The impact of the finite-size effects on the vacuum free energy density of full QCD with $N_f$ massless flavors in the presence of a 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 $R$. The calculation is performed in the one-loop approximation improved by accounting for mixing of the quark and gluon quasizero modes with normal modes, with the use of the $\zeta$-function regularization. It is indicated that, under plausible assumptions 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 the QCD vacuum based on the domain wall network representation of the mean field, the existence of the minimum may prevent infinite growth of individual domains, 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.

中文翻译：

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平均场 QCD 中的能量驱动无序

有限尺寸效应对全 QCD 真空自由能密度的影响 $N_F$研究了在均质（反）自双阿贝尔背景胶子场存在下的无质量风味。四维球面域的零温度自由能密度计算为背景场强的函数$乙$ 和域半径 $\mathrm{电阻}$. 该计算是在通过考虑夸克和胶子准塞罗模式与正常模式的混合而改进的单循环近似中进行的，使用$\zeta$- 函数正则化。这表明，在对混合特征的合理假设下，自由能密度的量子校正具有最小值，作为$乙$ 和 $\mathrm{电阻}$. 在基于平均场的畴壁网络表示的 QCD 真空的平均场方法中，最小值的存在可以防止单个域的无限增长，从而保护真空免受远程排序，因此作为畴壁网络的统计集合中无序的起源，由主导规范场配置的整体自由能的最小化驱动。