Editing Kinetic term (section)

=== Multi-field kinetic terms ===

In a theory with multiple fields of the same type, such as multiple scalars or multiple fermions, their kinetic and mass terms can be grouped together into kinetic and mass [[matrix (mathematics)|matrices]]. For example, for a set of real scalar fields grouped into a [[vector (mathematics and physics)|vector]] <math>\Phi = (\phi_1, \cdots, \phi_n)</math> one can write the kinetic and mass terms as

:<math>
\mathcal L = \frac{1}{2}\Phi_i K_{ij}\Phi_j - \frac{1}{2}\Phi_i M_{ij}\Phi_j,
</math>

where <math>K_{ij}</math> and <math>M_{ij}</math> must be [[hermitian matrix|hermitian]] and [[definite matrix|positive-definite]]. Similar expressions exist for fermions. The kinetic matrix can always be brought into a canonical [[diagonal matrix|diagonal]] form while also [[diagonalizable matrix|diagonalizing]] the mass matrix.<ref>{{cite journal|last1=Gedalin|first1=E.|authorlink1=|last2=Moalem|first2=A.|authorlink2=|last3=Razdolskaya|first3=L.|authorlink3=|date=2001|title=Pseudoscalar meson mixing in effective field theory|url=|journal=Phys. Rev. D|volume=64|issue=|pages=076007|doi=10.1103/PhysRevD.64.076007|pmid=|arxiv=hep-ph/0106301|s2cid=|access-date=}}</ref> This is achieved by first diagonalizing the kinetic matrix, then rescaling the fields such that all the kinetic terms are canonically normalized, making the matrix proportional to the [[identity matrix]]. The mass matrix can then be diagonalized, with this second diagonalization not affecting the kinetic matrix as it is proportional to the identity.

It is not always desirable to diagonalize the kinetic and mass matrices as this may end up mixing up interactions in the full theory. For example, propagation of [[neutrino]]s is calculated in the mass [[basis (linear algebra)|basis]], which diagonalizes the kinetic and mass matrices.<ref name="Schwartz"/>{{rp|601–602}} However, the interactions that create neutrinos are written in the [[flavour (particle physics)|flavour]] basis, which instead diagonalizes the coupling of neutrinos to the [[W and Z bosons|W bosons]]. Calculations for each process are done in each respective basis. The disparity between these two basis gives rise to [[neutrino oscillation]]s.

Another example occurs when one has two abelian gauge bosons, where such theories often give rise to a kinetic mixing term.<ref>{{cite book|last1=Bauer|first1=M.|author-link1=|last2=Plehn|first2=T.|author-link2=|date=2019|title=Springer Yet Another Introduction to Dark Matter: The Particle Physics Approach|url=|doi=|location=|publisher=Springer|chapter=4|page=92|isbn=978-3030162337}}</ref> This is a term of the form <math>\epsilon F_{\mu\nu}\tilde F^{\mu\nu}</math>,{{refn|group=nb|Usually <math>\epsilon</math> is taken to be a small parameter so that kinetic mixing is a perturbative interaction between the two gauge bosons.}} which has the effect of converting one gauge boson into another as it propagates. It could be eliminated by diagonalizing the kinetic terms, however this can mix up interactions. Such kinetic mixing is common in the [[phenomenology (physics)|phenomenology]] of [[dark photon]]s.<ref>{{cite journal|last1=Caputo|first1=A.|authorlink1=|last2=Millar|first2=A.J.|authorlink2=|last3=O'Hare|first3=C.A.|authorlink3=|last4=Vitagliano|first4=E.|authorlink4=|date=2021|title=Dark photon limits: A handbook|url=|journal=Phys. Rev. D|volume=104|issue=9|pages=095029|doi=|pmid=|arxiv=2105.04565|s2cid=|access-date=}}</ref>

More general kinetic terms can also occur in scalar field theories in the form of [[non-linear sigma model]]s. In that case the kinetic matrix is replaced by a [[function (mathematics)|function]] of the fields themselves <math>K_{ij}\rightarrow g_{ij}(\phi)</math>.<ref>{{cite book|last1=Freedman|first1=D.Z.|author-link1=Daniel Z. Freedman|last2=Van Proeyen|first2=A.|author-link2=|date=2012|title=Supergravity|url=|doi=|location=Cambridge|publisher=Cambridge University Press|chapter=7|page=163–166|isbn=978-0521194013}}</ref> In these models, this function behaves as a [[metric tensor (general relativity)|metric]] on a [[manifold]], known as a scalar manifold, for which the scalars act as [[coordinate system|coordinates]]. A [[Taylor series|Taylor expansion]] around the [[flat manifold|flat metric]] returns the regular bilinear kinetic terms together with a series of interaction terms.