A Question about dilaton monopole interaction derivation

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I am trying to understand how one derives the dilaton monopole interaction. In "Black holes and membranes in higher-dimensional theories with dilaton fields", Gibbons and Maeda mentioned that one could obtain the dilaton monopole interaction as such:

Dilaton monopole interaction derivation by Gibbons and Maeda.


where the action is given by

The action.


However, I do not understand their reasoning for introducing ##\Psi## to define ##\Sigma## in order to derive Eq. (4.8). Could someone explain it?
 
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If you look at the coefficient of (∇φ)2 in their equation 2.1, you'll see it's minus the square of the field redefinition factor. So they must be aiming for a dilaton kinetic term with a coefficient of 1.
 
mitchell porter said:
If you look at the coefficient of (∇φ)2 in their equation 2.1, you'll see it's minus the square of the field redefinition factor. So they must be aiming for a dilaton kinetic term with a coefficient of 1.
Still, how do they get ##\Sigma## from ##\Psi##? Did they just consider the asymptotic behaviour of ##\Psi## and define ##\Sigma## as such?
 
4.7, 4.8 are the same form as 4.5, 4.4, which describe electric charge and electrostatic potential. The reasoning would appear to be exactly analogous.
 
https://arxiv.org/pdf/2503.09804 From the abstract: ... Our derivation uses both EE and the Newtonian approximation of EE in Part I, to describe semi-classically in Part II the advection of DM, created at the level of the universe, into galaxies and clusters thereof. This advection happens proportional with their own classically generated gravitational field g, due to self-interaction of the gravitational field. It is based on the universal formula ρD =λgg′2 for the densityρ D of DM...
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