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Reading about the minimal supersymmetric standard model (MSSM) I have found that an explicit breaking of supersymmetry is considered, given by the term $$\cal{L}_{soft}$$ where soft supersymmetry breaking terms are included, that is, those that break supersymmetry without spoiling the cancellation of quadratic divergences.

I also have read that supersymmetry is considered to be spontaneously broken fundamentally, and that the explicit breaking term in the MSSM can be traced back to this spontaneous breaking in the more general theory.

My question is, why don't we consider spontaneous breaking of supersymmetry directly in the MSSM?

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Well, you do but the details are generally irrelevant for very low energies, like the LHC. The rational is as this:

  1. SUSY partners have to be heavier than regular matter, otherwise they would have been observed $\Rightarrow$ if realised in Nature, SUSY is broken.
  2. Like any other symmetry, if spontaneously broken leaves a massless (Nambu-)Goldstone mode. In the SUSY case it cannot be a boson, as the generating charges are fermionic quantities, but a fermion called Goldstino. Now, even though it might difficult to produce it we do not observe this massless fermion $\Rightarrow$ SUSY breaking mechanism has to be more intricate than the ones usually presented in intro books (D and F term breaking), just like the pion is not a massless Goldstone boson for the Chiral symmetry breaking, as the dynamics in Nature is more intricate than that.
  3. If realised in Nature, the quantum for the gravitational field (Graviton) should have a fermionic field, the Gravitino, of spin 3/2. There are many cosmological reasons why it can't be massless, so it too should be heavy. Furthermore, extending SUSY to allow for Gravitino is the same to say you allow for Supergravity, as the Gravitino works as the gauge field for localised SUSY transformations.

So a SUSY breaking theory as to account for different masses for fermions and scalars, heavy gravitino, absence of goldstino.

Supergravity has, in fact, all the nice ingredients to generate soft masses. It so happens that in Gravity the gravitino mass [1] specifies the soft scalar masses [2] although the specifications for the exact values is highly dependent on the correct theory of quantum gravity (the so called UV completion of Supergravity). Theory dependent formulae and predictions can be made, and are in fact made from String Theory for example. But in general we are not sure what is the correct theory of quantum gravity, and as such the generated soft masses from SUpergravity can be taken to be quite free if you are not doing String Theory.

Now, as many SUSY books and notes start from an MSSM study (Aitchison, Martin, etc [3]) without committing to a specific theory of quantum gravity, the specifications of the SUSY breaking terms are taken to be quite loose, and as such one can study a wider array of SUSY models. This is why, in the end, SUSY phenomenologists (like the ones cited above) take an explicit breaking Lagrangian and do very little assumptions about the structure of the soft couplings.

Summing up:

  1. Soft masses are indeed expected to be originated from Supergravity (probably an unavoidable extensions of regular SUSY).
  2. The details are very complicated and it's easier to work with the expected end results instead of deriving it.
  3. For low energy particle phenomenology and dynamics Supergravity theory does not play a role at all.
  4. You only need to carry about the specifications of Supergravity if you are working with String Theory or other UV completions of Supergravity or you care about the specific structures and relations appearing in $\mathcal{L}_{soft}$.

So, in general, you do work with the spontaneously broken SUSY, although with the general results which looks very like explicit breaking.

References:

[1] The gravitino, being the gauge field of Supergravity, assimilates the goldstino when it gets a mass, just like in the SM the $W$ and $Z$ eat the would-be-goldstone bosons.

[2] http://arxiv.org/abs/hep-ph/9707209

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