View source for Dilaton

{{Short description|Hypothetical particle}}
{{Use American English|date = March 2019}}
In [[particle physics]], the hypothetical '''dilaton''' is a particle of a [[scalar field]] <math>\varphi</math> that appears in theories with [[Dimension (mathematics and physics)#Additional dimensions|extra dimensions]] when the volume of the compactified dimensions varies.  It appears as a [[radion (physics)|radion]] in [[Kaluza–Klein theory]]'s [[compactification (physics)|compactification]]s of extra [[dimension]]s.  In [[Brans–Dicke theory]] of gravity, [[Newton's constant]] is not presumed to be constant but instead 1/''G'' is replaced by a [[scalar field]] <math>\varphi</math> and the associated particle is the dilaton.

== 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&nbsp;dimensions. However, [[M-theory]] in 11 dimensions does not include the dilaton in its spectrum unless [[Compactification (physics)|compactified]]. The dilaton in [[type IIA string theory]] parallels the [[radion (physics)|radion]] of M-theory compactified over a circle, and the dilaton in &nbsp;{{nobr| E{{sub|8}} × E{{sub|8}} }}&nbsp; string theory parallels the radion for the [[Hořava–Witten model]]. (For more on the M-theory origin of the dilaton, see Berman & Perry (2006).<ref>{{cite journal |first1=David S. |last1=Berman |first2=Malcolm J. |last2=Perry |date=6 April 2006 |title=M-theory and the string genus expansion |journal=[[Physics Letters B]] |volume=635 |issue=2–3 |pages=131–135 |arxiv=hep-th/0601141 |doi=10.1016/j.physletb.2006.02.038}}</ref>)

In [[string theory]], there is also a dilaton in the [[worldsheet]] CFT – [[two-dimensional conformal field theory]]. The [[exponential function|exponential]] of its [[vacuum expectation value]] determines the [[coupling constant]] {{mvar|g}} and the [[Euler characteristic]] {{nobr|&thinsp; {{math|χ {{=}} 2 − 2''g''}} &thinsp;}} as <math display=inline>\; \tfrac{1}{2\pi} \int R = \chi \;</math> for compact worldsheets by the [[Gauss–Bonnet theorem]], where the genus {{mvar|g}} counts the number of handles and thus the number of loops or string interactions described by a specific worldsheet.

:<math> g = \exp \left( \langle \varphi \rangle \right) </math>

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 (physics)|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 theory|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|reason=Unclear.|date=May 2018}}

== The dilaton in quantum gravity ==
The dilaton made its first appearance in [[Kaluza–Klein theory]], a five-dimensional theory that combined [[gravitation]] and [[electromagnetism]]. It appears in [[string theory]]. However, it has become central to the lower-dimensional many-bodied gravity problem<ref>{{cite journal|last1=Ohta|first1=Tadayuki|last2=Mann|first2=Robert|title=Canonical reduction of two-dimensional gravity for particle dynamics|journal=[[Classical and Quantum Gravity]]|volume=13|issue=9|pages=2585–2602|date=1996|doi=10.1088/0264-9381/13/9/022|arxiv=gr-qc/9605004|bibcode = 1996CQGra..13.2585O |s2cid=5245516}}</ref> based on the field theoretic approach of [[Roman Jackiw]]. The impetus arose from the fact that complete analytical solutions for the metric of a covariant ''N''-body system have proven elusive in general relativity. To simplify the problem, the number of dimensions was lowered to 1&nbsp;+&nbsp;1 – one spatial dimension and one temporal dimension. This model problem, known as [[R = T model|''R''&nbsp;=&nbsp;''T'' theory]],<ref>{{cite journal|last1=Sikkema|first1=A E|last2=Mann|first2=R B|title=Gravitation and cosmology in (1&nbsp;+&nbsp;1) dimensions|journal=[[Classical and Quantum Gravity]]|volume=8|issue=1|pages=219–235|date=1991|doi=10.1088/0264-9381/8/1/022|bibcode = 1991CQGra...8..219S |s2cid=250910547 }}</ref> as opposed to the general ''G''&nbsp;=&nbsp;''T'' theory, was amenable to exact solutions in terms of a generalization of the [[Lambert W function]].  Also, the field equation governing the dilaton, derived from [[differential geometry]], as the [[Schrödinger equation]] could be amenable to quantization.<ref>{{cite journal|author1=Farrugia|author2=Mann|author3=Scott|doi=10.1088/0264-9381/24/18/006|journal=[[Classical and Quantum Gravity]] |volume=24|title=''N''-body Gravity and the Schroedinger Equation|issue=18|pages=4647–4659|date=2007|arxiv=gr-qc/0611144|bibcode = 2007CQGra..24.4647F |s2cid=119365501}}</ref>

This combines gravity, quantization, and even the electromagnetic interaction, promising ingredients of a fundamental physical theory. This outcome revealed a previously unknown and already existing natural link between general relativity and quantum mechanics. There lacks clarity in the generalization of this theory to 3&nbsp;+&nbsp;1 dimensions. However, a recent derivation in 3&nbsp;+&nbsp;1 dimensions under the right coordinate conditions  yields a formulation similar to the earlier 1&nbsp;+&nbsp;1, a dilaton field governed by the [[logarithmic Schrödinger equation]]<ref>{{cite journal|last1=Scott|first1=T.C.|last2=Zhang|first2=Xiangdong|last3=Mann|first3=Robert|last4=Fee|first4=G.J.|title=Canonical reduction for dilatonic gravity in 3 + 1 dimensions|journal=[[Physical Review D]]|volume=93|issue=8|pages=084017|date=2016|doi=10.1103/PhysRevD.93.084017|bibcode = 2016PhRvD..93h4017S|arxiv=1605.03431}}</ref> that is seen in [[condensed matter physics]] and [[Superfluidity|superfluids]].  The field equations are amenable to such a generalization, as shown with the inclusion of a one-graviton process,<ref>{{cite journal|last1=Mann|first1=R B|last2=Ohta|first2=T|title=Exact solution for the metric and the motion of two bodies in (1&nbsp;+&nbsp;1)-dimensional gravity|journal=[[Physical Review|Phys. Rev. D]]|volume=55|issue=8|pages=4723–4747|date=1997|doi=10.1103/PhysRevD.55.4723|arxiv=gr-qc/9611008|bibcode=1997PhRvD..55.4723M|s2cid=119083668}}</ref> and yield the correct Newtonian limit in ''d'' dimensions, but only with a dilaton.  Furthermore, some speculate on the view of the apparent resemblance between the dilaton and the [[Higgs boson]].<ref>{{cite journal|last1=Bellazzini|first1=B.|last2=Csaki|first2=C.|last3=Hubisz|first3=J.|last4=Serra|first4=J.|last5=Terning|first5=J.|title=A higgs-like dilaton|journal=Eur. Phys. J. C|volume=73|number=2|pages=2333|date=2013|doi=10.1140/epjc/s10052-013-2333-x|bibcode=2013EPJC...73.2333B|arxiv=1209.3299|s2cid=118416422}}</ref>  However, there needs more experimentation to resolve the relationship between these two particles.  Finally, since this theory can combine gravitational, electromagnetic, and quantum effects, their coupling could potentially lead to a means of testing the theory through cosmology and experimentation.

== Dilaton action ==
The dilaton-gravity action is

:<math>\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].</math>

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

== See also ==
* [[CGHS model]]
* [[R = T model]]
* [[Quantum gravity#The dilaton|Quantum gravity]]
* [[List of hypothetical particles]]

== Citations ==
{{Reflist|25em}}

== References ==
{{refbegin|colwidth=25em|small=yes}}
* {{cite journal |first=Y. |last=Fujii |year=2003 |title=Mass of the dilaton and the cosmological constant |journal=[[Progress of Theoretical Physics]] |volume=110 |issue=3 |pages=433–439 |doi=10.1143/PTP.110.433 |arxiv=gr-qc/0212030 |bibcode=2003PThPh.110..433F|s2cid=119396089 }}
* {{cite journal |first1=M. |last1=Hayashi |first2=T. |last2=Watanabe |first3=I. |last3=Aizawa |first4=K. |last4=Aketo |year=2003 |title=Dilatonic inflation and SUSY breaking in string-inspired supergravity |journal=[[Modern Physics Letters A]] |volume=18 |issue=39 |pages=2785–2793 |doi=10.1142/S0217732303012465 |arxiv=hep-ph/0303029 |bibcode=2003MPLA...18.2785H |s2cid=7974289 }}
* {{cite journal |first1=F. |last1=Alvarenge |first2=A. |last2=Batista |first3=J. |last3=Fabris |year=2005 |title=Does quantum cosmology predict a constant dilatonic field? |journal=[[International Journal of Modern Physics D]] |volume=14 |issue=2 |pages=291–307 |doi=10.1142/S0218271805005955 |arxiv=gr-qc/0404034 |bibcode=2005IJMPD..14..291A |s2cid=119530947 }}
* {{cite arXiv |first1=H. |last1=Lu |first2=Z. |last2=Huang |first3=W. |last3=Fang |first4=K. |last4=Zhang |year=2004 |title=Dark energy and dilaton cosmology |eprint=hep-th/0409309}}
* {{cite book |first=Paul S. |last=Wesson |author-link=Paul S. Wesson |year=1999 |title=Space-Time-Matter, Modern Kaluza-Klein Theory |url=https://archive.org/details/spacetimematterm0000wess |url-access=registration |publisher=[[World Scientific]] |location=Singapore |isbn=978-981-02-3588-8 |page=[https://archive.org/details/spacetimematterm0000wess/page/31 31]}}
* {{cite journal |last1=Wang |first1=C.H.-T. |last2=Rodrigues |first2=D.P.F. |year=2018 |title=Closing the gaps in quantum space and time: Conformally augmented gauge  structure of gravitation |doi=10.1103/PhysRevD.98.124041 |journal=[[Physical Review]] D |volume=98 |page=124041|hdl=2164/11713 |hdl-access=free }}
* {{cite journal |last1=Wang, C. H.-T. |last2=Stankiewicz, M. |year=2020 |title=Quantization of time and the big bang via scale-invariant loop gravity |journal=[[Physics Letters]] B |volume=800 |page=135106 |doi=10.1016/j.physletb.2019.135106|hdl=2164/13314 |hdl-access=free }}
{{refend}}

{{String theory topics |state=collapsed}}
{{Particles}}
{{Theories of gravitation|selected=GR alternatives}}

[[Category:Hypothetical elementary particles]]
[[Category:String theory]]
[[Category:Supersymmetry]]