Abstract
An impact of the finite size effects on the vacuum free energy density of full QCD with $N_{\rm f}$ massless flavors in the presence of homogeneous (anti-)self-dual Abelian background gluon field is studied. The zero temperature free energy density of the four-dimensional spherical domain is computed as a function of the background field strength $B$ and domain radius $R$. Calculation is performed in the one-loop approximation improved by accounting for mixing of the quark and gluon quasi-zero modes with normal modes, with the use of the $\zeta$-function regularization. It is indicated that, under plausible assumption on the character of the mixing, the quantum correction to the free energy density has a minimum as a function of $B$ and $R$. Within the mean field approach to QCD vacuum based on domain wall network representation of the mean field, an existence of the minimum may prevent infinite growth of individual domain, thus protecting the vacuum from the long-range ordering, and, hence, serving as the origin of disorder in the statistical ensemble of domain wall networks, driven by the minimization of the overall free energy of the dominant gauge field configurations.
Highlights
It is generally accepted that the physical QCD vacuum can be characterized by various gluon, quark, and mixed condensates
The condensates have played an important role in understanding the basic features of hadron physics
The lowest-dimension condensates hg2F2i, hðg2F FÞ2i, and hψψi are relevant to the anomalous breakdown of scale and UAð1Þ symmetries, and the spontaneous breaking of chiral SULðNfÞ × SURðNfÞ symmetry
Summary
It is generally accepted that the physical QCD vacuum can be characterized by various gluon, quark, and mixed condensates. The picture of the QCD vacuum based on the Abelian (anti-)self-dual mean field turned out to be convenient for exposing a catalyzing impact of a strong electromagnetic field on quark deconfinement [14,18,29,30,31] On the whole, these rather satisfactory phenomenological applications inspire the task of clearing up the mechanism behind the balance between the competitive tendencies for a long-range order in the ground state and disorder that may originate from two complementary origins: the topologically stable defects in the background field and the existence of a minimum of the effective action with respect to the size of the regions of homogeneity (domain size). The main difficulty is combination of finite size and background field, and their simultaneous treatment is the result by itself
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