Pion () and scalar diquark (qq) masses as functions of . The straight line is mass = m — 2. The arrow is at = m/2

Donald Sinclair, Argonne National Laboratory

Research Objectives
At high baryon number density, QCD is believed to enter a phase with a diquark condensate. Since no practical method is known to simulate QCD at finite chemical potential for baryon/quark number, we are studying a simple model — QCD where the color group is SU(2) rather than SU(3) — at finite quark number chemical potential (). We will simulate true (SU(3)) QCD with two flavors at finite isospin density. A charged pion condensate is expected to form for large enough chemical potential, giving rise to spontaneous breakdown of isospin and Goldstone bosons.

Computational Approach
To measure diquark condensation, we include a small diquark source term, as well as the standard Dirac mass term. This model has a real positive fermion pfaffian, allowing us to simulate it by standard techniques. We perform simulations to determine the phase structure of this model with four flavors of light dynamical quarks. We measure the diquark and chiral condensates and the quark number density on a small (8⁴) lattice to map the phase diagram. This is repeated on a larger (12³ X 24) lattice, where we also measure the spectrum of Goldstone bosons in each phase. Upon completion of these simulations at a relatively large quark mass, we will move to a smaller quark mass with its more complex spectrum of Goldstone and pseudo-Goldstone bosons.

We use standard hybrid molecular-dynamics simulations with gaussian noise for the (pseudo) fermions, allowing us to "tune" the number of flavors of staggered fermions. This works since the Dirac matrix with Majorana and Dirac masses is positive definite and the pfaffian is always positive. We use the Verlet method, modified to keep the errors O(dt²) in the presence of the noisy fermions, to discretize molecular dynamic time. We perform the required inversion of the Dirac matrix at each update using the standard conjugate gradient algorithm.

Accomplishments
Our preliminary simulations have identified a phase with a diquark condensate, as has recently been suggested for QCD. It does, however, have differences. At zero chemical potential its "baryon," a two-quark state, is in the same multiplet as the pion and is a Goldstone boson in the chiral limit. In addition, the observed diquark condensate is a color singlet, and gives rise to a true Goldstone mode. In the QCD case, any diquark condensate has color, and thus the "broken" symmetry would be expected to be realized as a Higgs phenomenon.

Significance
Studies of QCD at finite baryon/quark number density (nuclear matter) have potential relevance to the physics of neutron stars. Studies of QCD at finite density and finite temperature are relevant to the physics of the early Universe. In addition, they are expected to be relevant to the physics of relativistic heavy ion collisions, which will soon be observed at RHIC, and which are already being observed at CERN, yielding preliminary evidence for a quark-gluon plasma. Finally, they would greatly enhance our knowledge of the structure of QCD.

QCD at finite chemical potential for isospin maps one surface of the phase diagram for nuclear matter (nuclei and neutron stars). It should exhibit spontaneous breakdown of isospin symmetry with a charged pion condensate and a true Goldstone boson. Our simulations will probe this behavior. Some of these properties are expected to survive to finite chemical potential for baryon number: nuclear matter has both finite isospin density and finite baryon number density.

Publications
S. J. Hands, J. B. Kogut, S. E. Morrison, and D. K. Sinclair, "Two-color QCD at finite fundamental quark-number density and related theories," Argonne National Laboratory technical report No. ANL-HEP-CP-00-110; hep-lat/0010028 (2000).

J.-F. Lagaë and D. K. Sinclair, "High temperature meson propagators with domain-wall quarks," Nucl. Phys. B (Proc. Suppl.) 83—84, 405 (2000).

J.-F. Lagaë and D. K. Sinclair, "Domain wall fermions at finite temperature," Nucl. Phys. B (Proc. Suppl.) 73, 450 (1999).

http://www.hep.anl.gov/dks/NERSC2000/