# Aspects of the QCD *θ*-vacuum

## Abstract

This paper addresses two aspects concerning the *θ*-vacuum of Quantum Chro-modynamics. First, large-*N*_{c} chiral perturbation theory is used to calculate the first two non-trivial cumulants of the distribution of the winding number, i.e. the topological susceptibility, *χ*_{top}, and the fourth cumulant, *c*_{4}, up to next-to-leading order. Their large-*N*_{c} scaling is discussed, and compared to lattice results. It is found that \( {\chi}_{\mathrm{top}} = \mathcal{O}\left({N}_c^0\right) \), as known before, and \( {c}_4 = \mathcal{O}\left({N}_c^{-3}\right) \), correcting the assumption of \( \mathcal{O}\left({N}_c^{-2}\right) \) in the literature. Second, we discuss the properties of QCD at *θ* ∼ *π* using chiral perturbation theory for the case of 2 + 1 light flavors, i.e. by taking the strange quark mass heavier than the degenerate up and down quark masses. It is shown that — in accordance with previous findings for *N*_{f} = 2 and *N*_{f} = 3 mass-degenerate flavors — in the region *θ* ∼ *π* two vacuum states coexist, which become degenerate at *θ* = *π*. The wall tension of the energy barrier between these degenerate vacua is determined as well as the decay rate of a false vacuum.

## Keywords

Chiral Lagrangians Effective Field Theories 1/N Expansion## Notes

### **Open Access**

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