How Kaluza–Klein Reduction Produces a Dilaton Field

Kaluza–Klein dimensional reduction explains how a theory in more dimensions can look like a lower-dimensional theory at long distances. In the simplest example, five-dimensional gravity is compactified on a circle. The resulting four-dimensional theory contains gravity, a gauge field, and a scalar field that tracks the circle’s size.

That scalar is usually called the radion. It is sometimes called a dilaton in Kaluza–Klein discussions, but that name can be confusing: the radion is not automatically the dilaton that appears in string theory.

The basic idea: hide a dimension on a circle

Let the fifth coordinate be $y$, identified under $y\sim y+2\pi R$. A circle this small cannot be resolved by low-energy experiments. At energies well below the compactification scale, one often keeps only fields independent of $y$, called the zero modes.

The five-dimensional metric has more components than a four-dimensional metric. From a four-dimensional viewpoint, those components separate into:

• a four-dimensional metric $g_{\mu\nu}$;
• a four-dimensional vector field $A_\mu$;
• a scalar field $\phi$.

The vector comes from metric components with one ordinary spacetime index and one compact-direction index. The scalar comes from the compact-direction metric component, which measures the circle’s size.

A five-to-four-dimensional metric ansatz

One useful convention, chosen so that $g_{\mu\nu}$ is in the four-dimensional Einstein frame, is

$$ds_5^2=e^{-2\phi/\sqrt{3}}g_{\mu\nu}dx^\mu dx^\nu+e^{4\phi/\sqrt{3}}(dy+2A_\mu dx^\mu)^2.$$

Here $\phi$ is normalized so that its kinetic term has the standard Einstein-frame normalization used below. The numerical factors depend on convention; other references may rescale $\phi$ or $A_\mu$ and consequently write different coefficients.

If $y$ has period $2\pi R$, the proper circumference of the circle at fixed four-dimensional position is

$$L=2\pi R e^{2\phi/\sqrt{3}}.$$

Thus changing $\phi$ changes the physical compactification radius. A constant shift in $\phi$ corresponds to changing that radius.

What the reduced action contains

Start with the five-dimensional Einstein–Hilbert action and substitute the metric ansatz. For fields independent of $y$, integrating over the circle gives, up to an overall gravitational prefactor and boundary terms,

$$S_4\propto\int d^4x\sqrt{-g}\left[R-2(\partial\phi)^2-e^{2\sqrt{3}\phi}F_{\mu\nu}F^{\mu\nu}\right],$$

where $F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$. This convention uses a metric term $(dy+2A)^2$; a different normalization of $A_\mu$ changes the coefficient multiplying $F^2$.

The three terms have direct interpretations: four-dimensional gravity, a scalar kinetic term for the radion, and a gauge-field kinetic term. The factor involving $\phi$ means the gauge coupling depends on the radion’s value. The reduced theory is therefore not simply Einstein gravity plus an unrelated electromagnetic field.

Why the vector behaves like a gauge field

A shift of the circle coordinate that depends on spacetime, $y\mapsto y-2\lambda(x)$, changes the one-form combination in the metric as

$$dy+2A_\mu dx^\mu\mapsto dy+2(A_\mu+\partial_\mu\lambda)dx^\mu.$$

This is the familiar form of a gauge transformation, $A_\mu\mapsto A_\mu+\partial_\mu\lambda$. The gauge symmetry is inherited from coordinate freedom in the higher-dimensional theory. In the original Kaluza–Klein proposal, this is how electromagnetism can arise from gravity in one extra dimension.

Worked example: the radius and the Kaluza–Klein scale

Take a background with constant $\phi=\phi_0$ and set $A_\mu=0$. If the coordinate period is $2\pi R$, the physical radius is

$$R_{\mathrm{phys}}=R e^{2\phi_0/\sqrt{3}}.$$

A field that varies around the circle has Fourier modes proportional to $e^{iny/R}$, where $n$ is an integer. Its momentum along the circle has magnitude $|n|/R_{\mathrm{phys}}$ when measured using the local metric induced on the four-dimensional slices. Because the ansatz uses the four-dimensional Einstein-frame metric $g_{\mu\nu}$, the corresponding Einstein-frame mass parameter is instead

$$m_{n,E}=\frac{|n|e^{-\sqrt{3}\phi_0}}{R}=\frac{|n|e^{-\phi_0/\sqrt{3}}}{R_{\mathrm{phys}}}.$$

Thus the inverse proper radius gives the scale in the induced four-dimensional metric, while the Einstein-frame mass includes the additional Weyl factor.

For example, if the physical radius is $10^{-18}$ metres, the first nonzero mode has energy scale $\hbar c/R_{\mathrm{phys}}$, about $197$ GeV when measured in the induced four-dimensional metric (or in the Einstein frame if $\phi_0=0$). At energies much lower than this, the nonzero modes are difficult to excite, and the zero-mode description can be a useful approximation.

This estimate assumes a simple circle and a fixed background radius. Interactions, other fields, and the precise definition of four-dimensional masses can modify details, but the inverse-radius scale is the key point: a smaller circle pushes the Kaluza–Klein modes to higher energies.

Common mistakes

• Calling every radion the string dilaton. The radion measures the size of an extra dimension. A string dilaton controls the string coupling; a particular model may contain both fields, and they may mix.
• Assuming reduction means the extra dimension vanishes. The compact dimension remains in the higher-dimensional description. Low-energy observers simply fail to resolve its structure.
• Dropping the scalar without explanation. Fixing the radius can be a useful approximation, but it requires a mechanism or assumption that stabilizes the radion. Otherwise it is a dynamical field.
• Treating the zero-mode theory as valid at every energy. Near or above the inverse-radius scale, nonzero Kaluza–Klein modes can no longer be ignored.
• Comparing formulas without checking conventions. Rescaling the scalar or vector changes displayed exponents and coefficients. Compare the metric ansatz and field normalizations before concluding that two reductions disagree.

A circle reduction is the simplest illustration, not the only possibility. Reductions on more complicated spaces can produce several scalar fields, more gauge fields, and additional interactions determined by the geometry of the compact dimensions.