Why should it be allowed to set the einbein to unity?

Question

The Polyakov action for a massive free point particle with worldline $\gamma$ is given by

$$ S[\gamma] = \frac{1}{2}\int_\gamma e \biggl(\frac{1}{e^2}\dot{x}^2 - m^2\biggr)\mathrm{d}\tau $$

where $e$ is the so-called einbein, which may be interpreted as a one-dimensional worldline metric $\sqrt{g_{\tau\tau}}$. As is discussed, for example, in this P.SE question, it is often casually stated that the reparametrization invariance of the Polyakov action allows us to (gauge-)fix $e = 1$.

Yet, by inspection, we see that the einbein must have mass dimension $-1$, and indeed, it is also sometimes set to be $e = \frac{1}{m}$ when dealing with massive particles. The question is therefore:

Why would it be allowed to set a dimensionful quantity to unity? Would we not have to introduce some kind of mass scale (which may drop out of physical observables)? Note that there is no natural mass scale in our theory if $m = 0$.

Answer 1

Setting the einbein to $1$ corresponds to a diffeomorphism of the metric, as the einbein is given by $e_{\tau\tau}=\sqrt{g_{\tau\tau}}$, which can be easily deduced from the fact that a the vielbein is given as the transformation coefficients from the coordinate basis to a non-coordinate basis. Hence, the dimensionality of the einbein depends on that of the metric. From the requirement that the metric has mass dimension $-2$, it can only follow that the einbein has mass dimension $-1$. A reparametrization corresponds to multiplication with a dimensionless number, i.e. setting the einbein to unity actually means setting its numerical value to $1$ while keeping its dimension. No new mass scale has to be introduced.