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Possible color instabilities in abnormal quark matter
Volume 108B, number 6
PHYSICS LETTERS
4 February 1982
POSSIBLE COLOR INSTABILITIES IN ABNORMAL QUARK MATTER C.W. WONG Physics Department, University o f Califorma, Los Angeles, CA 90024, USA Received 15 August 1981
When the spatial wave function of a multiquark system is completely antlsymmetnc, the color-magnetic interaction energy may become attractive or may vanish. Under certain circumstances, this may cause a color-magnetic or color-electric collapse in the abnormal quark matter of singly occupied single-particle orbital states.
Under certain circumstances to be described below, the color magnetic (CM) interaction in quark matter may become attractive or may simply vanish. This may cause the quark matter to collapse. The fact that there is no experimental indication of an unstable quark matter suggests that the conditions for a collapse are not normally satisfied. In potential models of quark dynamics, it is customary to include a CM interaction term proportional t o Z ~ i , k/~i, where k and e are the quark color and spin operators, respectively. This term can be used to account nicely for the mass splittlngs of observed hadrons [1 ]. In more comphcated multiquark systems, its contributions depend rather sensitively on the permutation symmetries of the states. This may be seen by examining the permutation operator for a quark pair
(1) where the superscripts C, S, 1, x denote the color, spin, isospln and spatial degrees of freedom. The following discussion will be limited to systems containing only u, d quarks which are of particular interest in nuclear physics. Since quarks are fermions, eq. (1) implies that CS_ -
(2)
.
consequently 2 el'"
*j, 2(1
+ T . *l)p~//.
(3)
The expectation value of eq. (3) m a system with n quarks can be evaluated easily in two special cases. First, when the spatial wave function is completely symmetric with Young symmetry [n] x, we get the well-known result [2]
-3-4 [n(n - 6) + S ( S + 1) + 3 r ( r +
1)]
.
(4)
This is positive for N I> 6. For these systems, the resuiting CM energy is repulsive if the associated radial matrix elements have the same positive sign as that m the nucleon. Another simple situation is that of a completely antisynunetric spatial wave function with Young symmetry [1 n ] x ' for which P~ = - 1 holds for all pairs. Then eq. (3) can be summed to
t
t t
"[ln]X
= ~4 lr - ~1n 2 + S ( S + 1 ) - 3 T ( T + 1)].
(5)
The color wave function has the usual symmetry [A 3 ] O where A = n/3 is the nucleon number. Hence the spin-isospln function must have the symmetry [A 3 ] ST" The supermultiple structure of [A 3 ] ST is identical to that for [A] ST, i.e. S = T with A / 2 >i T /> 0 or 1/2. As a result 4 n(Sn + 3) ~< ~ - ~
\
i
Zi~ / • ~/a] ) <
52 ~n
(6) 383
Volume 108B, number 6
PHYSICS LETTERS
This gives rise to an attractive potential energy if the average radial matrix element remains positive. Many different radial functions have been used in the literature to describe the color magnetic Interaction. One extreme assumption [3] is that it is simply a constant. Eq. (6) then gives a strong attraction which easily overwhelms the kinetic energy for large n. It also overwhelms the repulsion from the confinement potential, which is likely to be weaker than the n 4/3 dependence expected of an r 2 potential. A CM collapse then results. It may be argued that the assumption of a constant interaction is quite unrealistic. If so, any extra CM attraction found in light multlquark systems [4] using this interaction might also be somewhat suspect. A more realistic assumption for the radial function in the CM interaction might be the 8-function of the Breit-Fermi Interaction [5]. The radial matrix element is then nonzero only in relative S-states. Since the CM interaction of eq. (5) is an odd-state interaction, it must vanish exactly in this model. In other words, the CM repulsion of the [n] x configuration is completely turned off in the [1 n ] x configuration. Any collapse which might appear can only be due to the ~'i" ~'1 terms in the quark-quark potential. I shall refer to this possibility as a color-electric (or CE) instability. ACE instability does not develop in a normal Fermi gas m the presence of a repulsive confinement potential, because the repulsion usually gets stronger as the dimension of the system increases. The followlng argument suggests however that a related effect might appear when there is a special position-color correlation in finite multiquark systems. The state of symmetry [A 3] c contains n3(n6) pairs of quarks antisymmetric (symmetric) in their color labels, where n3=n(n+3)/6
,
n6=n(n-1)/2-n
3.
(7)
Consequently, the CE interaction energy is Zi
= -- -$n3I 3 + 3 n 6 1 6 ,
(8)
where/3(/6) is the average radial integral for the color 3(6) pairs. I f I 3 = 16 = 1, eq. (8) reduces to the familiar result of ( - 8 / 3 ) n i . Since I is negative in the presence of a "repulsive" confinement potential, the system will not collapse. However, i f I 6 <13, the leading term in eq. (8) is 384
4 February 1982
proportional to n2(I 6 - •3), and is attractive. This might occur if color 3 pairs congregate to the interior of the sytem, while color 6 pairs are further apart with quarks nearer to the surface. It would be interesting to determine if the CE condensation of DeTar [6] and the color correlation found by Harvey [7] are of this origin. We may note parenthetically that it is easy to construct completely antisymmetric spatial states [ln]x. Examples having even panty and the lowest kinetic energies are sp 2 for n = 3, sp2(s, d) 3 for n = 6, sp2(s, d) 6 for n = 9, sp3(s, d)5(p, t) 3 for n = 12, and the singly occupied Fermi sea m quark matter. This shows that the total kinetic energy grows no faster than n 4/3 at large n for relativistic quarks. The dynamical situation in the MIT (Massachusetts Institute of Technology) bag model [2,8] differs somewhat from that in potential models. The normally positive energy due to the confinement potential is replaced by a volume energy in the bag whose energy density B is a universal positive constant in all hadrons. In the absence of CM contributions, the energy density of the singly occupied quark gas of massless quarks is [9,10] E/V=B
+(k4/8rr2)[1 - (1 + a ) % 2/37r1,
(9)
where a = 2 In 2 represents a relativistic correction [10] to the Coulomb exchange energy [11 ] of a color-singlet 0.e. color-neutral) quark gas, and k F = (6zr2n/V)l/3 .
(10)
Eq. (9) shows a CE instability when the effective color coupling constant a s exceeds the critical value of Orsc r i t =
37r/[2(1 + a)] = 2 .
(11)
Although this estimate is very rough and depends for its validity on the complete absence of CM and other contributions, its value is uncomfortably close to the empirical value of 2.2 used in the MIT bag model [8]. It would be interesting to study how the CM and other contributions might affect this result. It should also be noted that when the coupling constant is as strong a s Otfrlt, the result shown in eq. (9) may no longer be reahstic in that higher-order interaction effects neglected here may drastically alter the situation. (The same difficulty also appears for the strong couphng constant appearing in the MIT bag model. However, the situation IS not so critical
Volume 108B, number 6
PHYSICS LETTERS
when the interaction energy is repulsive.) This points out the possible danger of relying too literally on phenomenological models in dealing with states o f unusual permutation symmetries. The observations made m this note are very speculative in nature, and are valid only under the very special circumstances described here. In particular, there is no experimental evidence that such collapses are occurring in nature. They do suggest, however, that the associated color correlational effects might play significant roles in the dynamics o f multiquark systems. I would like to thank Dr. S.A. Chin for the help with eq. (9). This work is supported in part b y NSF Contract No. PHY78-15811.
4 February 1982
References [ 1 ] A. De Rfijula, H. Georgi and S.L. Glashow, Phys. Rev. D12 (1975) 147. [2] K. Johnson, Acta Phys. Pol. B6 (1975) 865. [3] D.A. Liberman, Phys. Rev. D16 (1977) 1542. [4] I.T. Obukhovsky, V.G. Neudatchln, Yu F. Smirnov and Yu M. Tchuvd'sky, Phys. Lett. 88B (1977) 231. [5] D.J. Jackson, Lectures on the new particles, Proc. 1976 Summer Institute on Particle Physics, Stanford Linear Accelerator Center, Report 198 (1976). [6] C. DeTar, Phys. Rev. D17 (1978) 323. [7] M. Harvey, Nucl. Phys. A352 (1981) 326. [8] T. DeGrand et al., Phys. Rev. D12 (1975) 2060. [9] G. Baym and S.A. Chin, Phys. Lett. 62B (1976) 241. [10] S.A. Chin, unpublished. [11] A.L. Fetters and J.D. Walecka, Quantum theory of many-particle systems (McGraw-Hill, New York, 1971) p. 29.
385
PHYSICS LETTERS
4 February 1982
POSSIBLE COLOR INSTABILITIES IN ABNORMAL QUARK MATTER C.W. WONG Physics Department, University o f Califorma, Los Angeles, CA 90024, USA Received 15 August 1981
When the spatial wave function of a multiquark system is completely antlsymmetnc, the color-magnetic interaction energy may become attractive or may vanish. Under certain circumstances, this may cause a color-magnetic or color-electric collapse in the abnormal quark matter of singly occupied single-particle orbital states.
Under certain circumstances to be described below, the color magnetic (CM) interaction in quark matter may become attractive or may simply vanish. This may cause the quark matter to collapse. The fact that there is no experimental indication of an unstable quark matter suggests that the conditions for a collapse are not normally satisfied. In potential models of quark dynamics, it is customary to include a CM interaction term proportional t o Z ~ i , k/~i, where k and e are the quark color and spin operators, respectively. This term can be used to account nicely for the mass splittlngs of observed hadrons [1 ]. In more comphcated multiquark systems, its contributions depend rather sensitively on the permutation symmetries of the states. This may be seen by examining the permutation operator for a quark pair
(1) where the superscripts C, S, 1, x denote the color, spin, isospln and spatial degrees of freedom. The following discussion will be limited to systems containing only u, d quarks which are of particular interest in nuclear physics. Since quarks are fermions, eq. (1) implies that CS_ -
(2)
.
consequently 2 el'"
*j, 2(1
+ T . *l)p~//.
(3)
The expectation value of eq. (3) m a system with n quarks can be evaluated easily in two special cases. First, when the spatial wave function is completely symmetric with Young symmetry [n] x, we get the well-known result [2]
-3-4 [n(n - 6) + S ( S + 1) + 3 r ( r +
1)]
.
(4)
This is positive for N I> 6. For these systems, the resuiting CM energy is repulsive if the associated radial matrix elements have the same positive sign as that m the nucleon. Another simple situation is that of a completely antisynunetric spatial wave function with Young symmetry [1 n ] x ' for which P~ = - 1 holds for all pairs. Then eq. (3) can be summed to
t
t t
"[ln]X
= ~4 lr - ~1n 2 + S ( S + 1 ) - 3 T ( T + 1)].
(5)
The color wave function has the usual symmetry [A 3 ] O where A = n/3 is the nucleon number. Hence the spin-isospln function must have the symmetry [A 3 ] ST" The supermultiple structure of [A 3 ] ST is identical to that for [A] ST, i.e. S = T with A / 2 >i T /> 0 or 1/2. As a result 4 n(Sn + 3) ~< ~ - ~
\
i
Zi~ / • ~/a] ) <
52 ~n
(6) 383
Volume 108B, number 6
PHYSICS LETTERS
This gives rise to an attractive potential energy if the average radial matrix element remains positive. Many different radial functions have been used in the literature to describe the color magnetic Interaction. One extreme assumption [3] is that it is simply a constant. Eq. (6) then gives a strong attraction which easily overwhelms the kinetic energy for large n. It also overwhelms the repulsion from the confinement potential, which is likely to be weaker than the n 4/3 dependence expected of an r 2 potential. A CM collapse then results. It may be argued that the assumption of a constant interaction is quite unrealistic. If so, any extra CM attraction found in light multlquark systems [4] using this interaction might also be somewhat suspect. A more realistic assumption for the radial function in the CM interaction might be the 8-function of the Breit-Fermi Interaction [5]. The radial matrix element is then nonzero only in relative S-states. Since the CM interaction of eq. (5) is an odd-state interaction, it must vanish exactly in this model. In other words, the CM repulsion of the [n] x configuration is completely turned off in the [1 n ] x configuration. Any collapse which might appear can only be due to the ~'i" ~'1 terms in the quark-quark potential. I shall refer to this possibility as a color-electric (or CE) instability. ACE instability does not develop in a normal Fermi gas m the presence of a repulsive confinement potential, because the repulsion usually gets stronger as the dimension of the system increases. The followlng argument suggests however that a related effect might appear when there is a special position-color correlation in finite multiquark systems. The state of symmetry [A 3] c contains n3(n6) pairs of quarks antisymmetric (symmetric) in their color labels, where n3=n(n+3)/6
,
n6=n(n-1)/2-n
3.
(7)
Consequently, the CE interaction energy is Zi
= -- -$n3I 3 + 3 n 6 1 6 ,
(8)
where/3(/6) is the average radial integral for the color 3(6) pairs. I f I 3 = 16 = 1, eq. (8) reduces to the familiar result of ( - 8 / 3 ) n i . Since I is negative in the presence of a "repulsive" confinement potential, the system will not collapse. However, i f I 6 <13, the leading term in eq. (8) is 384
4 February 1982
proportional to n2(I 6 - •3), and is attractive. This might occur if color 3 pairs congregate to the interior of the sytem, while color 6 pairs are further apart with quarks nearer to the surface. It would be interesting to determine if the CE condensation of DeTar [6] and the color correlation found by Harvey [7] are of this origin. We may note parenthetically that it is easy to construct completely antisymmetric spatial states [ln]x. Examples having even panty and the lowest kinetic energies are sp 2 for n = 3, sp2(s, d) 3 for n = 6, sp2(s, d) 6 for n = 9, sp3(s, d)5(p, t) 3 for n = 12, and the singly occupied Fermi sea m quark matter. This shows that the total kinetic energy grows no faster than n 4/3 at large n for relativistic quarks. The dynamical situation in the MIT (Massachusetts Institute of Technology) bag model [2,8] differs somewhat from that in potential models. The normally positive energy due to the confinement potential is replaced by a volume energy in the bag whose energy density B is a universal positive constant in all hadrons. In the absence of CM contributions, the energy density of the singly occupied quark gas of massless quarks is [9,10] E/V=B
+(k4/8rr2)[1 - (1 + a ) % 2/37r1,
(9)
where a = 2 In 2 represents a relativistic correction [10] to the Coulomb exchange energy [11 ] of a color-singlet 0.e. color-neutral) quark gas, and k F = (6zr2n/V)l/3 .
(10)
Eq. (9) shows a CE instability when the effective color coupling constant a s exceeds the critical value of Orsc r i t =
37r/[2(1 + a)] = 2 .
(11)
Although this estimate is very rough and depends for its validity on the complete absence of CM and other contributions, its value is uncomfortably close to the empirical value of 2.2 used in the MIT bag model [8]. It would be interesting to study how the CM and other contributions might affect this result. It should also be noted that when the coupling constant is as strong a s Otfrlt, the result shown in eq. (9) may no longer be reahstic in that higher-order interaction effects neglected here may drastically alter the situation. (The same difficulty also appears for the strong couphng constant appearing in the MIT bag model. However, the situation IS not so critical
Volume 108B, number 6
PHYSICS LETTERS
when the interaction energy is repulsive.) This points out the possible danger of relying too literally on phenomenological models in dealing with states o f unusual permutation symmetries. The observations made m this note are very speculative in nature, and are valid only under the very special circumstances described here. In particular, there is no experimental evidence that such collapses are occurring in nature. They do suggest, however, that the associated color correlational effects might play significant roles in the dynamics o f multiquark systems. I would like to thank Dr. S.A. Chin for the help with eq. (9). This work is supported in part b y NSF Contract No. PHY78-15811.
4 February 1982
References [ 1 ] A. De Rfijula, H. Georgi and S.L. Glashow, Phys. Rev. D12 (1975) 147. [2] K. Johnson, Acta Phys. Pol. B6 (1975) 865. [3] D.A. Liberman, Phys. Rev. D16 (1977) 1542. [4] I.T. Obukhovsky, V.G. Neudatchln, Yu F. Smirnov and Yu M. Tchuvd'sky, Phys. Lett. 88B (1977) 231. [5] D.J. Jackson, Lectures on the new particles, Proc. 1976 Summer Institute on Particle Physics, Stanford Linear Accelerator Center, Report 198 (1976). [6] C. DeTar, Phys. Rev. D17 (1978) 323. [7] M. Harvey, Nucl. Phys. A352 (1981) 326. [8] T. DeGrand et al., Phys. Rev. D12 (1975) 2060. [9] G. Baym and S.A. Chin, Phys. Lett. 62B (1976) 241. [10] S.A. Chin, unpublished. [11] A.L. Fetters and J.D. Walecka, Quantum theory of many-particle systems (McGraw-Hill, New York, 1971) p. 29.
385