quantum field theory - Goldstone bosons, quark and gluon masses counting in color-flavor locking QCD

Consider QCD, with three flavors of massless quarks, we like to focus on the possible Cooper paired phases.

For 3 quarks $(u,c,d)$ and 3 colors $(r,g,b)$, the Cooper pairs cannot be flavor singlets, and both color and flavor symmetries are broken. The attractive channel favored by 1-gluon exchange is known as “color-flavor locking.” A condensate involving left-handed quarks alone locks $SU(3)_L$ flavor rotations to $SU(3)_{color}$, in the sense that the condensate is not symmetric under either alone, but is symmetric under the simultaneous $SU(3)_{L+color}$ rotations. A condensate involving right-handed quarks alone locks $SU(3)_R$ flavor rotations to $SU(3)_{color}$. Because color is vectorial, the result is to breaking chiral symmetry. Thus, in quark matter with three massless quarks, the $SU(3)_{color} \times SU(3)_L \times SU(3)_R \times U(1)_B$ (the last one is baryon) symmetry is broken down to the global diagonal $SU(3)_{color+L+R}$ group.

question:

1) How many quarks among nine ($(u,c,d) \times (r,g,b)$) have a dynamical energy gap? What are they?

2) How many among the eight gluons get a mass? What are they?

3) How many massless Nambu-Goldstone bosons there are? What are they? How to describe them?

Answer

These questions are answered in the original literature:

1) All quarks are gapped. The nine quarks arrange themselves into an octet with gap $\Delta$ and a singlet with gap $2\Delta$.

2) All gluons are gapped.

3) There is an octet of Goldstone bosons related to chiral symmetry breaking, and a singlet associated with $U(1)$ breaking.

Postscript:

i) When pair condensates form there is a gap in the excitation spectrum of single quarks (this is just regular BCS). However, the gapped excitations may be linear combinations of the microscopic quark fields. In the present case the nine types of quark fields ($N_c\times N_f=9$), form an octet and a singlet of an unbroken $SU(3)$ color-flavor symmetry.

ii) Pair condensation and the formation of a gap take place near the Fermi surface. There is no Fermi surface for anti-quarks (if $\mu$ is positive and large), and therefore no pairing and no gaps.

iii) There is both a $U(1)$ GB (associated with the broken $U(1)_B$) and a masssless $U(1)$ gauge boson (associated with the $U(1)_{Q}$ gauge symmetry that is not Higgsed).

iv) The [8] GB correspond to spontaneous breaking of chiral symmetry. In ordinary QCD these would be quark-anti-quark states, but at high density anti-quarks decouple. A detailed analysis shows that the GBs are predominantly 2-particle-2-hole states, $(qq)(\bar{q}\bar{q})$.