Eigenvalue correlations in QCD with a chemical potential

Abstract

We discuss a new Random Matrix Model for QCD with a chemical potential that is based on the symmetries of the Dirac operator and can be solved exactly for all eigenvalue correlations for any number of flavors. In the microscopic limit of small energy levels the results should be an accurate description of QCD. This new model can also be scaled so that all physical observables remain at their µ = 0 values until a first order chiral restoration transition is reached. This gives a more realistic model for the QCD phase diagram than previous RMM. We also mention how the model might aid in determining the phase diagram of QCD from future numerical simulations.It is well known that at zero chemical potential the low energy modes of QCD can be described by a chiral effective Lagrangian(1) In a finite volume and at low energy when 1/m π ≫ L (also known as the ǫ-regime) the kinetic term can be ignored and the theory reduces to simply an integration over the zero momentum part of the SU(N f ) matrix U that parameterizes the Goldstone manifold. The results then do not depend on the dynamical details of QCD but rely solely on the symmetries. In this limit the chiral effective theory can also be expressed as a Random Matrix Theory (RMT) with the appropriate chiral symmetry [1]. The partition function is given in terms of an integral over all (with a Dirac matrixThe choice of a complex matrix corresponds to a SU(N c ≥ 3) gauge field with ν the topological charge. In the microscopic limit of N going to infinity with N m held fixed the chiral RMT is equivalent to the static part of the chiral effective Lagrangian. The RMT form of the effective theory, however, is better suited for studying the eigenvalues of the Dirac operator.Here we want to examine the extension of the chiral RMT to include a baryon chemical potential. In this case a Hermitian term µγ 0 is added to the anti-Hermitian / D. The eigenvalues of the Dirac operator become spread out in the complex plane and this gives rise to a sign problem that hinders numerical investigations.One model considered previously by Stephanov [2] used the Dirac matrixwhere the chemical potential µ is multiplied by an identity matrix. This has the correct symmetries but the fixed form of the chemical potential term makes it very difficult to calculate the eigenvalues analytically. Also it has been shown that this model has an unphysical µ dependence when it is studied at larger µ [3].A new model for QCD with a chemical potential was introduced recently [4]. This uses a different form for the Dirac matrixwhere the chemical potential part is now also modeled with a Gaussian random matrix B. This model still has the correct symmetries and can be solved for all eigenvalue correlations analytically. Additionally we can remove the unphysical µ dependence at zero temperature giving a more physical model for chiral symmetry breaking with a chemical potential.

Partially quenched QCD with a chemical potential

“…For nonzero temperature this will not be the case and one can leave the functions a and b independent. As was mentioned previously [14], this could possibly be used to help map out the QCD phase diagram. A nonlinear sigma model is obtained by using a Hubbard-Stratonovitch transformation to break apart the four fermion terms and then integrate out the Grassmann variables.…”

“…For nonzero temperature this will not be the case and one can leave the functions a and b independent. As was mentioned previously [14], this could possibly be used to help map out the QCD phase diagram.…”

Mixing of orthogonal and skew-orthogonal polynomials and its relation to Wilson RMT

“…In the last decade, a second approach was pursued. A second chiral random matrix was introduced yielding the chiral analog of the Ginibre ensembles [8][9][10][11][12][13][14][15]. A quantitative analysis of the sign problem in Monte Carlo simulations was quite elusive until it was solved in χ RMT [6,7,16,17].…”