all loops have center of gravity at the origin, the expression for the geometry-dependent regularized Casimir energy then is: $$\Gamma_\Lambda = -\frac{1}{(4\pi)^2} \int_{1/\Lambda}^\infty \frac{ds}{s^5} \langle\Theta_V(sy,x)\rangle_y$$ where the mean value is over all unit loops, $sy$ is the loop $y(t)$ scaled pointwise around its center of gravity $\bar y$ by the factor $s$, and $$\Theta_V(sy,x) =
  \begin{cases}
    0 & \text{Loop does not pierce a surface}\\
    1 & \text{Loop pierces a surface.}
  \end{cases}$$

The problem with this approach is that the Casimir energy attributed to the surface of any single body goes to infinity as we send the energy regulator $\Lambda$ to infinity, due to the contribution from very short loops close to the surface. (This is, of course, a non-physical artefact related to the “geometry much larger than atomic scales” approximation.) In order to predict Casimir forces between different objects we are only interested in the dependency of the energy on the relative position of these objects. Therefore, it makes sense to modify this scheme in a way that (i) the contribution of each loop is taken into account relative to a configuration in which all objects are at infinite separation from one another, and (ii) the Casimir energy contribution attributed 