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Color-electric instabilities in baryons and dibaryons?
Volume 114B, number 4
PHYSICS LETTERS
29 July 1982
COLOR-ELECTRIC INSTABILITIES IN BARYONS AND DIBARYONS? K.F. LIU Department of Physics and Astronomy, University of Kentucky, Lexington, KY 40506, USA Received 3 December 1981 Revised manuscript received 20 April 1982
It is shown that the wavefunctions used in conjunction with Harvey's quark potential to study baryons and dibaryons are far from the variational equilibrium. Once self-consistency is imposed, we find that the color-electric interaction yields no configuration mixing in the nucleon. It is further reckoned, following an estimate, that configuration mixing in the sixquark system might be greatly reduced at equilibrium.
Attempts have been made to study the short-distance behavior o f the nucleon-nucleon interaction in terms o f the colored quarks and the gluons. It is believed that when the two nucleons merge, the dynamics may be delineated perturbatively with QCD inside a region which is in a phase delimitated by the nonperturbative "true" vacuum outside. It is in this spirit that the quark dynamics have been studied in both the potential models [ 1 - 6 ] and the MIT bag model [7]. The issue is rather involved in view of the complications due to the center of mass correction, the relativistic aspects, the confinement mechanisms and the pion cloud outside. Added to the complication is an intriguing calculation by Harvey [6] which shows a large configuration mixing between (ls) 6 configuration (six quarks in ls orbits) and (ls)4(lp) 2 configuration in which two quarks are orbitally excited to the lp state. This is caused largely by the color-electric interaction as opposed to the case o f strong colormagnetic attraction in the orbitaUy excited state [8]. This effect o f larger configuration mixing in sixquark systems has been magnified recently by Oka et al. [9] in a shell model study of Harvey's potential. In addition, they also found a large mixing between (ls) 3 configuration and ls(lp) 2 and (ls) 2 2s configurations in the baryon sector. As a result, the N and A masses are lowered by 540 MeV and 270 MeV, respectively. In this case, there is no hidden color channel. 222
All the quark pairs are in the color 3 representation. Actually, the baryon result [9] is easy to comprehend. It stems from the inconsistency between the wavefunctions and the interaction used. As was shown earlier [10], there is an approximate scaling rule between the sizes of mesons with different flavors. It is deduced from the observation that the excitation spectra of spin one mesons above their respective vector meson masses are roughly independent of the quark flavors. The most illuminating example is the nearly identical patterns of radial excitations between the heavy mesons cE and bb [11]. Described in a nonrelativistic potential model, it yields a scaling behavior for the characteristic lengths r 0 of mesons with different flavors r 0 = (u60)-1/z,
(1)
where/J is the reduced mass of the two quarks in a meson and 60 is the flavor-independent characteristic excitation energy. (For a linear potential, the energy difference between 2S and Is states is 1.756o.) This Size effect is also evidenced in the systematics of the fine splittings, the hyperf'me splittings and the leptonic widths in different mesons [10]. It follows from eq. (1) that the size, the quark mass and the excitation energy are not independent variables if one wants to fit the hadron sizes as well as their excitation spectra. It is found [12] that given 0 031-9163/82[0000-0000/$02.75 © 1982 North-Holland
Volume 114B, number 4
PHYSICS LETTERS
the proton charge radius at 0.81 fm and the excitation spectra of N, A and p, the u, d quark masses should be around 120 MeV. Now, in the non-relativistic revision of Harvey's potentials, the u, d quark masses are set to be 355 MeV, a factor of 3 larger than the masses deduced from scaling. This implies that the optimal size of the nucleon will be roughly a factor of x/'3 smaller than the observed nucleon size, provided the fit to the excitation spectrum of N is kept. In other words, there is bound to be an inconsistency between the hamiltonian and the wavefunctions which are used to generate the N, A excitation spectra and the proton charge radius with a quark mass as large as 355 MeV. The large energy shift ( - 5 4 0 MeV for N and - 2 7 0 MeV for A) found by Oka et al. [9] can therefore be interpreted as an attempt to reach self-consistency through the configuration mixing. To demonstrate this in more detail, we shall use the harmonic-oscillator wavefunctions to search for the variational minimum of Harvey's hamiltonian. Since our main interest lies in the color-electric part of the interaction, we shall suppress the complication due to the color-magnetic interaction. The colormagnetic interaction can be as important as the colorelectric interaction. It has been shown [8] that colormagnetic attraction due to the configuration mixing can be of the order of few hundred MeV. With the color-electric part only, Harvey's nonrelativistic hamiltonian is written as
g H = / ~ 2m •
p2
x;. x,.
2n~ + ~
2) + Br 2 + C,
(2)
(3)
with A = 3810 MeV, B = - 1 2 . 5 MeV fm -2, C = - 4 7 9 . 8 MeV and a = 0.8 fro. The ground-state baryons are in the [56, 0 +] multiplet of flavor-spin SU(6). The spatial wavefunction for N(A) can be written in the internal coordinates as: 1//3(56 , 0 +) = (p3/rr3/2)
~3,(56', 0 +) = (u5/X/~ 7r3/2 ) X (p2 + )~2 _ 3/u2)exp (_u202/2 _ u2)~2/2).
(6)
This can also be written in terms of the shell model wavefunctions [13] as ~3,(56, 0 +) = x/~-~(1 s) 2 2s + X/~(ls) (lp) 2.
(7)
The orthogonal combination corresponds to a state with internal motion in the ground state and the center of mass in the (2s) excitation, i.e. a spurious state. The [70, 0 +] multiplet with mixed spatial symmetry will not mix with the ground state through the colorelectric interaction. Thus, they are discarded. The diagonal matrix elements of Harvey's hamiltonian for ff3(56, 0 +) and ff3,(56', 0 +) are given in the following M 3 = (56, 0+1HI56, 0 +) = 3m + 3u2/2m
+ ¼(x i- xj ~3(3~/v 2 + c + AX3/2),
(8)
M 3, = (56', 0+1HI56 ', 0 +) = 3m + 5u2/2m
+ ~ X 3/2 + [2XT/2/u4(u2 +/32/2)21 X @8 + ~ v6/32 + ~ u4/34 + _~/)2/36 + s/38)] ),
(4)
where k = (1/X/~) (r 1 + r 2 - 2r3).(5)
(9)
where t3 = 1/a and X = u2/(v 2 + 2/32). Since all the quark pairs are in the color 3 representation, (Xi- ~ ) are equal to 8/3. The off-diagonal matrix element is also calculated which is M3, 3, = (56, 0+1HI56 ', 0 +) = - x / ~ u2/2m + ~(k i" kj)
X 3 [Vr3B/u 2 + vt~A(X 5/2 - X3/2)].
X exp(p2p2/2)exp(-u2X2/2),
p = ( l / x / ' 2 ) (r 1 - r2),
The variational parameter u is the inverse of the size parameter b used in the single-particle harmonic-oscillator wavefunction in ref. [6]• The next excited state can mix with the ground state is the [56', 0 +] multiplet with radial excitations in the internal coordinates. Its wavefunction is
+ -~(xi. xi> (158/~ 2 + 3c + A [~ X 7/2 - ~ X 5/2
%'
where Xi is the color SU(3) operator, m is the quark mass, n is the number of quarks and
Vii = A e x p ( - r 2 / a
29 July 1982
(10)
These matrix elements are plotted in fig. 1 as a function of the variational size parameter b. Indeed we find that the minimum o f M 3 occurs at a b (b = 0.305 fm) which is smaller than the proton charge ra-223
Volume 114B, number 4
PHYSICS LETTERS
I
I
I
3000
I
M3
.........
,ooo
M3, 3,
-
2000
---'"" ......
W
// 0
Il t
- -
- 1000
I;t 0.2
I OA
I 0.6
I 0.8
b (fro)
Fig. 1. The baryon masses M3 and M3' are plotted as a function of the variational size parameter b together with the offdiagonal matrix element M3,3,.
dins. One needs not be disturbed by the negative masses, since we are only interested in mass differences. One can always add a constant term to remove the negative masses. As shown in table I and fig. 1, the N(A) mass at minimum is ~ 2 . 3 GeV lower than that at b = 0.8 fm. This is much larger than AM 3 = 448 MeV which is the mass shift o f M 3 at b = 0.8 fm through configuration mixing. The latter agrees quite well with the spin averaged mass shift o f N and A as calculated by Oka et al. [9] which is 405 MeV. Notice that at equilibrium, the off-diagonal matrix element between 4 3 ( 5 6 , 0 +) a n d ~b3,(56 ' , 0 +) goes through zero. Therefore, there is no configuration mixing at this point. This is, however, a special case owing to the particular forms of the color-electric potential and the wavefunctions chosen. In general, it is non-zero
29 July 1982
but minimal at the variational equilibrium. As mentioned earlier, it is expected from the scaling rule in eq. (1) that the quark mass of 355 MeV itself will lower the nucleon size by a factor of ~ V ~ as compared to the charge radius of the proton. In addition, it is found that the excitation energy o f M 3, at equilibrium is 3.28 times that o f M 3 at b = 0.8 fm. Therefore, the characteristic excitation energy 6o at equilibrium is enlarged by the same factor as compared to that at b = 0.8 fro. This will, again according to eq. (1), lower the size by a factor of(3.28) 1/2. Together, we expect from the scaling that the equilibrium size will be 3.1 times smaller than b = 0.8 fm which puts the nucleon radius at 0.26 fm. This represents a good estimate to the actual equilibrium point at b = 0.305 fm. A question then arises: to what extent this spurious configuration mixing at the non-equilibrium point may be carried over to the calculations of the six-quark systems in refs. [6] and [9]. The ultimate answer lies, of course, in the detailed calculations like in refs. [6, 9] with the explicit hidden color channel coupling and performed at the equilibrium point instead of at b = 0.8 fm. In this letter, we shall use a statistical approximation to estimate the configuration mixing without an explicit treatment of the hidden color channels. For the ground-state configuration (ls) 6 , the radial matrix elements are the same for each pair of quarks. Therefore, the potential energy can be evaluated as
\i<]
]/
where I is the common two-body radial matrix element. Evaluated in a color irreducible representation of n quarks, the color matrix element gives
Table 1 The diagonal matrix elements of the three-quark states (M3, M3,) and the six-quark states (/1//6,M6,) are listed together with their respective off-diagonal matrix elements (M3 3,,M6 6') and mass shifts (AM3, AM6). They are calculated at different sizes. M3(M 6) has a minimum at b = 0.305 fm (0.331 fm).' b (fm)
M3 (MeV)
M3, (MeV)
M3, 3 , (MeV)
AM3
0.8 0.305 0.331
863.1 -1396.8
1591.5 992.2
726.0 0
-448.0 0
224
M6 (MeV)
M6,
Mfi,6 , (MeV)
AM6
(MeV)
1854.8 -1911.5 -1980.0
2839.4 863.8 681.5
900.9 203.8 437.6
-534.3 -14.9 -70.1
Volume 114B, number 4
PHYSICS LETTERS
29 July 1982
I
I
where C is the eigenvalue of the Casimir operator which is zero in the case of the color-singlet representation. For each pair in the (ls) 6 configuration, the average color strength is, therefore,
-
5000
I i
(ki" k/) 6 = ~(k/" X/> 3 .
/
It
I000
/'
/
!
(13)
For n = 6, this average gives - 1 6 / 1 5 which is 2/5 times that in the baryons, i.e.
M6
M6' ~ M6'6/, ""
......
I
2000
/
I
I
i / kk
.~J/
-~,.
/ t
(14) I
We shall calculate the mass of the six-quark state with (ls) 5 2s and ( l s ) 4 ( l p ) 2 configurations and the off. diagonal matrix element by neglecting color correlations in these cases and assuming that the same average color strength applies to each pair. The six-quark wavefunction with 2hco excitation should have the same combination of ( l p ) 2 mid 2s as in eq. (7) in order to be free from the spurious center of mass motion. Hence,
¢6,-
~( ls ) 5 2s + ~_~(1s)4(1p)2,
(15)
the masses of ~6 and ~b6, are then given as M 6 = (ff61Hlff6) = 6m + 15u2/4m + -~( ~ - X/.)6 X 15(3B/u 2 + C +AX3/2);
(16)
and M 6, = (~6,1Hlff6,) = 6m + 19u2/4m + ¼(Xi" Xj) 6 X (15C + 9(5B/v 2) + 6(3B/v 2) + A [6X 3/2 + -~X 7/2
3X5/2 + 5X3/2 + [8X7/2fi, a(v 2 + ~2/2)2] X (/)8+~v6/32+~271)a/~,4+~u2/~6+~f18)]}.
(17)
In the expression of M6, , there are 6 pair-wise interaction matrix elements of the kind in M 3 or M 6 and 9 matrix elements of the kind in M3,. The offdiagonal matrix element between ~6 and ~6' is M6, 6, = (•6,1H1¢6) = _vr~u2/2m + ~ (ki"
k/) 6
X 9 [v~B/u 2 + Vr3A/2(X 512 - X3/2)].
(18)
-I000
-2000 I
I
I
I
0.2
0.4
0.6
0.8
b (fro) Fig. 2. The dibaryon masses M 6 and M 6, and the off-diagonal matrix element M6, 6, are plotted as a function of b.
There are 9 pair-wise interaction matrix elements of the kind inM3, 3, o f e q . (10). The results of these matrix elements are plotted in fig. 2 as a function of b. It is found that the equilibrium point f o r M 6 is at b = 0.331 fm. The fact that it is slightly larger than that in the baryon case has to do with the relative kinetic energy between the two nucleons which is also included in the mass of the sixquark state. As tabulated in table 1, the mass shift z~/6 is - 5 3 4 MeV at b = 0.8 fm. This roughly agrees with the result o f - 7 7 0 MeV from ref. [10] if we allow some room for the color-magnetic interaction which is suppressed here. At b = 0.305 fm, the equilibrium size of N(A), the mass shift ZkM6 is only - 1 5 MeV and at b = 0.331 fm, the equilibrium point of (1 s) 6, the mass shift zXM6 is - 7 0 MeV. The small mass shift may be understood as follows: The off-diagonal matrix element M3, 3, is zero at equilibrium due to the cancellation between the kinetic energy and the potential energy terms. In the six-quark system, when the small difference in equilibrium sizes is neglected, the kinetic energy contribution remains the same as in the threequark case [see eq. (18)], while the potential contribution is changed by a factor of
225
Volume 114B, number 4
V6,6,/V3, 3, =
~ X 9
PHYSICS LETTERS
-g,
--6
(19)
which is the product o f the color strength ratio and the ratio o f the effective numbers of pairs. Given this ratio close to unity, the potential contribution will cancel the kinetic energy contribution to a large extent, hence a smaller off-diagonal matrix element is obtained. This result o f large cancellation will be altered when the color-magnetic interaction is introduced [8]. In conclusion, we have shown that the large configuration mixing found in the three-quark system is largely due to the lack o f consistency between the wavefunctions and the interaction used. When selfconsistency is imposed in the six-quark system, our estimate shows that the configuration mixing is greatly reduced. This is based on the assumption of averaged color strength. It needs further verification from a more realistic calculation with explicit coupling to the hidden color channel. Only then will we have the question of color-electric instability settle d.
226
29 July 1982
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13 ]
D.A. Liberman, Phys. Rev. D16 (1977) 1542. D. Robson, Nucl. Phys. A308 (1978) 381. M. Oka and K. Yazaki, Phys. Lett. 90B (1980)41. J.D. de Deus and J.E. Ribeiro, Phys. Rev. D21 (1980) 1251. C.S. Warke and R. Shanker, Phys. Rev. C21 (1980) 2643. M. Harvey, Nucl. Phys. A352 (1981) 301, 326. C.E. Detar, Phys. Rev. D17 (1978) 323;D19 (1979) 1451. I.T. Obukhovsky, V.G. Neudatchin, Yu.F. Smirnov and Yu.M. Tchuvil'sky, Phys. Lett. 88B (1979) 231. M. Oka, A. Arima, S. Ohta and K. Yazaki, Proc. ninth Intern. Conf. on High energy physics and nuclear structure (Versailles, 1981). K.F. Liu and C.W. Wong, Phys. Rev. D17 (1978) 2350. K.F. Liu and C.W. Wong, Phys. Lett. 73B (1978) 223. K.F. Liu and C.W. Wong, Phys. Rev. D21 (1980) 1350. See, for instance, F.E. Close, An introduction to quarks and protons (Academic Press, New York, 1979).
PHYSICS LETTERS
29 July 1982
COLOR-ELECTRIC INSTABILITIES IN BARYONS AND DIBARYONS? K.F. LIU Department of Physics and Astronomy, University of Kentucky, Lexington, KY 40506, USA Received 3 December 1981 Revised manuscript received 20 April 1982
It is shown that the wavefunctions used in conjunction with Harvey's quark potential to study baryons and dibaryons are far from the variational equilibrium. Once self-consistency is imposed, we find that the color-electric interaction yields no configuration mixing in the nucleon. It is further reckoned, following an estimate, that configuration mixing in the sixquark system might be greatly reduced at equilibrium.
Attempts have been made to study the short-distance behavior o f the nucleon-nucleon interaction in terms o f the colored quarks and the gluons. It is believed that when the two nucleons merge, the dynamics may be delineated perturbatively with QCD inside a region which is in a phase delimitated by the nonperturbative "true" vacuum outside. It is in this spirit that the quark dynamics have been studied in both the potential models [ 1 - 6 ] and the MIT bag model [7]. The issue is rather involved in view of the complications due to the center of mass correction, the relativistic aspects, the confinement mechanisms and the pion cloud outside. Added to the complication is an intriguing calculation by Harvey [6] which shows a large configuration mixing between (ls) 6 configuration (six quarks in ls orbits) and (ls)4(lp) 2 configuration in which two quarks are orbitally excited to the lp state. This is caused largely by the color-electric interaction as opposed to the case o f strong colormagnetic attraction in the orbitaUy excited state [8]. This effect o f larger configuration mixing in sixquark systems has been magnified recently by Oka et al. [9] in a shell model study of Harvey's potential. In addition, they also found a large mixing between (ls) 3 configuration and ls(lp) 2 and (ls) 2 2s configurations in the baryon sector. As a result, the N and A masses are lowered by 540 MeV and 270 MeV, respectively. In this case, there is no hidden color channel. 222
All the quark pairs are in the color 3 representation. Actually, the baryon result [9] is easy to comprehend. It stems from the inconsistency between the wavefunctions and the interaction used. As was shown earlier [10], there is an approximate scaling rule between the sizes of mesons with different flavors. It is deduced from the observation that the excitation spectra of spin one mesons above their respective vector meson masses are roughly independent of the quark flavors. The most illuminating example is the nearly identical patterns of radial excitations between the heavy mesons cE and bb [11]. Described in a nonrelativistic potential model, it yields a scaling behavior for the characteristic lengths r 0 of mesons with different flavors r 0 = (u60)-1/z,
(1)
where/J is the reduced mass of the two quarks in a meson and 60 is the flavor-independent characteristic excitation energy. (For a linear potential, the energy difference between 2S and Is states is 1.756o.) This Size effect is also evidenced in the systematics of the fine splittings, the hyperf'me splittings and the leptonic widths in different mesons [10]. It follows from eq. (1) that the size, the quark mass and the excitation energy are not independent variables if one wants to fit the hadron sizes as well as their excitation spectra. It is found [12] that given 0 031-9163/82[0000-0000/$02.75 © 1982 North-Holland
Volume 114B, number 4
PHYSICS LETTERS
the proton charge radius at 0.81 fm and the excitation spectra of N, A and p, the u, d quark masses should be around 120 MeV. Now, in the non-relativistic revision of Harvey's potentials, the u, d quark masses are set to be 355 MeV, a factor of 3 larger than the masses deduced from scaling. This implies that the optimal size of the nucleon will be roughly a factor of x/'3 smaller than the observed nucleon size, provided the fit to the excitation spectrum of N is kept. In other words, there is bound to be an inconsistency between the hamiltonian and the wavefunctions which are used to generate the N, A excitation spectra and the proton charge radius with a quark mass as large as 355 MeV. The large energy shift ( - 5 4 0 MeV for N and - 2 7 0 MeV for A) found by Oka et al. [9] can therefore be interpreted as an attempt to reach self-consistency through the configuration mixing. To demonstrate this in more detail, we shall use the harmonic-oscillator wavefunctions to search for the variational minimum of Harvey's hamiltonian. Since our main interest lies in the color-electric part of the interaction, we shall suppress the complication due to the color-magnetic interaction. The colormagnetic interaction can be as important as the colorelectric interaction. It has been shown [8] that colormagnetic attraction due to the configuration mixing can be of the order of few hundred MeV. With the color-electric part only, Harvey's nonrelativistic hamiltonian is written as
g H = / ~ 2m •
p2
x;. x,.
2n~ + ~
2) + Br 2 + C,
(2)
(3)
with A = 3810 MeV, B = - 1 2 . 5 MeV fm -2, C = - 4 7 9 . 8 MeV and a = 0.8 fro. The ground-state baryons are in the [56, 0 +] multiplet of flavor-spin SU(6). The spatial wavefunction for N(A) can be written in the internal coordinates as: 1//3(56 , 0 +) = (p3/rr3/2)
~3,(56', 0 +) = (u5/X/~ 7r3/2 ) X (p2 + )~2 _ 3/u2)exp (_u202/2 _ u2)~2/2).
(6)
This can also be written in terms of the shell model wavefunctions [13] as ~3,(56, 0 +) = x/~-~(1 s) 2 2s + X/~(ls) (lp) 2.
(7)
The orthogonal combination corresponds to a state with internal motion in the ground state and the center of mass in the (2s) excitation, i.e. a spurious state. The [70, 0 +] multiplet with mixed spatial symmetry will not mix with the ground state through the colorelectric interaction. Thus, they are discarded. The diagonal matrix elements of Harvey's hamiltonian for ff3(56, 0 +) and ff3,(56', 0 +) are given in the following M 3 = (56, 0+1HI56, 0 +) = 3m + 3u2/2m
+ ¼(x i- xj ~3(3~/v 2 + c + AX3/2),
(8)
M 3, = (56', 0+1HI56 ', 0 +) = 3m + 5u2/2m
+ ~ X 3/2 + [2XT/2/u4(u2 +/32/2)21 X @8 + ~ v6/32 + ~ u4/34 + _~/)2/36 + s/38)] ),
(4)
where k = (1/X/~) (r 1 + r 2 - 2r3).(5)
(9)
where t3 = 1/a and X = u2/(v 2 + 2/32). Since all the quark pairs are in the color 3 representation, (Xi- ~ ) are equal to 8/3. The off-diagonal matrix element is also calculated which is M3, 3, = (56, 0+1HI56 ', 0 +) = - x / ~ u2/2m + ~(k i" kj)
X 3 [Vr3B/u 2 + vt~A(X 5/2 - X3/2)].
X exp(p2p2/2)exp(-u2X2/2),
p = ( l / x / ' 2 ) (r 1 - r2),
The variational parameter u is the inverse of the size parameter b used in the single-particle harmonic-oscillator wavefunction in ref. [6]• The next excited state can mix with the ground state is the [56', 0 +] multiplet with radial excitations in the internal coordinates. Its wavefunction is
+ -~(xi. xi> (158/~ 2 + 3c + A [~ X 7/2 - ~ X 5/2
%'
where Xi is the color SU(3) operator, m is the quark mass, n is the number of quarks and
Vii = A e x p ( - r 2 / a
29 July 1982
(10)
These matrix elements are plotted in fig. 1 as a function of the variational size parameter b. Indeed we find that the minimum o f M 3 occurs at a b (b = 0.305 fm) which is smaller than the proton charge ra-223
Volume 114B, number 4
PHYSICS LETTERS
I
I
I
3000
I
M3
.........
,ooo
M3, 3,
-
2000
---'"" ......
W
// 0
Il t
- -
- 1000
I;t 0.2
I OA
I 0.6
I 0.8
b (fro)
Fig. 1. The baryon masses M3 and M3' are plotted as a function of the variational size parameter b together with the offdiagonal matrix element M3,3,.
dins. One needs not be disturbed by the negative masses, since we are only interested in mass differences. One can always add a constant term to remove the negative masses. As shown in table I and fig. 1, the N(A) mass at minimum is ~ 2 . 3 GeV lower than that at b = 0.8 fm. This is much larger than AM 3 = 448 MeV which is the mass shift o f M 3 at b = 0.8 fm through configuration mixing. The latter agrees quite well with the spin averaged mass shift o f N and A as calculated by Oka et al. [9] which is 405 MeV. Notice that at equilibrium, the off-diagonal matrix element between 4 3 ( 5 6 , 0 +) a n d ~b3,(56 ' , 0 +) goes through zero. Therefore, there is no configuration mixing at this point. This is, however, a special case owing to the particular forms of the color-electric potential and the wavefunctions chosen. In general, it is non-zero
29 July 1982
but minimal at the variational equilibrium. As mentioned earlier, it is expected from the scaling rule in eq. (1) that the quark mass of 355 MeV itself will lower the nucleon size by a factor of ~ V ~ as compared to the charge radius of the proton. In addition, it is found that the excitation energy o f M 3, at equilibrium is 3.28 times that o f M 3 at b = 0.8 fm. Therefore, the characteristic excitation energy 6o at equilibrium is enlarged by the same factor as compared to that at b = 0.8 fro. This will, again according to eq. (1), lower the size by a factor of(3.28) 1/2. Together, we expect from the scaling that the equilibrium size will be 3.1 times smaller than b = 0.8 fm which puts the nucleon radius at 0.26 fm. This represents a good estimate to the actual equilibrium point at b = 0.305 fm. A question then arises: to what extent this spurious configuration mixing at the non-equilibrium point may be carried over to the calculations of the six-quark systems in refs. [6] and [9]. The ultimate answer lies, of course, in the detailed calculations like in refs. [6, 9] with the explicit hidden color channel coupling and performed at the equilibrium point instead of at b = 0.8 fm. In this letter, we shall use a statistical approximation to estimate the configuration mixing without an explicit treatment of the hidden color channels. For the ground-state configuration (ls) 6 , the radial matrix elements are the same for each pair of quarks. Therefore, the potential energy can be evaluated as
\i<]
]/
where I is the common two-body radial matrix element. Evaluated in a color irreducible representation of n quarks, the color matrix element gives
Table 1 The diagonal matrix elements of the three-quark states (M3, M3,) and the six-quark states (/1//6,M6,) are listed together with their respective off-diagonal matrix elements (M3 3,,M6 6') and mass shifts (AM3, AM6). They are calculated at different sizes. M3(M 6) has a minimum at b = 0.305 fm (0.331 fm).' b (fm)
M3 (MeV)
M3, (MeV)
M3, 3 , (MeV)
AM3
0.8 0.305 0.331
863.1 -1396.8
1591.5 992.2
726.0 0
-448.0 0
224
M6 (MeV)
M6,
Mfi,6 , (MeV)
AM6
(MeV)
1854.8 -1911.5 -1980.0
2839.4 863.8 681.5
900.9 203.8 437.6
-534.3 -14.9 -70.1
Volume 114B, number 4
PHYSICS LETTERS
29 July 1982
I
I
where C is the eigenvalue of the Casimir operator which is zero in the case of the color-singlet representation. For each pair in the (ls) 6 configuration, the average color strength is, therefore,
-
5000
I i
(ki" k/) 6 = ~(k/" X/> 3 .
/
It
I000
/'
/
!
(13)
For n = 6, this average gives - 1 6 / 1 5 which is 2/5 times that in the baryons, i.e.
M6
M6' ~ M6'6/, ""
......
I
2000
/
I
I
i / kk
.~J/
-~,.
/ t
(14) I
We shall calculate the mass of the six-quark state with (ls) 5 2s and ( l s ) 4 ( l p ) 2 configurations and the off. diagonal matrix element by neglecting color correlations in these cases and assuming that the same average color strength applies to each pair. The six-quark wavefunction with 2hco excitation should have the same combination of ( l p ) 2 mid 2s as in eq. (7) in order to be free from the spurious center of mass motion. Hence,
¢6,-
~( ls ) 5 2s + ~_~(1s)4(1p)2,
(15)
the masses of ~6 and ~b6, are then given as M 6 = (ff61Hlff6) = 6m + 15u2/4m + -~( ~ - X/.)6 X 15(3B/u 2 + C +AX3/2);
(16)
and M 6, = (~6,1Hlff6,) = 6m + 19u2/4m + ¼(Xi" Xj) 6 X (15C + 9(5B/v 2) + 6(3B/v 2) + A [6X 3/2 + -~X 7/2
3X5/2 + 5X3/2 + [8X7/2fi, a(v 2 + ~2/2)2] X (/)8+~v6/32+~271)a/~,4+~u2/~6+~f18)]}.
(17)
In the expression of M6, , there are 6 pair-wise interaction matrix elements of the kind in M 3 or M 6 and 9 matrix elements of the kind in M3,. The offdiagonal matrix element between ~6 and ~6' is M6, 6, = (•6,1H1¢6) = _vr~u2/2m + ~ (ki"
k/) 6
X 9 [v~B/u 2 + Vr3A/2(X 512 - X3/2)].
(18)
-I000
-2000 I
I
I
I
0.2
0.4
0.6
0.8
b (fro) Fig. 2. The dibaryon masses M 6 and M 6, and the off-diagonal matrix element M6, 6, are plotted as a function of b.
There are 9 pair-wise interaction matrix elements of the kind inM3, 3, o f e q . (10). The results of these matrix elements are plotted in fig. 2 as a function of b. It is found that the equilibrium point f o r M 6 is at b = 0.331 fm. The fact that it is slightly larger than that in the baryon case has to do with the relative kinetic energy between the two nucleons which is also included in the mass of the sixquark state. As tabulated in table 1, the mass shift z~/6 is - 5 3 4 MeV at b = 0.8 fm. This roughly agrees with the result o f - 7 7 0 MeV from ref. [10] if we allow some room for the color-magnetic interaction which is suppressed here. At b = 0.305 fm, the equilibrium size of N(A), the mass shift ZkM6 is only - 1 5 MeV and at b = 0.331 fm, the equilibrium point of (1 s) 6, the mass shift zXM6 is - 7 0 MeV. The small mass shift may be understood as follows: The off-diagonal matrix element M3, 3, is zero at equilibrium due to the cancellation between the kinetic energy and the potential energy terms. In the six-quark system, when the small difference in equilibrium sizes is neglected, the kinetic energy contribution remains the same as in the threequark case [see eq. (18)], while the potential contribution is changed by a factor of
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Volume 114B, number 4
V6,6,/V3, 3, =
~ X 9
PHYSICS LETTERS
-g,
--6
(19)
which is the product o f the color strength ratio and the ratio o f the effective numbers of pairs. Given this ratio close to unity, the potential contribution will cancel the kinetic energy contribution to a large extent, hence a smaller off-diagonal matrix element is obtained. This result o f large cancellation will be altered when the color-magnetic interaction is introduced [8]. In conclusion, we have shown that the large configuration mixing found in the three-quark system is largely due to the lack o f consistency between the wavefunctions and the interaction used. When selfconsistency is imposed in the six-quark system, our estimate shows that the configuration mixing is greatly reduced. This is based on the assumption of averaged color strength. It needs further verification from a more realistic calculation with explicit coupling to the hidden color channel. Only then will we have the question of color-electric instability settle d.
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29 July 1982
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13 ]
D.A. Liberman, Phys. Rev. D16 (1977) 1542. D. Robson, Nucl. Phys. A308 (1978) 381. M. Oka and K. Yazaki, Phys. Lett. 90B (1980)41. J.D. de Deus and J.E. Ribeiro, Phys. Rev. D21 (1980) 1251. C.S. Warke and R. Shanker, Phys. Rev. C21 (1980) 2643. M. Harvey, Nucl. Phys. A352 (1981) 301, 326. C.E. Detar, Phys. Rev. D17 (1978) 323;D19 (1979) 1451. I.T. Obukhovsky, V.G. Neudatchin, Yu.F. Smirnov and Yu.M. Tchuvil'sky, Phys. Lett. 88B (1979) 231. M. Oka, A. Arima, S. Ohta and K. Yazaki, Proc. ninth Intern. Conf. on High energy physics and nuclear structure (Versailles, 1981). K.F. Liu and C.W. Wong, Phys. Rev. D17 (1978) 2350. K.F. Liu and C.W. Wong, Phys. Lett. 73B (1978) 223. K.F. Liu and C.W. Wong, Phys. Rev. D21 (1980) 1350. See, for instance, F.E. Close, An introduction to quarks and protons (Academic Press, New York, 1979).