Why do we expect quark matter to be a color superconductor?

Superconductivity of electrons and quarks

The reason why we expect quarks to behave like electrons, forming Cooper pairs which produce a superconducting state, is that the mechanism that causes the pairing is very general. All you need is a high density of Fermions (quarks and electrons are both Fermions) with an attractive interaction. Let us see how this works.

Fermions in general

The Fermi sea for free fermions

Fermi sea diagram
Fermions are particles that obey the Pauli exclusion principle, which says that no two fermions can be in the same state. So as you add more and more fermions to a finite-sized box, you have to put them in higher and higher momentum states. For non-interacting fermions at zero temperature you would just end up with a Fermi sea of filled states: all states with energy less than the Fermi energy Ef = µ are filled, and all states above Ef are empty.

(The filled negative energy states form the Dirac sea: removing a particle from the Dirac sea creates an antiparticle. The filled positive energy states form the Fermi sea: removing a particle from the Fermi sea creates a hole.)

But if there is an attractive interaction between the fermions then things are very different. The fermions near the Fermi surface pair up to form lots of Cooper pairs, which settle down in a "condensate". This state, a condensate of Cooper pairs, always forms because you can show that it has lower free energy than the simple Fermi sea depicted above. This was first explained by Bardeen, Cooper, and Schrieffer (BCS).

It is actually quite easy to understand intuitively why a condensate of Cooper pairs forms. The system tries to minimize its "free energy" F = E - µN, where E is the total energy of the system, µ is the chemical potential for quarks, and N is the number of fermions. The Fermi surface is defined by the Fermi energy Ef = µ, at which, if we ignore the attractive interaction, the free energy is minimized, so adding or subtracting a single particle costs zero free energy. For example, adding a particle costs energy Ef because that is the lowest unoccupied state, but it increases fermion number N by 1, so F is unchanged. Now switch on a weak attractive interaction. It still costs no free energy to add a pair of particles (or holes) close to the Fermi surface, but the attractive interaction between them then lowers the free energy of the system. Many such pairs will therefore be created in the modes near the Fermi surface, and these pairs, being bosonic, will form a condensate. The ground state will be a superposition of states with all numbers of pairs, breaking the fermion number symmetry.

Superconductivity of electrons

In the case of electrons, their dominant interaction is electrostatic repulsion, and it is only the presence of a background lattice of positively charged ions in a metal that allows additional attractive phonon-mediated interactions to exist. The resultant Cooper pairing is rather fragile, and easily disrupted by thermal fluctuations, hence metals only become superconducting at very low temperatures. The condensate of Cooper pairs of electrons is charged, and as a result the photon, which couples to electric charge, becomes massive. Superconducting metals therefore contain neither electric nor magnetic fields. A perfect conductor cannot contain electric fields (the charges would rearrange themselves to cancel it), but the special thing about a superconductor is that it expels magnetic fields as well: the Meissner effect.

Color Superconductivity of quarks

For quarks things are very different. The dominant interaction between quarks is the strong interaction, described by QCD, which is very attractive in some channels (after all, QCD binds quarks together to form baryons). This leads us to expect that quarks will form Cooper pairs very readily and that quark matter will generically acquire a condensate of Cooper pairs. Since pairs of quarks cannot be color-neutral, the resulting condensate will break the local color symmetry, making the gluons massive. We call this color superconductivity. Note that the quark pairs play the same role here as the Higgs particle does in the standard model: the color-superconducting phase can be thought of as the Higgs phase of QCD.

Color superconducting phases

Color superconducting quark matter can come in a rich multiplicity of different possible phases, based on different pairing patterns of the quarks. This is possible because quarks come in three different colors, and at the density of a compact star core we expect three different flavors: up, down, and strange. Recent work has concentrated on calculating which type of pairing is favored at which density. This is a complicated problem, in which we must take into account the requirement that bulk matter be neutral with respect to both electric and color charge, as well as equilibration under the weak interaction processes that can turn one quark flavor into another, and finally the strange quark mass. The results so far, starting at the highest densities and working down, are roughly this:

So we know what phase is favored in the limit of infinite density, but the nature of the pairing in quark matter at realistic neutron-star densities is still a vigorously debated question.

For more detailed information, try these review articles.

The phase diagram of QCD, by S. Hands

Color superconductivity in dense quark matter by M. Alford, K. Rajagopal, T. Schaefer, A. Schmitt
Color superconducting quark matter, by M. Alford

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