Heavy-tailed chiral random matrix theory

Takuya Kanazawa

doi:10.1007/jhep05(2016)166

Abstract

We study an unconventional chiral random matrix model with a heavy-tailed probabilistic weight. The model is shown to exhibit chiral symmetry breaking with no bilinear condensate, in analogy to the Stern phase of QCD. We solve the model analytically and obtain the microscopic spectral density and the smallest eigenvalue distribution for an arbitrary number of flavors and arbitrary quark masses. Exotic behaviors such as non-decoupling of heavy flavors and a power-law tail of the smallest eigenvalue distribution are illustrated.

Highlights

Applications of chRMT to QCD have been mostly limited to the hadronic phase with ψψ = 0
One can imagine a situation where a chiral condensate is forbidden by an anomaly-free discrete subgroup of U(1)A and the spontaneous breaking SU(Nf )R × SU(Nf )L → SU(Nf )V is driven by a quartic condensate. (This pattern of symmetry breaking was studied by Dashen long time ago [42].) While this exotic phase that we call the Stern phase is ruled out by rigorous QCD inequalities at vanishing baryon density [43], there are arguments in favor of the Stern phase at finite density
We show that our chRMT with N × N random matrices reproduces, in the large-N limit, the finite-volume partition function of the Stern phase with K > 4 in the ε-regime. (Here we label the Stern phase with an index K that specifies the unbroken subgroup of U(1)A [53].) This implies that all infinitely many sum rules for the Dirac eigenvalues in the Stern phase [48, 53] are obeyed by microscopic eigenvalues of random matrices in this chRMT

Summary

Definition of the model and the large-N limit

Where X is a complex N × N random matrix and dX denotes the flat Cartesian measure. This integral converges for arbitrary N ≥ 1 and Nf ≥ 0. The weight (2.1) has three interesting properties: (i) it is invariant under unitary rotations X → V1XV2 for V1,2 ∈ U(N ), (ii) the matrix elements are statistically correlated, and (iii) the distribution is heavy tailed, i.e., it does not decay exponentially for large matrix elements This random matrix ensemble can be seen as an unquenched generalization of previous RMTs [29, 30, 32, 34, 35, 37] that had a heavy-tailed weight similar to (2.1) but with no determinants. Spontaneous symmetry breaking driven by higher-order condensates It follows from the coincidence of mass dependence between chRMT and QCD that infinitely many spectral sum rules of the Dirac operator in the Stern phase [48, 53] can be reproduced exactly from chRMT. A similar observation was made in chRMT for Wilson fermions [56]

Microscopic spectral density

Chiral limit

Large-mass limit

Smallest eigenvalue distribution

Conclusions and outlook