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\section*{standard model of particle physics}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{fields_and_quanta}{}\paragraph*{{Fields and quanta}}\label{fields_and_quanta}

[[!include fields and quanta - table]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{GaugeGroup}{Gauge group}\dotfill \pageref*{GaugeGroup} \linebreak
\noindent\hyperlink{TensionWithExperiment}{Tension with experiment and Incompleteness}\dotfill \pageref*{TensionWithExperiment} \linebreak
\noindent\hyperlink{variations_and_generalizations}{Variations and generalizations}\dotfill \pageref*{variations_and_generalizations} \linebreak
\noindent\hyperlink{kaluzaklein_theory}{Kaluza--Klein theory}\dotfill \pageref*{kaluzaklein_theory} \linebreak
\noindent\hyperlink{guts}{GUTs}\dotfill \pageref*{guts} \linebreak
\noindent\hyperlink{NCGeometry}{Noncommutative geometry}\dotfill \pageref*{NCGeometry} \linebreak
\noindent\hyperlink{string_theory}{String theory}\dotfill \pageref*{string_theory} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \textbf{standard [[model (in particle physics)|model]] of [[particle physics]]} is a [[model (in particle physics)]]: a [[quantum field theory]] that describes the \emph{[[fundamental particles]]} currently experimentally known, containing

\begin{itemize}%
\item \textbf{[[matter]]}: [[electrons]], [[neutrinos]], [[quarks]] and other [[fermions]],

\end{itemize}
as well as three of the four \emph{fundamental [[forces]]} as currently known, which, somewhat roughly, are

\begin{itemize}%
\item \textbf{[[force]]}: the [[electroweak field|electroweak force]] (including [[electromagnetism]]) and the [[strong nuclear force]].

\end{itemize}
It is defined as a [[local Lagrangian]] [[field (physics)|field]] [[theory (physics)|theory]] which is an [[Einstein-Maxwell-Yang-Mills-Dirac-Higgs theory]].

The main ingredient missing from the standard model is the [[quantum physics|quantum]] version of the field of [[gravity]]. For decades, a large part of theoretical physics has been absorbed with attempts to understand how this last of the known fundamental forces might fit into the picture.

As a [[quantum field theory]], the standard model is in particular a [[Yang-Mills theory|Yang–Mills]] [[gauge theory]] with [[spinors in Yang-Mills theory|spinors in Yang–Mills theory]].

Although there are several approaches to formulate a mathematically precise definition of what a [[quantum field theory]] is, there is no rigorous formulation (yet) that comprises the whole standard model.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{GaugeGroup}{}\subsubsection*{{Gauge group}}\label{GaugeGroup}

The \emph{exact} [[gauge group]] of the standard model is not quite the [[product group]]

\begin{displaymath}
U(1) \times SU(2) \times SU(3)
\end{displaymath}
of the [[circle group]] with [[special unitary groups]], but is the [[quotient group]]

\begin{equation}
G_{SM} 
  \;=\;
  \big(
    U(1) \times SU(2) \times SU(3)
  \big)
  / \mathbb{Z}_6
\label{ExactGSM}\end{equation}
of that by a [[cyclic group]] $\mathbb{Z}_6 \subset U(1) \times SU(2) \times SU(3)$ which is the [[subgroup]] [[generators and relations|generated]] from an element of the form

\begin{displaymath}
(q_6, q_2 \mathbf{1}_2, q_3 \mathbf{1}_3) \;\in\;
  U(1) \times SU(2) \times SU(3)
  \,,
\end{displaymath}
where $q_n \in U(1)$ denotes an $n$th [[primitive root of unity]], i.e.

\begin{displaymath}
\left(
    e^{2 \pi i \tfrac{1}{6}}
    \;,\; 
    e^{2 \pi i \tfrac{1}{2}} \mathbf{1}_2\;,\; 
    e^{2 \pi i \tfrac{1}{3}} \mathbf{1}_3
  \right) 
  \;\in\;
  U(1) \times SU(2) \times SU(3)
  \,.
\end{displaymath}
(See \hyperlink{BaezHuerta09}{Baez-Huerta 09, p. 33-36} for a fairly comprehensive discussion. See also e.g. \hyperlink{HMY13}{HMY 13, p. 2}.)

Strikingly, this exact gauge group \eqref{ExactGSM} of the standard model of particle physics happens to coincide with\ldots{}

\begin{enumerate}%
\item \ldots{}the group

\begin{displaymath}
G_{SM} 
    \;\simeq\; 
   S\big(U(2) \times U(3)\big) 
    \;\subset\; 
   SU(5)
\end{displaymath}
of [[determinant]]-1 elements in the [[direct product group]] $U(2) \times U(3)$, which makes manifest that the standard model gauge group is a [[subgroup]] of the [[simple Lie group]] [[special unitary group|SU(5)]].

This is the basis of ``[[grand unified theories]]'' ([[GUT]]), speculative extensions of the standard model trying to understand its gauge group as being a [[spontaneously broken symmetry|spontaneously broken]] \emph{[[simple Lie group]]}-[[symmetry]].


\item \ldots{}the [[subgroup]] of the [[Jordan algebra]] [[automorphism group]] of the [[octonions|octonionic]] [[Albert algebra]] that ``[[stabilizer subgroup|stabilizes]] a 4d sub-[[Minkowski spacetime]]'' (see \href{Albert+algebra#StabilizerOf4dMinkowskiInsideOctonionicAlbertAlgebra}{there} for details) -- this is part of ongoing speculation that exceptional [[octonion|octonionic]] structures might be behind the standard model.



\end{enumerate}
\hypertarget{TensionWithExperiment}{}\subsection*{{Tension with experiment and Incompleteness}}\label{TensionWithExperiment}

The standard model of particle physics is supported by a tremendous level of confirmation by [[experiment]], lately by the [[LHC]] experiment. But there are indictations of discrepancies, pointing to ``new physics''. See at

\begin{itemize}%
\item [[flavour anomaly]]


\item [[anomalous magnetic moment]] -- \href{anomalous+magnetic+moment#Anomalies}{anomalies}.



\end{itemize}
Not quite in tension with the standard model, but still possibly pointing to unaccounted effects is the

\begin{itemize}%
\item [[Higgs field]]-\href{Higgs+field#MassAndVacuumInstability}{metastability}.

\end{itemize}
Then there are experimental phenomena which are known to be implied by the standard model by way of computer simulatrion ([[lattice QCD]]), but which are not conceptually understood:

\begin{itemize}%
\item [[confinement]] ([[mass gap problem]])

\end{itemize}
Finally there is the evident and notorious fact that the standard model simply does not include any [[quantum field theory]] of the fourth known fundamental [[force]], [[gravity]]. See at

\begin{itemize}%
\item [[quantum gravity]]

\end{itemize}
for more on this.



\hypertarget{variations_and_generalizations}{}\subsection*{{Variations and generalizations}}\label{variations_and_generalizations}

There is a plethora of attempts and suggestions for variations and generalizations of the standard model into models that may resolve the tension with experiment mentioned \hyperlink{TensionWithExperiment}{above} and which may be conceptually more satisfying from the point of view of models in theoretical physics.

\hypertarget{kaluzaklein_theory}{}\subsubsection*{{Kaluza--Klein theory}}\label{kaluzaklein_theory}

Shortly after the conception of [[general relativity]], it was observed by Kaluza and Klein that the force of [[gravity]] alone may effectively appear -- if considered on a [[spacetime]] that is a [[bundle]] whose [[fiber]] has a tiny volume (as meaured by the [[Riemannian metric]]) -- as the field of gravity coupled to [[gauge field]]s on the base of the bundle. For details on this see [[Kaluza-Klein mechanism]].

The huge conceptual simplification that this observation suggested had excited theoreticians early on, but a problem of Kaluza--Klein models is that not only does the ``compactified'' theory of gravity as if by magic emulate [[gauge field]]s, but it also always contains further scalar fields that are \emph{not} experimentally observed.

For that reason interest in Kaluza--Klein theories had decreased in the middle of the last century. Physics departments saw a major revival of the idea when [[string theory]] (see below) gained interest, since that theory necessarily exhibits a Kaluza--Klein mechanism. Incidentally, the problem of the spurious fields -- the moduli -- was still present in this approach. For more on this see the entry [[landscape of string theory vacua]].

\hypertarget{guts}{}\subsubsection*{{GUTs}}\label{guts}

One of the oldest studies of variations of the standard model is the investigation of [[GUT|grand unified theories]] (GUTs), which are [[Yang-Mills theory|Yang–Mills theories]] that instead of the standard model gauge group have a bigger gauge group which is however a [[simple group]].

\hypertarget{NCGeometry}{}\subsubsection*{{Noncommutative geometry}}\label{NCGeometry}

A widespread perception is that some of the conceptual problems with the standard model point to the fact that some basic assumption of 20th century physics on the nature of reality is oversimplified. According to the approach of [[noncommutative geometry]], modeling [[spacetime]] as a smooth [[manifold]] is an oversimplification that makes itself felt when the quantization of the force of [[gravity]] becomes relevant.

In a class of ``noncommutative'' generalizations of the standard model, spacetime is therefore replaced more generally by a [[spectral triple]] that models a possibly ``noncommutative space''. One of the more successful approaches in this direction is the [[Connes-Lott-Chamseddine model]]. This effectively is a [[Kaluza-Klein theory]] (see above), but with the crucial difference that the fiber in the KK-picture is a highly non-classical non-commutative space, whose classical dimension is that of a point, but whose intrinsic dimension is 6. (This is incidentally the same value of the internal dimension as suggest by [[string theory]].)

For more on this see

\begin{itemize}%
\item [[Urs Schreiber]], \emph{Connes on Spectral Geometry of the Standard Model} (\href{http://golem.ph.utexas.edu/category/2006/09/connes_on_spectral_geometry_of.html}{part I}, \href{http://golem.ph.utexas.edu/category/2006/09/connes_on_spectral_geometry_of_1.html}{part II} \href{http://golem.ph.utexas.edu/category/2006/09/connes_on_spectral_geometry_of_2.html}{part III}, \href{http://golem.ph.utexas.edu/category/2006/09/connes_on_spectral_geometry_of_3.html}{part IV})

\end{itemize}
\hypertarget{string_theory}{}\subsubsection*{{String theory}}\label{string_theory}

A more drastic theoretical modifications to the standard model is proposed in the context of [[string theory]], where the entire concept of [[quantum field theory]] is proposed to be refined by something else. As opposed to [[GUTs]], this approach at least suggests a way in which also the fourth remaining force field of [[gravity]] could be incorporated into the picture.

See

\begin{itemize}%
\item [[intersecting D-brane models]]


\item [[G2-MSSM]]



\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[quantum field theory]]

\begin{itemize}%
\item [[perturbative quantum field theory]]


\item [[non-perturbative quantum field theory]]



\end{itemize}

\item [[forces]]

\begin{itemize}%
\item [[electromagnetism]]


\item [[strong nuclear force]]


\item [[weak nuclear force]]

\begin{itemize}%
\item [[CKM matrix]]

\end{itemize}


\end{itemize}

\item [[chiral fermions]]


\item [[generation of fundamental particles]]


\item [[Yang-Mills theory]]

\begin{itemize}%
\item [[QCD]], [[electroweak field]], [[electromagnetism]]

\end{itemize}

\item [[technicolor]], [[supersymmetry]]


\item [[Grand Unified Theory]], [[theory of everything]]


\item [[gauge coupling unification]]


\item relevant [[experiments]]

\begin{itemize}%
\item [[LHC experiment]]


\item [[LHCb experiment]]


\item [[Belle collaboration]]



\end{itemize}

\item anomalies

\begin{itemize}%
\item anomalous magnetic moment of the muonagnetic moment\#Anomalies)


\item [[flavour anomaly]]


\item [[V\_cb puzzle]]



\end{itemize}

\item [[phenomenology]], [[hierarchy problem]]


\item [[MSSM]]

\begin{itemize}%
\item [[G2-MSSM]]

\end{itemize}

\item [[intersecting D-brane model]]


\item [[standard model of cosmology]]


\item \href{string+theory+FAQ#InsightInParticlePhysics}{string theory FAQ -- Did string theory provide any insight relevant in experimental particle physics?}



\end{itemize}
[[!include standard model of fundamental physics - table]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[John Iliopoulos]], \emph{The making of the standard theory}, Adv.Ser.Direct.High Energy Phys. 26 (2016) 29-59 (\href{http://inspirehep.net/record/1497884}{spire:1497884}, \href{http://cds.cern.ch/record/2217096/files/9789814733519_0002.pdf}{pdf})

\end{itemize}
Textbook accounts include

\begin{itemize}%
\item W. Noel Cottingham, ; Derek A. Greenwood, \emph{An introduction to the standard model of particle physics}, Cambridge University Press 2012 (\href{https://doi.org/10.1017/CBO9780511791406}{doi:10.1017/CBO9780511791406}, \href{http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1126.81002&format=complete}{ZMATH entry})


\item [[Paul Langacker]], \emph{The Standard Model and Beyond}, CRC Press 2009, 2012, second edition 2017 (\href{http://inspirehep.net/record/846915/}{spire:846915}, \href{https://www.crcpress.com/The-Standard-Model-and-Beyond/Langacker/p/book/9781498763219}{publisher webpage}, \href{http://www.sns.ias.edu/~pgl/SMB/}{author webpage})


\item J D Vergados, \emph{The Standard Model and Beyond}, World Scientific 2017 (\href{https://www.worldscientific.com/worldscibooks/10.1142/10669}{doi:10.1142/10669})



\end{itemize}
Further review and outlook includes

\begin{itemize}%
\item [[Paul Langacker]], \emph{The standard model and beyond}, talk at TPFNP 2005 (\href{http://www.sns.ias.edu/~pgl/talks/neutron.pdf}{pdf}, [[Langacker05.pdf:file]])


\item Dmitry Kazakov, \emph{Beyond the Standard Model' 17} (\href{https://arxiv.org/abs/1807.00148}{arXiv:1807.00148})


\item Andrea Wulzer, \emph{Behind the Standard Model} (\href{https://arxiv.org/abs/1901.01017}{arXiv:1901.01017})



\end{itemize}
Review specifically with an eye towards [[grand unified theory]] is in

\begin{itemize}%
\item [[John Baez]], [[John Huerta]], \emph{The Algebra of Grand Unified Theories}, Bull.Am.Math.Soc.47:483-552,2010 (\href{http://arxiv.org/abs/0904.1556}{arXiv:0904.1556})

\end{itemize}
and specificlly with an eye towards [[intersecting D-brane models]] in

\begin{itemize}%
\item [[Luis Ibáñez]], [[Angel Uranga]], section 1 of \emph{[[String Theory and Particle Physics -- An Introduction to String Phenomenology]]}, Cambridge University Press 2012

\end{itemize}
See also

\begin{itemize}%
\item [[Alain Connes]], [[Matilde Marcolli]], chapter I, section 12 of \emph{[[Noncommutative Geometry, Quantum Fields and Motives]]}

\end{itemize}
A historical account is in

\begin{itemize}%
\item [[Tom Kibble]], \emph{The Standard Model of Particle Physics} (\href{http://arxiv.org/abs/1412.4094}{arXiv:1412.4094})

\end{itemize}
See also

\begin{itemize}%
\item Takaaki Hashimoto, Mamoru Matsunaga, Kenta Yamamoto, \emph{Quantization of hypercharge in gauge groups locally isomorphic but globally nonisomorphic to $SU(3)_c \times SU(2)_L \times U(1)_Y$} (\href{https://arxiv.org/abs/1302.0669}{arXiv:1302.0669})

\end{itemize}


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