Spectral methods for coupled channels with a mass gap

We develop a method to compute the vacuum polarization energy for coupled scalar fields with different masses scattering off a background potential in one space dimension. As an example we consider the vacuum polarization energy of a kinklike soliton built from two real scalar fields with different mass parameters.

Figure 1

The left and middle panels show the Jost determinants in the symmetric and antisymmetric channels for the potential, Eq. (15) with $v_{11}=4.0$, $v_{22}=1$, $v_{12}=0.5$ and $w=2.0$. The mass parameters are $m_1=1$ and $m_2=2$. The right panel shows their phases for the same set of parameters.

Figure 2

Same as Fig. 1 with $V_0\to -V_0$.

Figure 3

Real and imaginary parts and phases of the Jost determinants, Eq. (14), for $v_{11}=v_{12}=0$, $v_{22}=-2$ and $w=2$.

Figure 4

The real Jost determinant for $k=\mathrm{i}t$. The left and right panels corresponds to the parameters of Figs. 1 and 2, respectively. Note the different scales for the vertical axes.

Figure 5

Contour integrals $I_{\det }(\alpha )$ and $I_{\log }(\alpha )$ defined in Eq. (a2) for the model parameters of Fig. 2. The parameters for the line integral, Eq. (a1), are $k_0=1.2\mathrm{i}$ and $R=1$ (left and middle panel); $k_0=0.3+1.4\mathrm{i}$ and $R=0.6$ (right panel).