# Dilaton

In particle physics, the hypothetical dilaton particle, and scalar field, appears in theories with extra dimensions when the volume of the compactified dimensions varies. It appears as a radion in Kaluza–Klein theory's compactifications of extra dimensions. A particle of a scalar field Φ, a scalar field that always comes with gravity, and in a dynamical field the resulting dilaton particle parallels the graviton. For comparison, in standard general relativity, Newton's constant, or equivalently the Planck mass is a constant.

## Exposition

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 that 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, M-theory in 11 dimensions does not include the dilaton in its spectrum unless compactified. The dilaton in type IIA string theory parallels the radion of M-theory compactified over a circle, and the dilaton in E8 × E8 string theory parallels 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 - two-dimensional conformal field theory. The exponential of its vacuum expectation value determines the coupling constant g and the Euler characteristic χ = 2 − 2g as ∫R = 2πχ for compact worldsheets by the Gauss–Bonnet theorem, where the genus g counts the number of handles and thus the number of loops or string interactions described by a specific worldsheet.

${\displaystyle g=\exp(\langle \phi \rangle )}$

Therefore, the dynamic variable coupling constant in string theory contrasts the quantum field theory where it is constant. As long as supersymmetry is unbroken, such scalar fields can take arbitrary values 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 calculate 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 or here the dilatino, combines with the axion to form a complex scalar field[citation needed].

## Dilaton action

The dilaton-gravity action is

${\displaystyle \int d^{D}x{\sqrt {-g}}\left[{\frac {1}{2\kappa }}\left(\Phi R-\omega \left[\Phi \right]{\frac {g^{\mu \nu }\partial _{\mu }\Phi \partial _{\nu }\Phi }{\Phi }}\right)-V[\Phi ]\right]}$ .

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

## Notes

1. ^ David S. Berman, Malcolm J. Perry (2006), "M-theory and the string genus expansion"

## References

• Fujii, Y. (2003). "Mass of the dilaton and the cosmological constant". Prog. Theor. Phys. 110 (3): 433–439. arXiv:gr-qc/0212030. Bibcode:2003PThPh.110..433F. doi:10.1143/PTP.110.433.
• Hayashi, M.; Watanabe, T.; Aizawa, I. & Aketo, K. (2003). "Dilatonic Inflation and SUSY Breaking in String-inspired Supergravity". Modern Physics Letters A. 18 (39): 2785–2793. arXiv:hep-ph/0303029. Bibcode:2003MPLA...18.2785H. doi:10.1142/S0217732303012465.
• Alvarenge, F.; Batista, A. & Fabris, J. (2005). "Does Quantum Cosmology Predict a Constant Dilatonic Field". International Journal of Modern Physics D. 14 (2): 291–307. arXiv:gr-qc/0404034. Bibcode:2005IJMPD..14..291A. doi:10.1142/S0218271805005955.
• Lu, H.; Huang, Z.; Fang, W. & Zhang, K. (2004). "Dark Energy and Dilaton Cosmology". arXiv:hep-th/0409309.
• Wesson, Paul S. (1999). Space-Time-Matter, Modern Kaluza-Klein Theory. Singapore: World Scientific. p. 31. ISBN 981-02-3588-7.