Theoretical aspects of hybrid chiral bag models

In hybrid chiral bag models (HCBM's) the quarks are the source for the pion field outside the bag. If we want to solve this model with a classical external soliton solution and quantized fermions, it is necessary to evaluate the vacuum expectation values (VEV's) of those operators that contain fermion fields and appear in the boundary conditions. When the external solution is the so-called hedgehog solution, $\overrightarrow{\pi }(\overrightarrow{\mathrm{r}},t)=f_{\pi }\theta \mathrm{}\left(r\right)\mathrm{}\hat{r}$, the relevant VEV is $\frac{i}{16\pi }〈0\left|\oint \int d^{2}s[\bar{\psi },(\overrightarrow{\tau }·\hat{r}\left){\gamma }_5\exp \right(i\overrightarrow{\tau }·\hat{r}{\gamma }_5\theta \left)\psi \right]\right|0〉=\frac{sin2\theta }{16\pi \eta }+\frac{C_0\left(\theta \right)}{R},$ where $\eta$ is a cutoff parameter ($\eta \to 0$). To obtain this result we have used a multiple-reflection expansion of the Green's function, while $C_0\left(\theta \right)$ is evaluated numerically. We discuss the infinite contribution in the above VEV, and show that $\frac{4\pi C_0\left(\theta \right)}{R}$ is precisely the derivative of the Casimir energy with respect to $\theta$. We also discuss some solutions of the HCBM for bag radii varying from $0 \mathrm{to} \infty$.