Editing Axion (section)

== Phenomenology of the axion field ==
=== Searches <span class="anchor" id="K-S-V-Z-vs-D-F-S-Z-anchor"></span> ===
The axion models originally proposed by Wilczek and by Weinberg chose axion coupling strengths that were so strong that they would have already been detected in prior experiments. It had been thought that the [[Peccei–Quinn theory|Peccei–Quinn mechanism]] for solving the [[strong CP problem]] required such large couplings. However, it was soon realized that "invisible axions" with much smaller couplings also work. Two such classes of models are known in the literature as {{nowrap|KSVZ}} ([[Kim Jihn-eui|Kim]]–[[Mikhail Shifman|Shifman]]–[[Arkady Vainshtein|Vainshtein]]–{{nowrap|{{abbr|Zakharov|Valya Zakharov}})<ref name=Kim-1979>{{cite journal |last1=Kim |first1=J. E. |year=1979 |title=Weak-interaction singlet and strong CP invariance |journal=Physical Review Letters |volume=43 |issue=2 |pages=103–107 |bibcode=1979PhRvL..43..103K |doi=10.1103/PhysRevLett.43.103}}</ref><ref name=Shifman-etal-1980>{{cite journal |last1=Shifman |first1=M. |last2=Vainshtein |first2=A. |last3=Zakharov |first3=V. |year=1980 |journal=Nuclear Physics&nbsp;B |volume=166 |issue=3 |pages=493–506 |title=Can confinement ensure natural CP invariance of strong interactions? |doi=10.1016/0550-3213(80)90209-6 |bibcode=1980NuPhB.166..493S}}</ref>}} and {{nowrap|DFSZ}} ([[Michael Dine|Dine]]–[[Willy Fischler|Fischler]]–{{abbr|Srednicki|Mark Srednicki}}–{{nowrap|{{abbr|Zhitnitsky|Ariel R. Zhitnitsky}}).<ref>{{cite journal |last1=Dine |first1=M. |last2=Fischler |first2=W. |last3=Srednicki |first3=M. |year=1981 |journal=Physics Letters B |volume=104 |issue=3 |pages=199–202 |title=A simple solution to the strong CP problem with a harmless axion |doi=10.1016/0370-2693(81)90590-6 |bibcode=1981PhLB..104..199D}}</ref><ref>{{cite journal |last1=Zhitnitsky |first1=A. |year=1980 |journal=Soviet Journal of Nuclear Physics |language=en |volume=31 |page=260 |title=On possible suppression of the axion–hadron interactions |url= https://www.inp.nsk.su/images/preprint/1979_081.pdf}}</ref>}}

The very weakly coupled axion is also very light, because axion couplings and mass are proportional. Satisfaction with "invisible axions" changed when it was shown that any very light axion would have been overproduced in the early universe and therefore must be excluded.<ref name="auto"/><ref name="A cosmological bound on the invisib"/><ref name="The not-so-harmless axion"/>

=== Maxwell's equations with axion modifications ===
[[Pierre Sikivie]] computed how [[Maxwell's equations]] are modified in the presence of an axion in 1983.<ref>{{cite journal |last1=Sikivie |first1=P. |title=Experimental Tests of the 'Invisible' Axion |date=17 October 1983 |journal=Physical Review Letters |volume=51 |issue=16 |page=1413 |doi=10.1103/physrevlett.51.1415 |bibcode=1983PhRvL..51.1415S}}</ref> He showed that these axions could be detected on Earth by converting them to photons, using a strong magnetic field, motivating a number of experiments. For example, the [[Axion Dark Matter Experiment]] attempts to convert axion dark matter to microwave photons, the [[CERN Axion Solar Telescope]] attempt to convert axions that are produced in the Sun's core to X-rays, and other experiments search for axions produced in laser light.<ref>{{cite web |title=OSQAR |url=http://home.cern/about/experiments/osqar |publisher=CERN |date=2017 |access-date=3 October 2017}}</ref> As of the early 2020s, there are dozens of proposed or ongoing experiments searching for axion dark matter.<ref>{{cite arXiv |title=Axion Dark Matter |date=2022 |eprint=2203.14923 |last1=Adams |first1=C. B. |last2=Aggarwal |first2=N. |last3=Agrawal |first3=A. |last4=Balafendiev |first4=R. |last5=Bartram |first5=C. |last6=Baryakhtar |first6=M. |last7=Bekker |first7=H. |last8=Belov |first8=P. |last9=Berggren |first9=K. K. |last10=Berlin |first10=A. |last11=Boutan |first11=C. |last12=Bowring |first12=D. |last13=Budker |first13=D. |last14=Caldwell |first14=A. |last15=Carenza |first15=P. |last16=Carosi |first16=G. |last17=Cervantes |first17=R. |last18=Chakrabarty |first18=S. S. |last19=Chaudhuri |first19=S. |last20=Chen |first20=T. Y. |last21=Cheong |first21=S. |last22=Chou |first22=A. |last23=Co |first23=R. T. |last24=Conrad |first24=J. |last25=Croon |first25=D. |last26=D'Agnolo |first26=R. T. |last27=Demarteau |first27=M. |last28=DePorzio |first28=N. |last29=Descalle |first29=M. |last30=Desch |first30=K. |class=hep-ex |display-authors=1 }}</ref>

Treating the reduced Planck constant <math>\hbar</math>, speed of light <math>c</math>, and permittivity of free space <math>\varepsilon_0</math> all equivalent to 1, the electrodynamic equations are:
: {| class="wikitable" style="text-align: center;"
|-
! scope="col" style="width: 15em;" | Name
! scope="col" |  Equations
|-
! scope="row" | Gauss's law
| <math> \nabla \cdot \mathbf{E} = \rho - g_{a\gamma\gamma} \mathbf{B} \cdot \nabla a </math>
|-
! scope="row" | Gauss's law for magnetism
| <math> \nabla \cdot \mathbf{B}  = 0 </math>
|-
! scope="row" | Faraday's law
| <math> \nabla \times \mathbf{E} = - \dot{ \mathbf{B} } </math>
|-
! scope="row" | Ampère–Maxwell law
| <math>\quad \nabla \times \mathbf{B} = \dot{\mathbf{E}} + \mathbf{J} + g_{a\gamma\gamma} \left( \dot{a} \mathbf{B} - \mathbf{E} \times \nabla a \right) \quad</math>
|-
! scope="row" | Axion field's equation of motion
| <math> \ddot{a} - \nabla^2 a + m_a^2 a = -g_{a\gamma\gamma} \mathbf{E} \cdot \mathbf{B} </math>
|}

Above, a  dot above a variable denotes its time derivative; the dot spaced between variables is the [[vector dot product]]; the factor <math> g_{a\gamma\gamma} </math> is the axion-to-photon coupling constant.

Alternative forms of these equations have been proposed, which imply completely different physical signatures. For example, Visinelli wrote a set of equations that imposed duality symmetry, assuming the existence of [[magnetic monopole]]s.<ref>{{cite journal |last=Visinelli |first=L. |year=2013 |title=Axion-electromagnetic waves |journal=Modern Physics Letters&nbsp;A |volume=28 |number=35 |page=1350162 |doi=10.1142/S0217732313501629 |arxiv=1401.0709 |bibcode=2013MPLA...2850162V |s2cid=119221244 }}</ref> However, these alternative formulations are less theoretically motivated, and in many cases cannot even be derived from an [[action (physics)|action]].

=== Analogous effect for topological insulators ===
A term analogous to the one that would be added to [[Maxwell's equations]] to account for axions<ref>{{cite journal |last=Wilczek |first=Frank |title=Two applications of axion electrodynamics |journal=Physical Review Letters |date=4 May 1987 |volume=58 |issue=18 |pages=1799–1802 |doi=10.1103/PhysRevLett.58.1799 |pmid=10034541 |bibcode=1987PhRvL..58.1799W}}</ref> also appears in recent (2008) theoretical models for [[topological insulators]] giving an effective axion description of the electrodynamics of these materials.<ref>{{cite journal |last1=Qi |first1=Xiao-Liang |last2=Hughes |first2=Taylor L. |last3=Zhang |first3=Shou-Cheng |title=Topological field theory of time-reversal invariant insulators |date=24 November 2008 |journal=Physical Review&nbsp;B |volume=78 |issue=19 |page=195424 |doi=10.1103/PhysRevB.78.195424 |bibcode=2008PhRvB..78s5424Q |arxiv=0802.3537|s2cid=117659977 }}</ref>

This term leads to several interesting predicted properties including a quantized [[magnetoelectric effect]].<ref name="Franz 36">{{cite journal |doi=10.1103/Physics.1.36 |volume=1 |page=36 |last=Franz |first=Marcel |title=High-energy physics in a new guise |journal=Physics |date=24 November 2008 |bibcode=2008PhyOJ...1...36F |doi-access=free}}</ref> Evidence for this effect has been given in [[N. Peter Armitage|THz spectroscopy experiments]] performed at the [[Johns Hopkins University]] on quantum regime thin film topological insulators developed at [[Rutgers University]].<ref>{{cite journal |last1=Wu |first1=Liang |last2=Salehi |first2=M. |last3=Koirala |first3=N. |last4=Moon |first4=J. |last5=Oh |first5=S. |last6=Armitage |first6=N. P. |date=2 December 2016 |title=Quantized Faraday and Kerr rotation and axion electrodynamics of a 3D topological insulator |journal=Science |volume=354 |issue=6316 |pages=1124–1127 |arxiv=1603.04317 |bibcode=2016Sci...354.1124W |doi=10.1126/science.aaf5541 |pmid=27934759 |s2cid=25311729 }}</ref>

In 2019, a team at the [[Max Planck Institute for Chemical Physics of Solids]] published their detection of an [[Magnetic topological insulator#Axion coupling|axion insulator]] phase of a [[Weyl semimetal]] material.<ref>
{{cite journal
 |first1=J.    |last1=Gooth      |first2=B.   |last2=Bradlyn
 |first3=S.    |last3=Honnali    |first4=C.   |last4=Schindler
 |first5=N.    |last5=Kumar      |first6=J.   |last6=Noky
 |first7=Y.    |last7=Qi         |first8=C.   |last8=Shekhar
 |first9=Y.    |last9=Sun        |first10=Z.  |last10=Wang
 |first11=B. A.|last11=Bernevig  |first12=C.  |last12=Felser 
 |display-authors=6
 |date=7 October 2019
 |title=Axionic charge-density wave in the Weyl semimetal (TaSe<sub>4</sub>)<sub>2</sub>I
 |journal=Nature
 |volume=575  |issue=7782   |pages=315–319
 |doi=10.1038/s41586-019-1630-4  |pmid=31590178    |arxiv=1906.04510
 |bibcode=2019Natur.575..315G    |s2cid=184487056
}}</ref> In the axion insulator phase, the material has an axion-like [[quasiparticle]] – an excitation of electrons that behave together as an axion – and its discovery demonstrates the consistency of axion electrodynamics as a description of the interaction of axion-like particles with electromagnetic fields. In this way, the discovery of axion-like quasiparticles in axion insulators provides motivation to use axion electrodynamics to search for the axion itself.<ref>
{{cite web
 |last=Fore
 |first=Meredith
 |date=22 November 2019
 |title=Physicists have finally seen traces of a long-sought particle. Here's why that's a Big Deal.
 |website=Live Science
 |publisher=Future US, Inc.
 |url=https://www.livescience.com/axion-found-in-weyl-semimetal.html
 |access-date=25 February 2020
}}</ref>