# Dilaton

### Dilaton

In particle physics, the dilaton is a hypothetical particle that appears in theories with extra dimensions when the volume of the compactified dimensions is allowed to vary. It appears for instance in Kaluza–Klein theory's compactifications of extra dimensions. It is a particle of a scalar field Φ, a scalar field that always comes with gravity. For comparison, in standard general relativity, Newton's constant, or equivalently the Planck mass is a constant. If this constant is promoted to a dynamical field, the result is the dilaton.

In Kaluza–Klein theories, after dimensional reduction, the effective Planck mass varies as some power of the volume of compactified space. This is why volume can turn out as a dilaton in the lower-dimensional effective theory.

Although string theory naturally incorporates Kaluza–Klein theory (which first introduced the dilaton), perturbative string theories, such as type I string theory, type II string theory and heterotic string theory, already contain the dilaton in the maximal number of 10 dimensions. However, on the other hand, M-theory in 11 dimensions does not include the dilaton in its spectrum unless it is compactified. In fact, the dilaton in type IIA string theory is actually the radion of M-theory compactified over a circle, while the dilaton in E8 × E8 string theory is the radion for the Hořava–Witten model. (For more on the M-theory origin of the dilaton, see [1].)

In string theory, there is also a dilaton in the worldsheet CFT(Conformal field theory). The exponential of its vacuum expectation value determines the coupling constant g, as ∫R = 2πχ for compact worldsheets by the Gauss–Bonnet theorem and the Euler characteristic χ = 2 − 2g, where g is the genus that counts the number of handles and thus the number of loops or string interactions described by a specific worldsheet.

g = \exp(\langle \phi \rangle)

Therefore the coupling constant is a dynamical variable in string theory, unlike the case of quantum field theory where it is constant. As long as supersymmetry is unbroken, such scalar fields can take arbitrary values (they are moduli). However, supersymmetry breaking usually creates a potential energy for the scalar fields and the scalar fields localize near a minimum whose position should in principle be calculable in string theory.

The dilaton acts like a Brans–Dicke scalar, with the effective Planck scale depending upon both the string scale and the dilaton field.

In supersymmetry, the superpartner of the dilaton is called the dilatino, and the dilaton combines with the axion to form a complex scalar field.

## Dilaton action

The dilaton-gravity action is

\int d^Dx \sqrt{-g} \left[ \frac{1}{2\kappa} \left( \Phi R - \omega\left[ \Phi \right]\frac{g^{\mu\nu}\partial_\mu \Phi \partial_\nu \Phi}{\Phi} \right) - V[\Phi] \right].

This is more general than Brans–Dicke in vacuum in that we have a dilaton potential.