Dimensional analysis of an equation

[Safinaz]

Homework Statement

In extra dimensions models like ADD model [paper][1] the relation between the extra dimensions Planck scale $M_{p_l(5)}$, the 4-dimensional Planck scale $M_{p_l}$, and the size of the extra dimension $R$ for a single extra dimension ( $n=1$ ) is given by:

Relevant Equations

$M^2_{p_l} \sim M^3_{p_l(5)} ~ R$

The units in this equation are equal on the right and the left-hand side of the equation since in natural units, the length unit = GeV$^{-1}$.

Now my question about this [paper][2], where it's a 5D Supergravity model and alternatively the above relation is given by :

$M^2_{p_l} \sim M^3_{p_l(5)} ~ R^3$

I wonder is there units or dimensional balance between the RHS and the LHS in this equation? And if not, is this equation any more valid, since any physical equation should have correct dimensional analysis.

Any help is appreciated!

[1]: https://arxiv.org/abs/hep-ph/9803315
[2]: https://arxiv.org/abs/2312.09166

 

[Steve4Physics]

Relevant Equations: $M^2_{p_l} \sim M^3_{p_l(5)} ~ R$

.
Now my question about this [paper][2], where it's a 5D Supergravity model and alternatively the above relation is given by :

$M^2_{p_l} \sim M^3_{p_l(5)} ~ R^3$

An area I know little about, so apologies if this is off the mark.

In your 2nd paper, equation (3) is: $M^2_{Pl} \sim M^{2+n}_{Pl(4+n)} R^n$
With n=1: $M^2_{Pl} \sim M^3_{Pl(5)} R$
With n=3: $M^2_{Pl} \sim M^5_{Pl(7_)} R^3$

It is not clear where your (inhomogeous) equation $M^2_{p_l} \sim M^3_{p_l(5)} ~ R^3$ comes from.