Physics – High Energy Physics – High Energy Physics - Lattice
Scientific paper
2010-11-23
PoS Lattice2010:092,2010
Physics
High Energy Physics
High Energy Physics - Lattice
7 pages, 4 figures. Talk presented at the XXVIII International Symposium on Lattice Field Theory, Lattice2010, Villasimius, It
Scientific paper
The QCD partition function for the Wilson Dirac operator, $D_W$, at nonzero lattice spacing $a$ can be expressed in terms of a chiral Lagrangian as a systematic expansion in the quark mass, the momentum and $a^2$. Starting from this chiral Lagrangian we obtain an analytical expression for the spectral density of $\gamma_5 (D_W+m)$ in the microscopic domain. It is shown that the $\gamma_5$-Hermiticity of the Dirac operator necessarily leads to a coefficient of the $a^2$ term that is consistent with the existence of an Aoki phase. The transition to the Aoki phase is explained, and the interplay of the index of $D_W$ and nonzero $a$ is discussed. We formulate a random matrix theory for the Wilson Dirac operator with index $\nu$ (which, in the continuum limit, becomes equal to the topological charge of gauge field configurations). It is shown by an explicit calculation that this random matrix theory reproduces the $a^2$-dependence of the chiral Lagrangian in the microscopic domain, and that the sign of the $a^2$-term is directly related to the $\gamma_5$-Hermiticity of $D_W$.
Akemann Gernot
Damgaard Poul H.
Splittorff Kim
Verbaarschot Jacobus J. M.
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