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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.

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  • $\begingroup$ What is your definition of "Higgs mechanism" that you are not satisfied with a VEV developing and breaking an existing gauge symmetry to a smaller one? $\endgroup$ – ACuriousMind Jun 12 '17 at 10:57
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The Higgs field simply is a mode of an open string transforming as a spacetime scalar, an open string that is stretched from the stack of $N-M$ branes to the stack of $M$ branes. When the geometric distance of these two stacks is $L$, then the minimum mass of the open string connected them is $LT$ where $T$ is the string tension, basically a linear mass density of the fundamental strings. The mass proportional to $L$ is exactly the same formula as in the Higgs mechanism – the identification is that $L$ is the vev of the Higgs boson transforming in the bifundamental representation $(N-M,M)$.

You should use string theory to quantize the open strings connecting the two stacks in detail. The separation of the branes will give you the basic contribution $(LT)^2$ to the squared mass of the open string states, and the remaining terms in the Fourier decomposition of $X,\theta$ on the string work just like for any open string. Note that in the Higgs mechanism, the W-boson mass is $gv$, also linear in $v$ which is analogous to my $L$.

The full effective field theory Lagrangians that make it clear that it's the Higgs mechanism are very clear in type II string theory. In that basic version of superstring theory, the scalar fields – including the bifundamental Higgs bosons – are a part of the supersymmetric multiplet that also contains the gauge fields. So the terms $(D_\mu H)^2$ with the covariant derivatives are obtained in the effective action for the very same reason why the effective action also produces the Yang-Mills $F_{\mu\nu}F^{\mu\nu}$. You just rename some components of $A_\mu$ to the scalar fields, and the off-block-diagonal ones will be the Higgses that break the $U(N)$ to the product group.

I have some trouble to believe that the textbooks you mention don't discuss these issues in detail but even if they don't, it's not hard for you to rediscover all these things explicitly. They work in the most natural way you could imagine.

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