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QCD at finite chemical potential
Nuclear Physics B (Proc. Suppl.) 9 (1989) 347-350 North-Holland, Amsterdam
347
QCD AT FINITE CHEMICAL POTENTIAL Eduardo MENDEL * FB-8 Physik, Universit~t Oldenburg, 2900 Oldenburg, F.R. Germany The Baryon density in terms of the Chemical Potential # is calculated on a large 164 lattice aiming at weaker couplings with finer spacings. The fermionic determinant is also included in a new approximate scheme. The onset of the thermodynamic quantities with # seems to be still lower than at the nucleon mass.
I.
INTRODUCTION
The aim in this field of research is to be able to get a description of hadronic matter at finite baryon densities from first principles in QCD. Once one has obtained properly thermodynamic quantities such as the baryon density, the fermionic contribution to the energy density or the chiral condensate one can also look at hadronic quark distributions by measuring baryon current correlation functions. It is interesting to obtain these quantities for the normal phase and for the expected high density transition to deconfinement. Soon after implementing a chemical potential coupled to the baryon current on the lattice 1,2 which gave the proper continuum limit for the free Fermi gas 3 it was realized by several authors 1,4,5 that the onset with # of the thermodynamic quantities was earlier than at the expected m,~/3, as if controlled by a Goldstone mass m=/2. These first results, which have been interpreted by comparing the simulations to a free Fermi gas at some effective mass 5, were obtained on quite coarse lattices and in the quenched approximation. Several studies done by Dagotto et al. 6 on small lattices using dimer methods in the strong coupling limit of SU(N) or by Gibbs 7 exactly for the one dimensional U(1) case, showed that the determinant was important in these cases to obtain the expected results. The fact that the fermion determinant is complex at finite # makes it very hard to be included with usual Monte Carlo techniques 8,9 due to the probability weight interpretation. There have been attempts to include it with complex Langevin or as part of the operator expectation using Lanczos algorithm for the de*Humboldt Research Fellow 0920-5632/89/$03.50 (~ Elsevier Science Publishers B.V. (North-Itolland Physics Publishing Division)
terrninant 7,10 which show that the crucial phase 11 seems to fluctuate wildly, rendering it hard to be used as a weight for the gauge configurations. Recently Barbout et al. 12 have made progress by considering the Z3 symmetry to dampen the noise and fixing the particle number. They obtain results at strong coupling, compatible with mean field on yet small lattices. Contrasting to the emphasis on the determinant to explain the earlier than expected onset, we have argued 5 that one needs to consider fine enough lattices and that it is possible that as one tends to the weakcoupling scaling regime the determinant could get less and less important. (In the free limit it trivially cancels for any expectation value). We have shown that even for the free Fermi gas there are huge corrections to the thermodynamic quantities, corresponding to a distorted spectrum of states due to the coarseness of the lattice in the time direction. Therefore we decided to work on a quite fine lattice of 164 where the corrections are around 20 % and also considering an innovative approximation for the determinant which consists in an expansion starting down from saturation in the winding in T. Recently there has been an interesting work by Bili~ and Demeterfl 13 where they show that for the one dimensional SU(N) (in contrast to U(N)) for fine enough lattices the quenched and exact results converge to the same answer, in accordance to our previously exposed ideas. In the next section we present our first results at a coupling of 6.0 on a large lattice. In section 3 we introduce the new approximation for the determinant and discuss the results with its inclusion.
E. Mendel / QCD at finite chemical potential
348
2.
QUENCHED RESULTS ON A LARGE LATTICE
In an earlier paper5 we showed that the thermodynamic quantities as function of chemical potential/~ have strong distortions even for a free Fermi gas due to a2 effects and that it is very important to consider a fine enough lattice in the time direction. The distorted energy spectrum could create in this way unphysical states that could contribute to the baryon current so as to produce its early onset with #. To be able to interpret our results for the interacting case we developed the formalism to compare those results with the curves we obtain for the free Fermi gas on the lattice, at some effective mass comparable to the expected relevant masses like the nucleon one. Remember that the conserved baryon current does not renormalize and therefore can be looked at the quark level or at the nucleon level where the system can be seen as a weakly interacting gas of nucleons. Assuming a free Fermi gas could be dangerous at densities comparable with nuclear densities as we know from the difficulties encountered in statistical mechanics to treat a Fermi gas with an attractive interaction. With this precaution in mind, we will use the expression for the free Fermi gas density at a given lattice size N~ x N ~ and temperature 1/8 which can be written in terms of
< J > = 1/83 F ( # ~ , m ~ , N ~ , N I )
(2.1)
where the function gets independent of N~ and Ni as one tends to the continuum thermodynamic limit. In the interactive case we need to calculate expectation values of current correlation functions or of the current density itself:
J(~)
= ~ r , ( ~ ) (e~'2.u.,4x.+4 - e
X=+4U~',4)C.)
My previous quenched simulations5 had been done on a smaller 8 x (14 × 8 ~) lattice at a coupling of 5.7 and at a weaker 6.0. At coupling 5.7 we found, in accordance with other authors, that the thermodynamic quantities seemed to behave as if controlled by a mass m~/2 instead of the expected ran~3. On the other hand at coupling 6.0, which for this lattice size is almost on top of the finite temperature deconfinement transition, we found that the onset for the current with/~ was compatible with ran~3. This made us think that possibly as we would move to weaker couplings and finer lattices the determinant could become less important and also the distortions in the spectrum due to the coarseness of the lattice would diminish ( they were 100% for the free Fermi gas on a lattice of this size). We showed that it was crucial to have enough lattice points in the time direction to approach the continuum limit. g
16"-lattice free case curves m=o.o
/ / /
I/
A
c;7~ o
o,o
d. 1
d.2
d.3
0'.*
0.5
/z
Figure 1: Results for the density versus # and relevant free curves at masses 0, m~/2 and m J 3 .
(2.2)
After integrating over the fermion fields X we see that we have to invert a nonhermitian matrix M to obtain the propagators and also calculate its determinant. The method used to obtain the thousands of propagators needed, for a given field [7, to calculate the thermodynamic quantities and the current correlation functions is a very efficient combination of a one-step second order pseudofermion and a heatbath, which runs 5 times faster than usual but needs a big memory to store the matrix to be inverted, M M t.
Therefore this year we have started simulations on a 164 lattice for couplings of 6.0, 6.3 and 6.6. Up to now we have some results for a coupling 6.0 at two quark mass values. For the higher quark mass of .04, Fig.(1), the current density starts for small # compatible with the nucleon mass as obtained with other methods 14,15 (tuna = 1.0, a -1 = 1.7GeV and rn,~ = .48 ) and then, for a # possibly consistent with some phenomenological estimates for the expected high density transition to deconfinement it grows to a value in agreement with
E. Mendel/QCD at finite chemical potential
o
! :::1:/;
i
1..,°.,o.
"
| /.;'
free cose curves
m=O.O
/:;' ....
Xo -~o d
c.o
g.~
0'.3
d.2
o'4
0.5
/z
Figure 2: Same as Fig.(1) for a lower quark mass• a massless fermion as expected in the chiral symmetric phase. But this graph is clearly also consistent with the possibility for the jump to occur at m~/2. To be able to decide among these possibilities we lowered the quark mass so as to have a much lighter pion mass which could trigger earlier the onset. At the lower quark mass of .01 (for which m , = .73 and m~ = .24 ) the results, in Fig.(2), show again a density consistent to a massless fermion at high #, but at a # in the range from m , / 2 to m J 3 show high instability with a density clearly higher than the one corresponding to a free nucleon gas. At a # in this range, equals 0.2, out of six configurations calculated so far only one has a low density value close to the nucleon one. For this lighter quark mass the determinant is expected to play a greater role 13 as larger loops get enhanced and we will consider it in an approximate scheme in the next section. 3.
CONSIDERING THE DETERMINANT
We have been searching for a method that would tell us which configurations have a higher weight due to the Fermion determinant without having to calculate it fully. To first approximation we will assume that all configurations give the same value for the determinant, when we omit all the loops that wind around the time direction. This condition corresponds to our intuition that only the winding loops could make the difference
349
for the thermodynamic quantities, the contractable ones would contribute mainly to a coupling renormalization. Furthermore we will represent all the contractable loops by a geometrically averaged eigenvalue A so that it can be inserted to the appropriate power in an expansion of the determinant in the winding loops. The loops that wind around the lattice saturate the determinant when each of these loops is fixed in space and winds 3 times in time (in both senses). As soon as we allow some displacement in space we have to loose 3 windings somewhere else (using Z3 invariance) so that it seems reasonable to consider first this family of loops and then expand in loops that propagate more and more in space (loosing factors e3~" each time we have to break up positive winding loops). Barbour 12 is doing an expansion from the other end of few windings which seems to work well at strong coupling where the coefficients decay exponentially but for intermediate couplings they seem to remain constant for a while and only then decay. A nice feature of our expansion is that for the first family, of straight loops, the approximation to the determinant can be written in terms of the one dimensional problem, which in turn can be expressed in terms of traces of Polyakov loops and so is easy to calculate,
det-
1]-[(A3n+ITr(P3) ~
norm. ~-~
-~Tr(P2)Tr(P)
(3.1)
+ ~(Tr(P))3) .
Here A is the above mentioned average eigenvector, the normalization is taken from the one dimensional case and P is an expression in the Polyakov line L on a lattice with n time steps:
P -- polynomial(.)(2m) + e~nL + e-~"L t
(3.2)
This expression can be calculated for each configuration, with a fixed A chosen in the range of geometric averaged eigenvalues for the strong and weak coupling limits, and seems to be fairly stable in the complex plane when properly normalized. Including this determinant as part of the operator seems to improve the results presented in Fig.(2), by lowering the density at a # of 0.2 by a 10%. Note also that there seems to be a correlation between the configuration with a low density and a high weight obtained for it with this method.
350
E. Mendel / QCD at finite chemical potential
For a # of 0.3 which is above m , / 3 including this determinant increases the quenched results by a 10%. Several improvements can still be made for this approximation to the determinant like choosing ,k as average from the actual configurations or expanding to the next orders allowing then displacements in space of the loops and so interactions among them. Note also that when we average with this determinant over the three Z3 transformed configurations we get a stable answer with a very small imaginary piece.
4. CONCLUSIONS We have presented the results for a larger lattice at a coupling of 6.0 in the quenched case and by including a new expansion to first approximation for the determinant and find that the onset for the density seems to be still earlier than the one expected from a free Fermi gas (this comparison could be a wrong assumption), of effective mass m,/3. The determinant probably plays an important role at intermediate couplings and further study is needed to improve on the newly introduced scheme to estimate it. One discouraging fact is that there are very few configurations that give a low density consistent whith the nucleon onset and therefore if we consider the determinant as part of the operator, to weight each configuration, we will get very poor statistics. Therefore one should try to include the determinant to some approximation in the generation of the gauge configurations and here a fast scheme like the one introduced above could be very helpful. We will also compute the thermodynamic quantities for the configurations at weaker couplings ( 6.3 and 6.6) to check scaling and to study the behaviour at the higher temperatures. At those temperatures we also hope to see a good signal for the density correlation functions which for this low T was very weak. These further studies should help in the approach to describe properly finite baryon density systems from first principles in lattice QCD.
ACKNOWLEDGEMENTS I would like to thank E. Hilf for his support and the Humboldt Foundation. Also to J. Hoek and the Edinburgh group for the gauge configurations and to the GSI Computer Center for their excellent facilities. REFERENCES 1. J. Kogut, H. Matsuoka, M. Stone, H.W. Wyld, S. Shenker, J. Shigemitsu and D.K. Sinclair, Nucl. Phys. B225 (1983) 93. 2. P. Hasenfratz and F. Karsch, Phys. Lett. 125B
(1983) 308. 3. R.V. Gavai and A. Ostendorf, Phys. Lett. 132B (1983) 137; R.V. Gavai, Phys. Rev. D32 (1985) 519. 4. I. Barbour, N. Behilil, E. Dagotto, F. Karsch, A. Moreo, M. Stone and H.W. Wyld, Nucl. Phys. B275 (1986) 296. 5. E. Mendel, NucI. Phys. B4 (Proc. Suppl.) (1988) 308. 6. E. Dagotto, A. Moreo, and U. Wolff, Phys. Rev. Lett. 57 (1986) 1292. 7. P.E. Gibbs, Phys. Lett. B172 (1986) 53 and Phys. Lett. B182 (1986) 369. 8. J. Engels and H. Satz, Phys. Lett. 159B (1985) 151. 9. F. Karsch and H.W. Wyld, Phys. Rev. Lett. 55 (1985) 2242. 10. A. Vladikas, Nucl. Phys. B4 (Proc. Suppl.) (1988) 322. 11. A. Gocksch, San Diego Preprints, UCSD-PTH87/13, Aug. 1987 and UCSD-PTH-88/02, March
1988. 12. I. Barbour, C. Daviesand Z. Sabeur, U. of Glasgow Preprint, GUTPA/88/7-2, July 1988. 13. N. 8ili( and K. Demeterfi, Zagreb Preprint, RBITP-2-88, July 1988. 14. D. 8arkai, K. Moriarty and C. Rebbi, Phys. Lett.
156B (1985) 385. 15. K. Bowler, C. Chalmers, R. Kenway, G. Pawley and D. Roweth, Nucl. Phys. B284 (1987) 299.
347
QCD AT FINITE CHEMICAL POTENTIAL Eduardo MENDEL * FB-8 Physik, Universit~t Oldenburg, 2900 Oldenburg, F.R. Germany The Baryon density in terms of the Chemical Potential # is calculated on a large 164 lattice aiming at weaker couplings with finer spacings. The fermionic determinant is also included in a new approximate scheme. The onset of the thermodynamic quantities with # seems to be still lower than at the nucleon mass.
I.
INTRODUCTION
The aim in this field of research is to be able to get a description of hadronic matter at finite baryon densities from first principles in QCD. Once one has obtained properly thermodynamic quantities such as the baryon density, the fermionic contribution to the energy density or the chiral condensate one can also look at hadronic quark distributions by measuring baryon current correlation functions. It is interesting to obtain these quantities for the normal phase and for the expected high density transition to deconfinement. Soon after implementing a chemical potential coupled to the baryon current on the lattice 1,2 which gave the proper continuum limit for the free Fermi gas 3 it was realized by several authors 1,4,5 that the onset with # of the thermodynamic quantities was earlier than at the expected m,~/3, as if controlled by a Goldstone mass m=/2. These first results, which have been interpreted by comparing the simulations to a free Fermi gas at some effective mass 5, were obtained on quite coarse lattices and in the quenched approximation. Several studies done by Dagotto et al. 6 on small lattices using dimer methods in the strong coupling limit of SU(N) or by Gibbs 7 exactly for the one dimensional U(1) case, showed that the determinant was important in these cases to obtain the expected results. The fact that the fermion determinant is complex at finite # makes it very hard to be included with usual Monte Carlo techniques 8,9 due to the probability weight interpretation. There have been attempts to include it with complex Langevin or as part of the operator expectation using Lanczos algorithm for the de*Humboldt Research Fellow 0920-5632/89/$03.50 (~ Elsevier Science Publishers B.V. (North-Itolland Physics Publishing Division)
terrninant 7,10 which show that the crucial phase 11 seems to fluctuate wildly, rendering it hard to be used as a weight for the gauge configurations. Recently Barbout et al. 12 have made progress by considering the Z3 symmetry to dampen the noise and fixing the particle number. They obtain results at strong coupling, compatible with mean field on yet small lattices. Contrasting to the emphasis on the determinant to explain the earlier than expected onset, we have argued 5 that one needs to consider fine enough lattices and that it is possible that as one tends to the weakcoupling scaling regime the determinant could get less and less important. (In the free limit it trivially cancels for any expectation value). We have shown that even for the free Fermi gas there are huge corrections to the thermodynamic quantities, corresponding to a distorted spectrum of states due to the coarseness of the lattice in the time direction. Therefore we decided to work on a quite fine lattice of 164 where the corrections are around 20 % and also considering an innovative approximation for the determinant which consists in an expansion starting down from saturation in the winding in T. Recently there has been an interesting work by Bili~ and Demeterfl 13 where they show that for the one dimensional SU(N) (in contrast to U(N)) for fine enough lattices the quenched and exact results converge to the same answer, in accordance to our previously exposed ideas. In the next section we present our first results at a coupling of 6.0 on a large lattice. In section 3 we introduce the new approximation for the determinant and discuss the results with its inclusion.
E. Mendel / QCD at finite chemical potential
348
2.
QUENCHED RESULTS ON A LARGE LATTICE
In an earlier paper5 we showed that the thermodynamic quantities as function of chemical potential/~ have strong distortions even for a free Fermi gas due to a2 effects and that it is very important to consider a fine enough lattice in the time direction. The distorted energy spectrum could create in this way unphysical states that could contribute to the baryon current so as to produce its early onset with #. To be able to interpret our results for the interacting case we developed the formalism to compare those results with the curves we obtain for the free Fermi gas on the lattice, at some effective mass comparable to the expected relevant masses like the nucleon one. Remember that the conserved baryon current does not renormalize and therefore can be looked at the quark level or at the nucleon level where the system can be seen as a weakly interacting gas of nucleons. Assuming a free Fermi gas could be dangerous at densities comparable with nuclear densities as we know from the difficulties encountered in statistical mechanics to treat a Fermi gas with an attractive interaction. With this precaution in mind, we will use the expression for the free Fermi gas density at a given lattice size N~ x N ~ and temperature 1/8 which can be written in terms of
< J > = 1/83 F ( # ~ , m ~ , N ~ , N I )
(2.1)
where the function gets independent of N~ and Ni as one tends to the continuum thermodynamic limit. In the interactive case we need to calculate expectation values of current correlation functions or of the current density itself:
J(~)
= ~ r , ( ~ ) (e~'2.u.,4x.+4 - e
X=+4U~',4)C.)
My previous quenched simulations5 had been done on a smaller 8 x (14 × 8 ~) lattice at a coupling of 5.7 and at a weaker 6.0. At coupling 5.7 we found, in accordance with other authors, that the thermodynamic quantities seemed to behave as if controlled by a mass m~/2 instead of the expected ran~3. On the other hand at coupling 6.0, which for this lattice size is almost on top of the finite temperature deconfinement transition, we found that the onset for the current with/~ was compatible with ran~3. This made us think that possibly as we would move to weaker couplings and finer lattices the determinant could become less important and also the distortions in the spectrum due to the coarseness of the lattice would diminish ( they were 100% for the free Fermi gas on a lattice of this size). We showed that it was crucial to have enough lattice points in the time direction to approach the continuum limit. g
16"-lattice free case curves m=o.o
/ / /
I/
A
c;7~ o
o,o
d. 1
d.2
d.3
0'.*
0.5
/z
Figure 1: Results for the density versus # and relevant free curves at masses 0, m~/2 and m J 3 .
(2.2)
After integrating over the fermion fields X we see that we have to invert a nonhermitian matrix M to obtain the propagators and also calculate its determinant. The method used to obtain the thousands of propagators needed, for a given field [7, to calculate the thermodynamic quantities and the current correlation functions is a very efficient combination of a one-step second order pseudofermion and a heatbath, which runs 5 times faster than usual but needs a big memory to store the matrix to be inverted, M M t.
Therefore this year we have started simulations on a 164 lattice for couplings of 6.0, 6.3 and 6.6. Up to now we have some results for a coupling 6.0 at two quark mass values. For the higher quark mass of .04, Fig.(1), the current density starts for small # compatible with the nucleon mass as obtained with other methods 14,15 (tuna = 1.0, a -1 = 1.7GeV and rn,~ = .48 ) and then, for a # possibly consistent with some phenomenological estimates for the expected high density transition to deconfinement it grows to a value in agreement with
E. Mendel/QCD at finite chemical potential
o
! :::1:/;
i
1..,°.,o.
"
| /.;'
free cose curves
m=O.O
/:;' ....
Xo -~o d
c.o
g.~
0'.3
d.2
o'4
0.5
/z
Figure 2: Same as Fig.(1) for a lower quark mass• a massless fermion as expected in the chiral symmetric phase. But this graph is clearly also consistent with the possibility for the jump to occur at m~/2. To be able to decide among these possibilities we lowered the quark mass so as to have a much lighter pion mass which could trigger earlier the onset. At the lower quark mass of .01 (for which m , = .73 and m~ = .24 ) the results, in Fig.(2), show again a density consistent to a massless fermion at high #, but at a # in the range from m , / 2 to m J 3 show high instability with a density clearly higher than the one corresponding to a free nucleon gas. At a # in this range, equals 0.2, out of six configurations calculated so far only one has a low density value close to the nucleon one. For this lighter quark mass the determinant is expected to play a greater role 13 as larger loops get enhanced and we will consider it in an approximate scheme in the next section. 3.
CONSIDERING THE DETERMINANT
We have been searching for a method that would tell us which configurations have a higher weight due to the Fermion determinant without having to calculate it fully. To first approximation we will assume that all configurations give the same value for the determinant, when we omit all the loops that wind around the time direction. This condition corresponds to our intuition that only the winding loops could make the difference
349
for the thermodynamic quantities, the contractable ones would contribute mainly to a coupling renormalization. Furthermore we will represent all the contractable loops by a geometrically averaged eigenvalue A so that it can be inserted to the appropriate power in an expansion of the determinant in the winding loops. The loops that wind around the lattice saturate the determinant when each of these loops is fixed in space and winds 3 times in time (in both senses). As soon as we allow some displacement in space we have to loose 3 windings somewhere else (using Z3 invariance) so that it seems reasonable to consider first this family of loops and then expand in loops that propagate more and more in space (loosing factors e3~" each time we have to break up positive winding loops). Barbour 12 is doing an expansion from the other end of few windings which seems to work well at strong coupling where the coefficients decay exponentially but for intermediate couplings they seem to remain constant for a while and only then decay. A nice feature of our expansion is that for the first family, of straight loops, the approximation to the determinant can be written in terms of the one dimensional problem, which in turn can be expressed in terms of traces of Polyakov loops and so is easy to calculate,
det-
1]-[(A3n+ITr(P3) ~
norm. ~-~
-~Tr(P2)Tr(P)
(3.1)
+ ~(Tr(P))3) .
Here A is the above mentioned average eigenvector, the normalization is taken from the one dimensional case and P is an expression in the Polyakov line L on a lattice with n time steps:
P -- polynomial(.)(2m) + e~nL + e-~"L t
(3.2)
This expression can be calculated for each configuration, with a fixed A chosen in the range of geometric averaged eigenvalues for the strong and weak coupling limits, and seems to be fairly stable in the complex plane when properly normalized. Including this determinant as part of the operator seems to improve the results presented in Fig.(2), by lowering the density at a # of 0.2 by a 10%. Note also that there seems to be a correlation between the configuration with a low density and a high weight obtained for it with this method.
350
E. Mendel / QCD at finite chemical potential
For a # of 0.3 which is above m , / 3 including this determinant increases the quenched results by a 10%. Several improvements can still be made for this approximation to the determinant like choosing ,k as average from the actual configurations or expanding to the next orders allowing then displacements in space of the loops and so interactions among them. Note also that when we average with this determinant over the three Z3 transformed configurations we get a stable answer with a very small imaginary piece.
4. CONCLUSIONS We have presented the results for a larger lattice at a coupling of 6.0 in the quenched case and by including a new expansion to first approximation for the determinant and find that the onset for the density seems to be still earlier than the one expected from a free Fermi gas (this comparison could be a wrong assumption), of effective mass m,/3. The determinant probably plays an important role at intermediate couplings and further study is needed to improve on the newly introduced scheme to estimate it. One discouraging fact is that there are very few configurations that give a low density consistent whith the nucleon onset and therefore if we consider the determinant as part of the operator, to weight each configuration, we will get very poor statistics. Therefore one should try to include the determinant to some approximation in the generation of the gauge configurations and here a fast scheme like the one introduced above could be very helpful. We will also compute the thermodynamic quantities for the configurations at weaker couplings ( 6.3 and 6.6) to check scaling and to study the behaviour at the higher temperatures. At those temperatures we also hope to see a good signal for the density correlation functions which for this low T was very weak. These further studies should help in the approach to describe properly finite baryon density systems from first principles in lattice QCD.
ACKNOWLEDGEMENTS I would like to thank E. Hilf for his support and the Humboldt Foundation. Also to J. Hoek and the Edinburgh group for the gauge configurations and to the GSI Computer Center for their excellent facilities. REFERENCES 1. J. Kogut, H. Matsuoka, M. Stone, H.W. Wyld, S. Shenker, J. Shigemitsu and D.K. Sinclair, Nucl. Phys. B225 (1983) 93. 2. P. Hasenfratz and F. Karsch, Phys. Lett. 125B
(1983) 308. 3. R.V. Gavai and A. Ostendorf, Phys. Lett. 132B (1983) 137; R.V. Gavai, Phys. Rev. D32 (1985) 519. 4. I. Barbour, N. Behilil, E. Dagotto, F. Karsch, A. Moreo, M. Stone and H.W. Wyld, Nucl. Phys. B275 (1986) 296. 5. E. Mendel, NucI. Phys. B4 (Proc. Suppl.) (1988) 308. 6. E. Dagotto, A. Moreo, and U. Wolff, Phys. Rev. Lett. 57 (1986) 1292. 7. P.E. Gibbs, Phys. Lett. B172 (1986) 53 and Phys. Lett. B182 (1986) 369. 8. J. Engels and H. Satz, Phys. Lett. 159B (1985) 151. 9. F. Karsch and H.W. Wyld, Phys. Rev. Lett. 55 (1985) 2242. 10. A. Vladikas, Nucl. Phys. B4 (Proc. Suppl.) (1988) 322. 11. A. Gocksch, San Diego Preprints, UCSD-PTH87/13, Aug. 1987 and UCSD-PTH-88/02, March
1988. 12. I. Barbour, C. Daviesand Z. Sabeur, U. of Glasgow Preprint, GUTPA/88/7-2, July 1988. 13. N. 8ili( and K. Demeterfi, Zagreb Preprint, RBITP-2-88, July 1988. 14. D. 8arkai, K. Moriarty and C. Rebbi, Phys. Lett.
156B (1985) 385. 15. K. Bowler, C. Chalmers, R. Kenway, G. Pawley and D. Roweth, Nucl. Phys. B284 (1987) 299.