigenstate (qSs((Cq32)sa3)), which is estimated to be uud 8E = l A X 10 -2
uus
uuc
uuQ (mQ = ~)
11.1
68.7
137 (MeV).
(11)
These values are again very small values in comparison with the quark mass differences. In particular the u - d mass difference effect is negligible and hereafter we treat u and d as identical particles. The values (11) except for the uud case, however, are not so sufficiently small that they cannot be neglected in evaluating the masses of the ground states. Now we evaluate the masses with the following hamiltonian:
2 . . H = H b o + ,>~/(
%
3 l r . --r!,
s~ sj ] +go83(ri- r j ) ~ i m / l ,
(12)
where the one gluon exchange potential -2as/3r and a phenomenological spin spin potential si.s / are treated perturbatively. One obtains
M = V0 + m(2/x + l/y) + ~ coo (xl/2 + [(2y + 1)/3] 1/2} 4 ( m c o o ~ 1/2 [ ([ 4X1/2 -SC~s~--~ ! x -1/4 1 + 2 3 ( 2 y + x ) ] l / 2 + x l / 2
)1/2]
[rncoo]3/2 1 ( 4xl/2 )3/2] +go~-] - ~ [ x5/2($1"$2) +xl/4y((s1 +S2)'S3) [3(2y + 1)] 1/2 +xl/2 ' where m = rn u, x = m/rnc,, y = m/m~, coo = x / ~
(13)
and
413
Volume 79B, number 4,5
PftYSICS LETTERS
4 December 1978
Table 1 Calculated baryon masses, using the following values for the parameters: m u = 336 MeV, m s = 614 MeV, m c = 1855 MeV, Vo = 740.6 MeV, wo = 354.6 MeV, % = 0.446 and go = 8.45. The masses o f A c and E c are taken from ref. [31.
Table 2 Tile estimated parameters of" the weak form factors, F(q 2 ) = fexp [ - ( K / 6 , / ~ ) q 2 1 , for B 1 -~ B2 £v decays. B 1
B2
Baryon
Mass (MeV)
Experimental value
N E A
938 940 1189-1197 1115.6 -+ 0.05 1315-1321 9 2260 -+ 10 1230 - 1234 1382- 1387 1532-1535 1672.2 + 0.4 2400-2500 (?) ?
p A G0 +
Ec Ac A Y~'(E*) "~* Iz E"* c .~
939 (input) 1184 1118.6 1330 2401 2260 (input) 1232 (input) 1379 1525 1672.2 (input) 2482 3629
n E.~)20
-
3711
9
~C
zC,o C
= !
f o r N , A a n d A c' for E, v E
4
~'
C
1
1.18 1
p
)2-
n
G-
A
X~-
.-0
Ac Ac E÷ C 12C
A E° A )2
Ac
n
)2~
p
1.28
1.57 1.57
C
A
((s 1 + s 2 ) ' s 3 ) = 0
and d e c u p l e t b a r y o n s ,
K
Ac
c
(S 1 "S 2) = --a3
f
0.944
0.910
0.992
1.07
0.991
1.11
0.991
1.53
0.971
0.805
f o r N , A, a n d A c,
=-1
for E, Z and E c '
_ 1
for d e c u p l e t b a r y o n s .
When we use m u = 3 3 6 MeV and the masses o f N, 2x, f2, and A c [3] as i n p u t , u n d e r the r e q u i r e m e n t % < I, we can well r e p r o d u c e the masses o f the g r o u n d states as s h o w n in table 1. The p a r a m e t e r m s = 6 1 4 MeV seems to be a little larger t h a n the usual one (m s ~ m u + ( 1 5 0 - 2 0 0 MeV)). Nevertheless, o u r value for m s is c o n s i s t e n t l y u n d e r s t o o d in the following c o n s i d e r a t i o n : The suppression f a c t o r x 1/2 + [(2y + 1)/31 1/2 o f the h a r m o n i c oscillator t e r m and the e n h a n c e m e n t f a c t o r x -1/4 o f the 1/r t e r m , b o t h o f w h i c h are neglected in the usual analysis, reduce the masses o f the b a r y o n s w i t h an s-quark. In o r d e r to c o m p e n s a t e those c o n t r i b u t i o n s m s m u s t be larger t h a n the usual values. The difference b e t w e e n o u r analysis and the usual one like t h a t o f ref. [21 appears in the mass differences b e t w e e n c h a r m e d b a r y o n s , w h i c h is r e p r e s e n t e d b y
m2\
27r ]
l-y
c
1/2
=141MeV(160MeV),
(14)
1 + [ 3 ( 2 y + I)1
where the value derived in ref. [2] is q u o t e d in the last p a r e n t h e s e s a n d the f a c t o r [4/{1 + [3(2y + 1)] 1/2)3/2 is the n e w l y derived f a c t o r in our m o d e l , w h i c h is ~ 1.5 for the c h a r m e d b a r y o n s , whereas it is ~ 1.16 for the strange b a r y o n s . Now, we will e s t i m a t e the s y m m e t r y b r e a k i n g effects o f the wave f u n c t i o n s (4) in evaluating w e a k f o r m factors. T h e overlap integral in the weak f o r m factor is p a r a m e t r i z e d in the form F ( q 2) =
fd3x d 3 y qS~q~i exp [iqri]
= f(x, y ) exp [--K(x, y)q2/(6x/%-~)].
The values o f f ( x , y ) and K(x, y ) for various processes are given in table 2. E x c e p t for the decays o f c h a r m e d 414
(15)
Volume 79B, number 4,5
PHYSICS LETTERS
4 December 1978
B \\ B ~1~,1=0 0 " 0 a~l,=L~a
(I)
%
[3
1
I,=L
1,~1
~
......
(2)
a -t,-I-O.
(I)
(L+I)
Fig. 2. Configurations of baryon excited states, where Ii and 12, respectively, stand for the angular momenta of the twoidentical-quark system and that of quark 13.
baryons the breaking effects seem to be very small +s. In conclusion we may say that the SU2N ® 0 3 symmetry is quite good as an approximate symmetry for lowlying baryons, whatever large mass difference between quarks exists. We briefly point out the effect of the breaking in high mass baryon states. The X-type interaction is known to decrease with energy in the Regge pole model and the dual topological model. If it is true, at very high mass, where aM(M) >> (n IHx In) , the eigenfunctions defined in eqs. (1) and (2) become almost real wave functions and the SU2N ® 0 3 symmetry is significantly broken. Then states with the same angular momentum L split into (L + 1) states shown in fig. 2. In the A and ~2 (also A c and Zc) trajectories the (L + 1)th configuration of fig. 2 has the lowest mass and the first has the highest mass, whereas the former has the highest and the latter the lowest in the -- (Ec) trajectories. It is expected that only the state described by the (L + l)th configuration in the A and Z (A c and V~c) trajectories can couple with NK systems, because the angular momentum of the identical quark system in the (L + 1)th configuration (I1 = 0) coincides with that of the ground state (N), while in other configurations the l l ' s are non-zero. Finally we want to mention that, if the situation discussed in this paper is not observed, the confining potential must have a certain quark mass dependence. In the case that the spring constant of the ith quark is proportional to the quark mass (k i -- rnik 0 with k 0 = const.), which was proposed by Hirata et al. [4] in the evaluation of meson masses, no energy differences between the configurations in fig. 2 appear. *s We will discuss the details in a forthcoming paper. References
[l I N. Isgur and G. Karl, Phys. Lett. 74B (1978) 353. [21 A. De Rujula, If. Georgi and S.L. Glashow, Phys. Rev. D12 (1975) 147. [3] E.G. Cazzoli et al., Phys. Rev. Lett. 34 (1975) 1125; B. Knapp et al., Phys. Rev. Lett. 37 (1976) 882. [41 K. Hirata, T. Kobayashi and N. Nakamaru, Phys. Rev. D, to be published.
415
UNIVERSAL
PItYSICS LETTERS
CONFINING
POTENTIAL
AND
SYMMETRY
4 December 1978
BREAKING
IN BARYONS
T. KOBAYASHI and Y. T A K A I W A i h~stitute for Physics, University of Tsukuba, Ibaraki, Japan 300-3I Received 10 July 1978
It is pointed out that in a universal confining potential model quark mass differences induce symmetry breaking in baryons. We show why SU8 ® 03 symmetry can be a very good symmetry in low-lying baryons and also that high-mass baryon resonances split into quite different types of baryon trajectories. The breaking effects in masses and form factors ot octet (JP = l/2 +) and decuplet (3/2 +) baryons are evaluated.
It is easily seen that SU2N ® 0 3 (the generalization of SU 6 ® 0 3 to N-flavour s y m m e t r y ) symmetric wave functions of baryons are n o t the eigenfunctions of the Schr6dinger equation with a universal (quark-mass-independent) potential because of the mass differences between the quark masses. In this article, as a simple example, we will consider this problem for b a r y o n s which are constructed from two identical quarks (a) and one nonidentical quark (~) with different nrasses (m a 4: rn~). The h a m i l t o n i a n for this system has a s y m m e t r y under the exchange of the two identical quarks only. Then the eigenfunctions of the hamiltonian have either of the following forms + ~, ~(1) (O~loQ j33 ;rl) = -0Xs(bs( (oe1 oQ)/33 ;t/),
( 1)
~(2) (oq o~2133;n ) = ~Xa ~a( [eq oQ]33;n),
(2)
where ~, X and ~b, respectively, stand for the color singlet, the flavour-and-spin and the space wave functions and the suffixes s and a represent symmetric and anti-symlnetric properties under the exchange of the two quarks in the parentheses, respectively , 2 hr general, the energy eigenvalues for qb(1) and q5(2) are different from each other, which was pointed out by Isgur and Karl [ 1 ] for excited b a r y o n states. The SU2N ® 0 3 symmetry, however, appears to be badly b r o k e n in these wave t'unctions. In order to clarify this situation we specify the problem to the SU 6 X 0 3 case, of which the generalization to the N-flavour case is straightforward. We can write the X'S in eqs. (1) and (2) in terlns of SU 6 irreducible representations: 1
Xs
_ 3-1/2
[X56(8) + X70s(8)
+
X70s(10)],
1 = 3-1/2
Xa
[X20(8) + X70a(8)
+
XT0a(10)]'
(3a)
for total quark spin l / 2 , and 3 = 3-1/2 Xs [X56(10) + ~
×70s(8)1
(3b)
for total quark spin 3/2. The suffixes A and D in XA(D) denote the dimensions of the SU 6 and SU 3 representations, respectively. Such a large SU 6 breaking as shown in eqs. (3) c a n n o t be accepted for low-lying baryons. That i Postdoctoral Fellow of the Japan Society for the Promotion of Science. +1 The suffixes of ee and t3 have no physical meaning at this stage. That is, they can be written as ~0) (al~2ee3 ;n) = r~Xs¢,s ({cqc~3}~z;n) and q~0)(~tee2a3;n) = ~Xses((a2a3}t~ l ;n) for ,l,0)(eeia2~3;n) and also ~l,(2)(alB2a3;n) and q~(2)(Blee2a3; n) for eo(2) (ce1c~2B3 ;n). (See fig. 1 .) • . *2 The exphc~t forms ot- the x , s are written down as x 1s , a = eeee~!o 1.a~, ) x~3 = c~al3~(3 ) , where the spin wave functions ~ for sz = 1/2 are 1 defined by ~ ) = (2~t; - l'~t - ~"* t)/,,/-6, ~l) = (tl~" - *t tI/x/2and ~(3) = (tt~ + ' t , t + *t t)/x/5. 411
Volume 79B, number 4,5
PItYSICS LETTERS
4 December 1978
is, we know that the classification of the ground states (1/2 +, 3/2 +) in terms of the {56}-dimensional representation and also the lp-wave states (l/2 , 3/2-, 5/2 -) in terms of {70} is quite well established. In order to reproduce the SU2N ® 0 3 symmetry a symmetrization of the wave function Xga is needed. The symmetrized wave function, however, is no more an eigenfunction of the original hamiltonian, because Xsgas ({al/32 }a3) and/or Xa 4~a(Ill/32 ] a3), which must be included in the symmetrized wave function, are not eigenfunctions. On the other hand, we know that the flavour exchange interactions, which are called non-planar or Xtype interactions in dual quark models, induce transitions between the eigenstates and non-eigenstates. On the assmnption that transitions between states with different radial and/or angular momentum quantum numbers (n) by the X-type interactions (Hx) are negligible, i.e., [(n'lHxln)/(E n, - E n)l "~ 1 for n'4, n, the eigenstate wave functions for the total hamiltonian including H x are well approximated by the following wave functions ,s :
vP(i) = 3 - 1/2 [~(i)
(Ctl a2~3 ;n) + ~(i) (o~1/32ct3 ;n) + c[~(i) (/31&2ct3 ;n)],
(4)
where i = 1 or 2. It is also easily seen that in the limit mc~ = m# they reduce to the right SU2N ® 0 3 wave functions. (We will explicitly show this later on.) In the present case (m~ 4- rn~) q,(i) still includes the breaking of SU2N ® 0 3. In terms of SU 6 symmetry we can manifestly see this by rewriting eq. (4) as
+ ~ ~{XT0s(a)l~s(12) - ½@23) - ½~s(31)l + ½x/5 XT0a(8)I~s(23) - @31)1} + ½r/{X70s(10 ) [gas(12) - ½~s(23) - ½gas(31)] + ½N/~ XT0a(10) [gas(23) - ~bs(31)] },
(S)
where gas(i/) - ~bs((~ict/)s~k ;n ) (i :¢:j :¢: k 4= i). The function xp(2) is given by tire replacement of X56 with X20 and the suffixes s and a with a and s, respectively. It is clear that in the limit mc~ = m# the {56} for the ground states w h e r e ~s(12) = 4~s(23) = ~s(31) and the {70} for the lp states where q~s(a)(12) + gas(a)(23) + gas(a)(31) = 0 are derived. The mixing parameters of the {56}, {20} and (70} in eq. (5) are calculated by
Cs6=l fd3xd3ylgas(12)+(Js(23)+~s(31)12,
C20=l fd3xd3y[(~a(12)+qSa(23)+gaa(31)12, (6)
GO(A)
= l f d3xd3y
{{gas(a)(12)
½gas(a)(23) - ½gas(a)(31)[2 + ~lgas(a)(23)
-
~)s(a)(31)12}'
In order to estimate the symmetry breaking in the ground state baryons, we use the harmonic oscillator approximation for the universal confining potential, that is ~a, 3 p2 3
"t i H h o = . =i~=l 2m
+ -~k.= ' i~1 [ri -
r0[2+V0,
(7)
where k and V0 are, respectively, a universal spring constant and a universal constant, and ro, r 1 , r 2 and r 3 with the constraint Z3q (ri - r0) = 0 are defined in fig. I in the Y-shaped baryon. The mixing parameters (7) for the
ground states are obtained as
C56 =~(1
+2X),
C70(A ) =}(1 - X ) ,
(8)
where
,3 The X-type interaction hamiltonian is effectively described by a sum of all permutations of the quarks. Besides the above assumption in the text one can see that H,,¢ qb(i)(al o~2~/3;n) = ep(i) (al c~2¢/3 ;n) + q~(i)(cqt32a3 ;n) + qs(i) (t31ee2ce3 ;n). Therefore HXq~(i) is again written by ,I'(i) itself. "~ #4 A different choice of potential like (1/2)k' Zi>ilr i - rfr z" which corresponds to A-shaped baryons also gives the same results derived from eq. ( 7 ) b y the replacement o f k ' w'ith k/3.
412
Volume 79B, number 4,5
PHYSICS LETTERS
4 December 1978
13
r~,
Fig, 1, Configuration of the ",'-shaped baryon, where x = - r a andy = $I ( r c q + r 3 should be taken as interrrc~}variagles, rc~2) -
~,
1613(2x + 1)] 1/2
)3/2,
X = 3(2x + 1) + 1013(2x + 1)] 1/2 +9 with x = nl~/rn~. One has the following values for the choice o f m u = 336, m d = 336 + 5, m s = 540 and nl c = 1650 in MeV, where the values o f m s and m e are taken from De Rujula et al. [2], uud C56 = ~ 1 C70(A) -- 2.3 X 10 -6
uus
uuc
uuQ (mQ = o~)
0.9961 1.97 X 10 -3
0.9737 1.31 X 10 - 2
0.9459 2.70 X 10 -2.
(9)
It is surprising that the mixing of {70} in the ground states is less than 6% even in the limit of the infinite mass difference (x = 0) and that the SU2N synrmetry is a remarkably good sylmnetry in the ground states. It is clear that the above values are quite insensitive to variation of the quark masses. This situation is also realized by evaluating the difference of the energy expectation values between the eigenstate (~s((C~l c~2)s33)) and the non-eigenstate (0s((%32)sC~3)), which is estimated to be 8 E = (Hho)non.eigenstat e -- 32 w c~ {1 + [(2x + 1)/3] 1/2} - V0 = ~(1 - X ) { [ 3 / ( l + 2x)] 1/2 -- 1}coa,
(10)
where c% = ~ ~ 4 0 0 - 5 0 0 MeV is expected from the mass difference between the ground states and the lpstate (Os((alaZ)s33)) and the non-eigenstate (qSs((Cq32)sa3)), which is estimated to be uud 8E = l A X 10 -2
uus
uuc
uuQ (mQ = ~)
11.1
68.7
137 (MeV).
(11)
These values are again very small values in comparison with the quark mass differences. In particular the u - d mass difference effect is negligible and hereafter we treat u and d as identical particles. The values (11) except for the uud case, however, are not so sufficiently small that they cannot be neglected in evaluating the masses of the ground states. Now we evaluate the masses with the following hamiltonian:
2 . . H = H b o + ,>~/(
%
3 l r . --r!,
s~ sj ] +go83(ri- r j ) ~ i m / l ,
(12)
where the one gluon exchange potential -2as/3r and a phenomenological spin spin potential si.s / are treated perturbatively. One obtains
M = V0 + m(2/x + l/y) + ~ coo (xl/2 + [(2y + 1)/3] 1/2} 4 ( m c o o ~ 1/2 [ ([ 4X1/2 -SC~s~--~ ! x -1/4 1 + 2 3 ( 2 y + x ) ] l / 2 + x l / 2
)1/2]
[rncoo]3/2 1 ( 4xl/2 )3/2] +go~-] - ~ [ x5/2($1"$2) +xl/4y((s1 +S2)'S3) [3(2y + 1)] 1/2 +xl/2 ' where m = rn u, x = m/rnc,, y = m/m~, coo = x / ~
(13)
and
413
Volume 79B, number 4,5
PftYSICS LETTERS
4 December 1978
Table 1 Calculated baryon masses, using the following values for the parameters: m u = 336 MeV, m s = 614 MeV, m c = 1855 MeV, Vo = 740.6 MeV, wo = 354.6 MeV, % = 0.446 and go = 8.45. The masses o f A c and E c are taken from ref. [31.
Table 2 Tile estimated parameters of" the weak form factors, F(q 2 ) = fexp [ - ( K / 6 , / ~ ) q 2 1 , for B 1 -~ B2 £v decays. B 1
B2
Baryon
Mass (MeV)
Experimental value
N E A
938 940 1189-1197 1115.6 -+ 0.05 1315-1321 9 2260 -+ 10 1230 - 1234 1382- 1387 1532-1535 1672.2 + 0.4 2400-2500 (?) ?
p A G0 +
Ec Ac A Y~'(E*) "~* Iz E"* c .~
939 (input) 1184 1118.6 1330 2401 2260 (input) 1232 (input) 1379 1525 1672.2 (input) 2482 3629
n E.~)20
-
3711
9
~C
zC,o C
= !
f o r N , A a n d A c' for E, v E
4
~'
C
1
1.18 1
p
)2-
n
G-
A
X~-
.-0
Ac Ac E÷ C 12C
A E° A )2
Ac
n
)2~
p
1.28
1.57 1.57
C
A
((s 1 + s 2 ) ' s 3 ) = 0
and d e c u p l e t b a r y o n s ,
K
Ac
c
(S 1 "S 2) = --a3
f
0.944
0.910
0.992
1.07
0.991
1.11
0.991
1.53
0.971
0.805
f o r N , A, a n d A c,
=-1
for E, Z and E c '
_ 1
for d e c u p l e t b a r y o n s .
When we use m u = 3 3 6 MeV and the masses o f N, 2x, f2, and A c [3] as i n p u t , u n d e r the r e q u i r e m e n t % < I, we can well r e p r o d u c e the masses o f the g r o u n d states as s h o w n in table 1. The p a r a m e t e r m s = 6 1 4 MeV seems to be a little larger t h a n the usual one (m s ~ m u + ( 1 5 0 - 2 0 0 MeV)). Nevertheless, o u r value for m s is c o n s i s t e n t l y u n d e r s t o o d in the following c o n s i d e r a t i o n : The suppression f a c t o r x 1/2 + [(2y + 1)/31 1/2 o f the h a r m o n i c oscillator t e r m and the e n h a n c e m e n t f a c t o r x -1/4 o f the 1/r t e r m , b o t h o f w h i c h are neglected in the usual analysis, reduce the masses o f the b a r y o n s w i t h an s-quark. In o r d e r to c o m p e n s a t e those c o n t r i b u t i o n s m s m u s t be larger t h a n the usual values. The difference b e t w e e n o u r analysis and the usual one like t h a t o f ref. [21 appears in the mass differences b e t w e e n c h a r m e d b a r y o n s , w h i c h is r e p r e s e n t e d b y
m2\
27r ]
l-y
c
1/2
=141MeV(160MeV),
(14)
1 + [ 3 ( 2 y + I)1
where the value derived in ref. [2] is q u o t e d in the last p a r e n t h e s e s a n d the f a c t o r [4/{1 + [3(2y + 1)] 1/2)3/2 is the n e w l y derived f a c t o r in our m o d e l , w h i c h is ~ 1.5 for the c h a r m e d b a r y o n s , whereas it is ~ 1.16 for the strange b a r y o n s . Now, we will e s t i m a t e the s y m m e t r y b r e a k i n g effects o f the wave f u n c t i o n s (4) in evaluating w e a k f o r m factors. T h e overlap integral in the weak f o r m factor is p a r a m e t r i z e d in the form F ( q 2) =
fd3x d 3 y qS~q~i exp [iqri]
= f(x, y ) exp [--K(x, y)q2/(6x/%-~)].
The values o f f ( x , y ) and K(x, y ) for various processes are given in table 2. E x c e p t for the decays o f c h a r m e d 414
(15)
Volume 79B, number 4,5
PHYSICS LETTERS
4 December 1978
B \\ B ~1~,1=0 0 " 0 a~l,=L~a
(I)
%
[3
1
I,=L
1,~1
~
......
(2)
a -t,-I-O.
(I)
(L+I)
Fig. 2. Configurations of baryon excited states, where Ii and 12, respectively, stand for the angular momenta of the twoidentical-quark system and that of quark 13.
baryons the breaking effects seem to be very small +s. In conclusion we may say that the SU2N ® 0 3 symmetry is quite good as an approximate symmetry for lowlying baryons, whatever large mass difference between quarks exists. We briefly point out the effect of the breaking in high mass baryon states. The X-type interaction is known to decrease with energy in the Regge pole model and the dual topological model. If it is true, at very high mass, where aM(M) >> (n IHx In) , the eigenfunctions defined in eqs. (1) and (2) become almost real wave functions and the SU2N ® 0 3 symmetry is significantly broken. Then states with the same angular momentum L split into (L + 1) states shown in fig. 2. In the A and ~2 (also A c and Zc) trajectories the (L + 1)th configuration of fig. 2 has the lowest mass and the first has the highest mass, whereas the former has the highest and the latter the lowest in the -- (Ec) trajectories. It is expected that only the state described by the (L + l)th configuration in the A and Z (A c and V~c) trajectories can couple with NK systems, because the angular momentum of the identical quark system in the (L + 1)th configuration (I1 = 0) coincides with that of the ground state (N), while in other configurations the l l ' s are non-zero. Finally we want to mention that, if the situation discussed in this paper is not observed, the confining potential must have a certain quark mass dependence. In the case that the spring constant of the ith quark is proportional to the quark mass (k i -- rnik 0 with k 0 = const.), which was proposed by Hirata et al. [4] in the evaluation of meson masses, no energy differences between the configurations in fig. 2 appear. *s We will discuss the details in a forthcoming paper. References
[l I N. Isgur and G. Karl, Phys. Lett. 74B (1978) 353. [21 A. De Rujula, If. Georgi and S.L. Glashow, Phys. Rev. D12 (1975) 147. [3] E.G. Cazzoli et al., Phys. Rev. Lett. 34 (1975) 1125; B. Knapp et al., Phys. Rev. Lett. 37 (1976) 882. [41 K. Hirata, T. Kobayashi and N. Nakamaru, Phys. Rev. D, to be published.
415