BLED WORKSHOPS
IN PHYSICS
VOL. 1, NO. 1
Proceedins of the Mini-Workshop
Few-Quark Problems (p. 12)
Bled, Slovenia, July 8-15, 2000
Nucleons or diquarks: Competition between
clustering and color superconductivity in quark
matter
Michael C. Birse?
Theoretical Physics Group, Department of Physics and Astronomy, University of
Manchester, Manchester M13 9PL, United Kingdom
Baryonic matter at high densities could be very different frommatter in ordinary
nuclei: manifest chiral symmetry might be restored, baryons might dissolve into
a fluid of free quarks, and quark pairs might condense to form a colour supercon-
ductor. Do these changes occur at sharp phase transitions or smooth cross-overs?
How many phase transitions are there? At what densities do they occur? So far
lattice QCD has provided no answers to these questions and we must rely on
models to explore the possible behaviour of strongly interacting matter at high
densities.
The possibility of colour superconductivity has recently attracted consider-
able attention following suggestions byWilczek, Shuryak and others that strongly
attractive quark-quark forces could lead to such a state with a large gap, of the
order of 100 MeV. (For reviews of this idea, see [1,2].) In recent work [3], we have
examined the competition between this pairing of quarks and the three-quark
clustering responsible for forming baryons at low densities. We have used a gen-
eralized Nambu–Jona-Lasinio model, which provides a quark-quark interaction
which is similar to that in the instanton-liquid model often used in studies of
colour superconductivity. Since there is no simple analogue of the BCS state for
composite fermions, we have gone back to a Cooper-type treatment, looking for
the instabilities of a Fermi gas of quarks.
The model includes interactions that generate bound diquarks states in both
scalar and axial diquark channels. The energies of these are found by solving
the quark-quark Bethe-Salpeter equations, which are straightforward algebraic
equations in this kind of model. The relativistic Faddeev equations for the nu-
cleon and 
 are constructed using the methods of Refs. [4,5] and, in particular,
[6]. These are integral equations which we solve iteratively using the method of
Malfleit and Tjon [7].
A sharp 3-momentum cut-off of about 600 MeV was used to regulate the
contact interaction since this is easy to combine with the effects of the Fermi sea.
However it should be noted that this choice is not covariant. The couplings in
the scalar and axial diquark channels are chosen to give the observed N and 

masses in vacuum. For a parameter set which gives a quark mass of 450 MeV, the
scalar and axial diquarks have masses of 635 MeV and 700 MeV.? E-mail: mike.birse@man.ac.uk
Competition between clustering and color superconductivity... 13
At finite density, the quark mass is reduced and vanishes at the chiral phase
transition, which occurs at a Fermi momentum of about 370 MeV for the parame-
ter sets we use. (For comparison, nuclear-matter density corresponds to kF = 270
MeV.) In addition, Pauli blocking provides a lower cut-off on the momenta of the
quarks in the Bethe-Salpeter and Faddeev equations.
We find that the nucleon remains bound with respect to the quark-diquark
threshold only up to nuclear matter density. Morover, except for densities below
about a quarter of that of nuclear matter, we find that it is energetically much
more favorable to form three diquarks rather than two nucleons. Hence inmodels
of this type, quark matter is more unstable against pairing (leading to a colour
superconductor) than it is against three-quark clustering, even at the density of
nuclear matter.
Clearly something important is missing from models of this type: confine-
ment. We need to examine whether extending the model to include confinement
at low densities also affects behaviour at high densities. Indeed, is it possible to
get a realistic phase diagram without confinement?
References
1. K. Rajagopal, Nucl. Phys. A661 (1999) 150 [hep-ph/9908360].
2. T. Schäfer, nucl-th/9911017.
3. S. Pepin, M. C. Birse, J. A. McGovern and N. R. Walet, Phys. Rev. C61 (2000) 055209
[hep-ph/9912475].
4. A. Buck, R. Alkofer and H. Reinhardt, Phys. Lett. B 286, 29 (1992).
5. S. Huang and J. Tjon, Phys. Rev. C 49, 1702 (1994).
6. N. Ishii, W. Bentz and K. Yazaki, Nucl. Phys. A587, 617 (1995).
7. R. A. Malfliet and J. A. Tjon, Nucl. Phys. A127, 161 (1969).