The Silver Blaze Problem in QCD

Abstract

This article provides a pedagogical introduction to the Silver Blaze problem. This problem refers to the difficulty of reconciling to perspectives on QCD with a chemical potential. The first is the phenomenological fact that at $T=0$ QCD remains in its ground state -- the vacuum -- with all physical observables unchanged whenever the magnitude of a chemical potential is less than some critical value. The second is the fact that in functional integral treatments, the inclusion of any nonzero chemical potential changes all eigenvalues of the Dirac operator for every gauge configuration, leading to a natural expectation that the functional determinants also changes, which leads to the expectation that physical observables should be altered. The problem amounts to explaining why nothing happens below the critical chemical potential. By focusing on the eigenvalues of $\gamma_0$ times the Dirac operator rather than the Dirac operator itself, it is possible to show that for QCD with two flavors and identical quark masses, an isospin chemical potential with a magnitude less than $m_\pi$ (and no baryon chemical potential), or a baryon chemical potential of less than $\frac{3}{2} m_\pi$ (and no isospin chemical potential), the functional integerals at $T=0$ themselves remain unchanged in all configurations that contribute to the functional integral with non-vanishing weight. However, for $\mu_{\rm crit}\mu_B > \frac{3}{2} m_\pi$, the Silver Blaze phenomenon arises due to functional determinants having nontrivial phases that lead to cancellations between different gauge configurations. The mechanism leading to such cancellations remains unknown.