I'm learning Arakelov theory on arithmetic surfaces and I have the following general question.

Let $K$ be a number field and consider its ring of integers $O_K$. Moreover let $S:=\operatorname{Spec} O_K$ and consider a regular, projective arithmetic surface $\pi: X\to S$. For every embedding $\sigma:K\to\mathbb C$ we have the fiber at infinity $X_\sigma$ which is a Riemann surface. If we fix a Kalher metric $\Omega_\sigma$ on each $X_\sigma$, then an Arakelov divisor can be written uniquely as: $$\hat D:=D+\sum_{\sigma}\alpha_\sigma X_\sigma$$

where $D$ is a usual divisor of $X$ and $\alpha_\sigma\in\mathbb R$.

I don't understand why the coefficients for the fibers at infinity are in $\mathbb R$ and not in $\mathbb Z$, why do we need this? Foe example consider the real vector $\varepsilon=(\varepsilon_\sigma)_\sigma$ where for every $\sigma$ the real number $\varepsilon_\sigma$ is small enough and construct the Arakelov divisor $$\hat D_\varepsilon:=D+\sum_{\sigma}(\alpha_\sigma+\varepsilon_\sigma) X_\sigma$$ (note that $D$ is fixed)

What are the geometrical differences between $\hat D$ and $\hat D_\varepsilon$?