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) F^{i}, 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 Dterm 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 Dterm 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 Fterm, 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 nonspontaneously, or explicitly,
by directly adding socalled "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.
Next:
Looking for supersymmetry >>
