Random Matrix Model of QCD at Finite Density and the Nature of the Quenched Limit

We use a random matrix model to study chiral symmetry breaking in QCD at finite chemical potential $\mu$. We solve the model and compute the eigenvalue density of the Dirac matrix on a complex plane. A naive “replica trick” fails for $\mu \ne 0$; we find that quenched QCD is not a simple $n\to 0$ limit of QCD with $n$ quarks. It is the limit of a theory with $2n$ quarks: $n$ quarks with original action and $n$ quarks with conjugate action. The results agree with earlier studies of lattice QCD at $\mu \ne 0$ and provide a simple analytical explanation of a long-standing puzzle.