8 Beyond the Standard Model
507
Fig. 8.7 The potential, familiar from the standard model, has a shape of a sombrero. In the figure,
its minima lie on a circle, each point along the circle, can be a basis for a ground state. The physics
around each is equivalent. In string theory and in supersymmetric systems flat potential arise
Fig. 8.8 The potential has two orthogonal flat directions. The landscape of the valleys is
generically more elaborate. Each point along the flat directions can serve as a basis of a ground
state. In many cases. The physics around each ground state is different
there is a large variety of 22 dimensional compact manifolds in the case of bosonic
strings (6 dimensional compact dimensions in the case of the supersymmetry),
which could accompany the four dimensional Minkowski space-time one is familiar
with. Each such compact manifold was called a string compactification. Next it
was suggested that actually all possible compactifications are nothing but different
solutions/ground states of a single string theory. Each of the solutions differing from
each other by the detailed values of the physical parameters it leads to. But the
ground states have also many common features, such as generically the same low
energy gauge group. To appreciate this consider the difference between the potential
of the Higgs particles which may describe the mass generation of the carriers of the
weak interactions and the effective potential leading to the ground states in string
theory. For the electro-weak interaction case the potential has the form of a Mexican
hat, the ground state may be described by any point in the valley (Fig. 8.7) but all
choices are equivalent, they describe exactly the same physics.
In the case at hand the valley of minima is flat and extends all the way to infinity.
Any point along the valley can be chosen as a ground state as they all have the
same energy and yet each describes different physics. The manifold describing the
infinity of these degenerate ground states is called the moduli space when they are
continuously connected (Fig. 8.8).
507
Fig. 8.7 The potential, familiar from the standard model, has a shape of a sombrero. In the figure,
its minima lie on a circle, each point along the circle, can be a basis for a ground state. The physics
around each is equivalent. In string theory and in supersymmetric systems flat potential arise
Fig. 8.8 The potential has two orthogonal flat directions. The landscape of the valleys is
generically more elaborate. Each point along the flat directions can serve as a basis of a ground
state. In many cases. The physics around each ground state is different
there is a large variety of 22 dimensional compact manifolds in the case of bosonic
strings (6 dimensional compact dimensions in the case of the supersymmetry),
which could accompany the four dimensional Minkowski space-time one is familiar
with. Each such compact manifold was called a string compactification. Next it
was suggested that actually all possible compactifications are nothing but different
solutions/ground states of a single string theory. Each of the solutions differing from
each other by the detailed values of the physical parameters it leads to. But the
ground states have also many common features, such as generically the same low
energy gauge group. To appreciate this consider the difference between the potential
of the Higgs particles which may describe the mass generation of the carriers of the
weak interactions and the effective potential leading to the ground states in string
theory. For the electro-weak interaction case the potential has the form of a Mexican
hat, the ground state may be described by any point in the valley (Fig. 8.7) but all
choices are equivalent, they describe exactly the same physics.
In the case at hand the valley of minima is flat and extends all the way to infinity.
Any point along the valley can be chosen as a ground state as they all have the
same energy and yet each describes different physics. The manifold describing the
infinity of these degenerate ground states is called the moduli space when they are
continuously connected (Fig. 8.8).