Editing Anomaly (physics) (section)

==Anomaly cancellation==
[[File:Triangle diagram.svg|left]]
Since cancelling anomalies is necessary for the consistency of gauge theories, such cancellations are  of central importance in constraining the fermion content of the [[standard model]], which is a chiral gauge theory.

For example, the vanishing of the [[mixed anomaly]] involving two SU(2) generators and one U(1) hypercharge constrains all charges in a fermion generation to add up to zero,<ref>Bouchiat, Cl,  Iliopoulos, J, and  Meyer, Ph (1972) . "An anomaly-free version of Weinberg's model." ''Physics Letters'' '''B38''', 519-523.</ref><ref>{{cite journal | last1 = Minahan | first1 = J. A. | last2 = Ramond | first2 = P. | last3 = Warner | first3 = R. C. | year = 1990 | title = Comment on anomaly cancellation in the standard model | journal = Phys. Rev. D | volume = 41 | issue = 2| pages = 715–716 | doi = 10.1103/PhysRevD.41.715 | pmid = 10012386 |bibcode = 1990PhRvD..41..715M }}</ref> and thereby dictates that the sum of the proton plus the sum of the electron vanish: the ''charges of quarks and leptons must be commensurate''.
Specifically, for two external gauge fields {{math|''W<sup>a</sup>''}}, {{math|''W<sup>b</sup>''}} and one hypercharge {{mvar|B}} at the vertices of the triangle diagram, cancellation of the triangle requires
:<math>\sum_{all ~doublets}\!\!\!\! \mathrm{Tr} ~T^a T^b Y \propto \delta^{ab}  \sum_{all ~doublets} Y=\sum_{all ~doublets} Q =0 ~,  </math>  
so, for each generation, the charges of the leptons and quarks are balanced, <math>-1+3\times\frac{2-1}{3}=0 </math>, whence {{math|1=''Q''<sub>p</sub> + ''Q''<sub>e</sub> = 0}}{{Citation needed|date=April 2020|reason=Why should that follow?}}.

The anomaly cancelation in SM was also used to predict a quark from 3rd generation, the [[top quark]].<ref>{{Cite book|last=Conlon|first=Joseph|url=https://www.taylorfrancis.com/books/9781482242492|title=Why String Theory?|date=2016-08-19|publisher=CRC Press|isbn=978-1-315-27236-8|edition=1|language=en|doi=10.1201/9781315272368|page=81}}</ref>

Further such mechanisms include:
* [[Axion]]
* [[Chern–Simons]]
* [[Green–Schwarz mechanism]]
* Liouville action