I do understand the intuitive logic behind the Higgs mechanism for parallel D-branes that are separated by a distance $x$, but I would like to see a more quantitative argument. Is it possible to explicitly find the potential of the scalar field and an exact expression of the resulting worldvolume theory?
What I do understand, is that if I separate $M$ D-branes from an initial stack of $N$ D-branes the gauge group will split in $U(N-M)\times U(M)$ as the gauge fields, that correspond to strings stretching from the remaining stack of $N-M$ branes to the stack of $M$ branes, become massive. In addition, the transverse scalars, corresponding to the position of the $M$ separated D-branes along the separated direction, will develop a vev (as is partially addressed in this thread).
I do understand the above logic sound like some 'Higs-like' mechanism breaking the original $U(N)$ gauge symmetry and giving mass to some of the gauge fields, but could someone show more quantitatively how it follows from the string description that this is indeed a Higgs mechanism? I cannot find such a derivation in any of the introductory books on String and Branes like Johnson, Kutasov and Giveon, Zwiebach or Becker and Schwarz.