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We know that no gauge symmetry can prevent a term $m_\phi^2|\phi|^2$ for a scalar field, and that, given the quadratic loop corrections, the natural scale is $m_\phi \sim M_P$. This is related to the hierarchy problem.

Now, I was thinking, is it fair to categorize the solutions to the hierachy problem into two sets:

  1. Solutions which transfer a symmetry to the scalar:

    • Supersymmetry, in which $\phi$ is protected via a symmetry transferred from a fermion.
    • Little Higgs, in which $\phi$ is a Goldstone-boson protected via a shift symmetry linearly realized from the original scalar that precipitated SSB.
    • Composite Higgs, in which the fundamental degrees of freedom at high energies are not scalar, but fermionic, and thus $m_\phi$ is protected.
  2. Solutions which lower the true $M_P$. i.e. the true $M_P \sim M_W$, but on the brane $M_P^\textrm{eff}\sim 10^{19}$ GeV, e.g. ADD and extra-dimensions.

Is this division of solutions accurate? Are there any solutions that don't fall into these categories? And are there any other ways to transfer symmetries to scalars?

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It's not obvious to me that the natural scale is the Planck mass. It's not even obvious that the scalar field variable is defined at that scale. – user1504 May 21 '13 at 19:00
You are referring to two of the solutions above? Models in which the true $M_P$ is lower, and composite Higgs models, respectively? – innisfree May 21 '13 at 19:08
I should have read more carefully. Nonetheless, it's not obvious to me that 'scalar is not fundamental' is a subclass of 'symmetry transfer'. – user1504 May 21 '13 at 20:22
At high energies, the composite Higgs fields fall apart into "quarks" of new strongly interacting gauge group. These fermions are protected from the Hierarchy problem by a chiral symmetry, so the loop corrections are trucated at the scale $f_\pi$ - the emergent scale of the new strong dynamics. So we've transferred the chiral symmetry to the scalar (I mean transfer in a informal sense). Maybe this is distinct, maybe my classification is too simple... – innisfree May 21 '13 at 20:41

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