Constructing Supersymmetric Models

   A supersymmetric particle model consists of a collection of particle supermultiplets and a set of potentials that describe the interactions between the particles. The three potentials relevant to supersymmetry are: the superpotential W, the Kähler potential K, and the potential V for the scalar fields in the theory, derived from W and K.
   For N=1 supersymmetry in four spacetime dimensions, the two possible types of supersymmetric particle multiplets are: the chiral multiplet, with a complex scalar field f with spin 0 and a chiral (that is, either right or left handed) fermion y with spin 1/2, and the vector multiplet, composed of a real (nonchiral) fermion l with spin 1/2 and a vector field A_m with spin 1.
    Local gauge symmetry can be combined with global supersymmetry relatively easily. If a gauge field transforms according to the rule

where L^a is an infinitesimal gauge parameter, and the coefficients f^abc are the structure constants of the group, then the spin 1/2 superpartner partner for the gauge field, called the gaugino, transforms as

    The chiral and vector multiplets by themselves describe massless noninteracting particles. A mass matrix M^ij for fermions, and Yukawa couplings y^ijk between fermions and scalars, can be added to theory as long as the action remains invariant under both gauge transformations and supersymmetry. The chiral multiplet contains an auxiliary field (with no kinetic term in the action) Fⁱ, but the equations of motion equate it to a derivative of the superpotential, with no dynamic evolution of its own.
    So in the end, in a model with several generations of chiral multiplets (f_i,y_i), the action with superpotential looks like

,

where the terms

are derivatives of the superpotential

with respect to the scalar fields.
    The situation for gauge fields is a little more complicated, but similar. The supersymmetry transformation rules for the gauge field and the gaugino require an auxiliary field D^a, where a labels a generator in the gauge algebra.
    The resulting scalar potential of the theory, which is important for understanding the ground state of the full theory, can be written

The D-term in this potential, from the gauge multiplet auxiliary field D^a, depends on the gauge coupling g and the gauge group generators T^a.
   The Kähler potential will come in later in the section on supergravity.

The Minimal Supersymmetric Standard Model

   If supersymmetry is to solve the gauge hierarchy problem in the Standard Model, then the Standard Model has to be derivable as a theory with supersymmetry. When all of the Standard Model fields are expressed in terms of chiral and gauge multiplets, and interactions terms are added, the resulting particle theory is called the Minimal Supersymmetric Standard Model, or MSSM for short.
   The particles predicted by the Minimal Supersymmetric Standard Model are all of the particles that are already observed in the Standard Model, plus one extra Higgs doublet, and the supersymmetry partners of those particles.
   Every chiral fermion in the Standard Model has a scalar superpartner; collectively these scalars are referred to as the sfermions, which divide like quarks and leptons into squarks and sleptons. The complex scalar Higgs SU(2) doublet from the Standard Model has a spin 1/2 superpartner called the Higgsino, as does the extra Higgs doublet that was made necessary by the supersymmetrization. The gauge bosons in the Standard Model have fermionic superpartners in the MSSM called gauginos.
   At this stage of the discussion, we're still in an imaginary supersymmetric world. The world we observe does not feature bosons and fermions all neatly paired up together as if they were ready to board Noah's Ark. So a realistic Minimal Supersymmetric Standard Model requires a realistic method of breaking supersymmetry while still preserving the effects of supersymmetry in stabilizing the sensitivity of the Higgs mass to quantum corrections.

Supersymmetry breaking and unification

   As was learned in the case of spontaneous symmetry breaking in the electroweak interactions, a theory can "have its symmetry cake and eat it too" by having a ground state that does not feature the full symmetry of the action.
   The superpotential for a supersymmetric theory yields a scalar potential V(f,f*) = |F|² + |D|²/2, which is either positive or zero. This means any ground state in the theory must have positive or zero energy. A supersymmetric vacuum has zero supercharge. But because the supersymmetry algebra relates the supercharge to the energy, so that the ground state energy can be rewritten as products of the supercharges, a vacuum with zero supercharge must also have zero energy. Therefore one can break supersymmetry spontaneously by adding terms to the action such that either |F| or |D| or both are nonzero.
    To break supersymmetry using the D-term from the gauge sector of the theory, a gauge term is added to the superpotential that is invariant under supersymmetry up to a total derivative. This turns out to require an extra unbroken U(1) gauge symmetry which is not present in the MSSM (and not observed in Nature). So this method requires looking for a theory beyond the Standard Model in which this extra U(1) field can live.
   To break supersymmetry using the F-term, one can add chiral multiplets that transform as singlets under the gauge symmetries in the theory. This method also requires extra fields not observed in Nature.
   It is also possible to break supersymmetry non-spontaneously, or explicitly, by directly adding so-called "soft terms" to the superpotential that give mass to the gauginos and scalars. "Soft" in this context means terms having mass dimension 2 or 3, to avoid quadratic divergences in the quantum corrections.
   Note that the is the only way to break global supersymmetry that is consistent with Standard Model physics is to add soft terms explicitly.
   But this is hardly a satisfactory way of resolving the gauge hierarchy problem, because instead of having to fine tune the theory to tame large quantum corrections to the Higgs mass, new arbitrary supersymmetry breaking parameters have to be added to the physics by hand. That is in effect passing the gauge hierarchy problem upstairs.
   If global supersymmetry doesn't work, then what about local supersymmetry, i.e. supergravity?
   Going from global to local supersymmetry means adding gravity to the theory. So supersymmetry starts to expand upward and involve unification very naturally. One desires to have a spontaneously broken supersymmetric theory, and then unification of elementary particle physics with gravity appears as a necessary ingredient.

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