nLab standard model of particle physics

Idea

The standard model of particle physics is a model (in particle physics): a quantum field theory that describes the fundamental particles currently experimentally known, containing

• matter: electrons, neutrinos, quarks and other fermions,

as well as three of the four fundamental forces as currently known, which, somewhat roughly, are

• force: the electroweak force (including electromagnetism) and the strong nuclear force.

It is defined as a local Lagrangian field theory which is an Einstein-Maxwell-Yang-Mills-Dirac-Higgs theory.

The main ingredient missing from the standard model is the 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 gauge theory with 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.

Properties

Gauge group

The exact gauge group of the standard model is not quite the product group

$$U(1) \times SU(2) \times SU(3)$$

of the circle group with special unitary groups, but is the quotient group

(1)$$G_{SM} \;=\; \big( U(1) \times SU(2) \times SU(3) \big) / \mathbb{Z}_6$$

of that by a cyclic group $\mathbb{Z}_6 \subset U(1) \times SU(2) \times SU(3)$ which is the subgroup generated from an element of the form

$$(q_6, q_2 \mathbf{1}_2, q_3 \mathbf{1}_3) \;\in\; U(1) \times SU(2) \times SU(3) \,,$$

where $q_n \in U(1)$ denotes an $n$th primitive root of unity, i.e.

$$\left( e^{2 \pi i \tfrac{1}{6}} \;,\; e^{2 \pi i \tfrac{1}{2}} \mathbf{1}_2\;,\; e^{2 \pi i \tfrac{1}{3}} \mathbf{1}_3 \right) \;\in\; U(1) \times SU(2) \times SU(3) \,.$$

(See Baez-Huerta 09, p. 33-36 for a fairly comprehensive discussion; also e.g. HMY 13, p. 2.

This is in the sense that all known elementary particles in the standard model are invariant under this $\mathbb{Z}/6$ subgroup. In principle it could be that as yet undetected heavy particles break this invariance after all, cf. Li & Xu 2024.)

Strikingly, this exact gauge group (1) of the standard model of particle physics happens to coincide with…

1. …the group

   $$G_{SM} \;\simeq\; S\big(U(2) \times U(3)\big) \;\subset\; SU(5)$$

   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 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 simple Lie group-symmetry.

2. …the subgroup of the Jordan algebra automorphism group of the octonionic Albert algebra that “stabilizes a 4d sub-Minkowski spacetime” (see there for details).

   More concretely, it is identified with the subgroup of Spin(9) which respects a splitting $\mathbb{H} \oplus \mathbb{H} \simeq_{\mathbb{R}} \mathbb{C} \oplus \mathbb{C}^3$ (Krasnov 19, see also at SO(10)-GUT)

This is part of ongoing speculation that octonionic exceptional structures might be behind the standard model.

Fermion content

flavors of fundamental fermions in the standard model of particle physics:
generation of fermions	1st generation	2nd generation	3d generation
quarks ($q$)
up-type	up quark ($u$)	charm quark ($c$)	top quark ($t$)
down-type	down quark ($d$)	strange quark ($s$)	bottom quark ($b$)
leptons
charged	electron	muon	tauon
neutral	electron neutrino	muon neutrino	tau neutrino
bound states:
mesons	light mesons: pion ($u d$) ρ-meson ($u d$) ω-meson ($u d$) f1-meson a1-meson	strange-mesons: ϕ-meson ($s \bar s$), kaon, K*-meson ($u s$, $d s$) eta-meson ($u u + d d + s s$) charmed heavy mesons: D-meson ($u c$, $d c$, $s c$) J/ψ-meson ($c \bar c$)	bottom heavy mesons: B-meson ($q b$) ϒ-meson ($b \bar b$)
baryons	nucleons: proton $(u u d)$ neutron $(u d d)$

Tension with experiment and Incompleteness

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

• flavour anomaly

• anomalous magnetic moment– anomalies.

Not quite in tension with the standard model, but still possibly pointing to unaccounted effects is the

• Higgs field-metastability.

Then there are experimental phenomena which are known to be implied by the standard model by way of computer simulation (lattice QCD), but which are not conceptually understood:

• confinement (mass gap problem)

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

• quantum gravity

for more on this.

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 above and which may be conceptually more satisfying from the point of view of models in theoretical physics.

Kaluza–Klein 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 fields 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 fields, but it also always contains further scalar fields that are 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.

GUTs

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

Noncommutative geometry

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

• Urs Schreiber, Connes on Spectral Geometry of the Standard Model (part I, part II part III, part IV)

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

• intersecting D-brane models

• G₂-MSSM

Related concepts

• quantum field theory

  • perturbative quantum field theory

  • non-perturbative quantum field theory

• forces

  • electromagnetism

  • strong nuclear force

  • weak nuclear force

    • CKM matrix

• chiral fermions

• chiral perturbation theory, quantum hadrodynamics

• flavorugeneration of fundamental particles

• Yang-Mills theory

  • QCD, electroweak field, electromagnetism

• technicolor, supersymmetry

• Grand Unified Theory, theory of everything

• gauge coupling unification

• relevant experiments

  • LHC experiment

  • LHCb experiment

  • Belle collaboration

• anomalies

  • anomalous magnetic moment of the muonagnetic moment#Anomalies)

  • flavour anomaly

  • V_cb puzzle

• phenomenology, hierarchy problem

• MSSM

  • G₂-MSSM

• intersecting D-brane model

• standard model of cosmology

• string theory FAQ – Did string theory provide any insight relevant in experimental particle physics?

standard model of particle physics and cosmology

theory:	Einstein-	Yang-Mills-	Dirac-	Higgs
	gravity	electroweak and strong nuclear force	fermionic matter	scalar field
field content:	vielbein field $e$	principal connection $\nabla$	spinor $\psi$	scalar field $H$
Lagrangian:	scalar curvature density	field strength squared	Dirac operator component density	field strength squared + potential density
$L =$	$R(e) vol(e) +$	$\langle F_\nabla \wedge \star_e F_\nabla\rangle +$	$(\psi , D_{(e,\nabla)} \psi) vol(e) +$	$\nabla \bar H \wedge \star_e \nabla H + \left(\lambda {\vert H\vert}^4 - \mu^2 {\vert H\vert}^2 \right) vol(e)$

References

General

Textbook accounts:

• W. Noel Cottingham, ; Derek A. Greenwood, An introduction to the standard model of particle physics, Cambridge University Press 2012 (doi:10.1017/CBO9780511791406, ZMATH entry)

• Paul Langacker, The Standard Model and Beyond, CRC Press 2009, 2012, second edition 2017 (spire:846915, publisher webpage, author webpage)

• J D Vergados, The Standard Model and Beyond, World Scientific 2017 (doi:10.1142/10669)

Review and introduction:

• Matthew Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press (2014) [ISBN:9781107034730, doi:10.1017/9781139540940, pdf]

• Gerd Rudolph, Matthias Schmidt, Section 7.7 of: Differential Geometry and Mathematical Physics Part II. Fibre Bundles, Topology and Gauge Fields, Springer 2017 (doi:10.1007/978-94-024-0959-8)

• Ben Gripaios, Lectures: From quantum mechanics to the Standard Model (arXiv:2005.06355)

• Fernando Quevedo, Andreas Schachner: Cambridge Lectures on The Standard Model [arXiv:2409.09211]

Mathematical (representation theoretic) review with an eye towards grand unified theory:

• Howard Georgi, Lie Algebras In Particle Physics, Westview Press (1999), CRC Press (2019) [doi:10.1201/9780429499210]

• John Baez, John Huerta, The Algebra of Grand Unified Theories, Bull. Am. Math. Soc. 47:483-552, 2010 (arXiv:0904.1556)

and with an eye towards intersecting D-brane modelsL

• Luis Ibáñez, Angel Uranga, section 1 of String Theory and Particle Physics – An Introduction to String Phenomenology, Cambridge University Press (2012)

and with emphasis of the effective field theories involved (chiral perturbation theory, Fermi’s theory):

• Luis Alvarez-Gaumé, Miguel A. Vázquez-Mozo, Lectures on Field Theory and the Standard Model: A Symmetry-Oriented Approach [arXiv:2306.08097]

More on the issue of the exact gauge group:

• Hao-Lin Li, Ling-Xiao Xu, Understanding the SM gauge group from SMEFT [arXiv:2404.04229]

On subgroups of Clifford algebras reminiscent of gauge groups of relevance in the standard model of particle physics and grand unified theories thereof:

• Robert A. Wilson, A group-theorist’s perspective on symmetry groups in physics (arXiv:2009.14613)

• Robert A. Wilson, On the Problem of Choosing Subgroups of Clifford Algebras for Applications in Fundamental Physics, Adv. Appl. Clifford Algebras 31, 59 (2021) (doi:10.1007/s00006-021-01160-5)

See also:

• Albert algebra – possible relation to color gauge structure

• Alain Connes, Matilde Marcolli, chapter I, section 12 of Noncommutative Geometry, Quantum Fields and Motives

Review with outlook “beyond the standard model”:

• Paul Langacker, The standard model and beyond, talk at TPFNP 2005 (pdf, Langacker05.pdf)

• Dmitry Kazakov, Beyond the Standard Model’ 17 (arXiv:1807.00148)

• Andrea Wulzer, Behind the Standard Model (arXiv:1901.01017)

• James Wells, The Once and Present Standard Model of Elementary Particle Physics (arxiv:1911.04604)

Anomaly cancellation

Discussion of anomaly cancellation on the standard model of particle physics:

• Takaaki Hashimoto, Mamoru Matsunaga, Kenta Yamamoto, Quantization of hypercharge in gauge groups locally isomorphic but globally nonisomorphic to $SU(3)_c \times SU(2)_L \times U(1)_Y$ (arXiv:1302.0669)

• Nakarin Lohitsiri, David Tong, Hypercharge Quantisation and Fermat’s Last Theorem (arXiv:1907.00514)

  (relating to Fermat's last theorem)

History

• James Bjorken, The November Revolution: A Theorist Reminisces, 1985 (spire:214067)

The big international conference of $[$1974$]$ in London was a turning point $[$…$]$ Ellis’ catalog well reflected the state of theoretical confusion and general disarray in trying to interpret the $e^+ e^-$ data. But in the midst of all of this was a talk by John Iliopoulos (I think I was there too). With passionate zealotry, he laid out with great accuracy what we call the standard model. Everything was there: proton decay, charm, the GIM mechanism of course, QCD, the $SU(2)\times U(1)$ electroweak theory, $SU(5)$ grand unification, Higgs, etc. It was all presented with absolute conviction and sounded at the time just a little mad, at least to me (I am a conservative).

• Tom Kibble, The Standard Model of Particle Physics (arXiv:1412.4094)

• John Iliopoulos, The making of the standard theory, Adv.Ser.Direct.High Energy Phys. 26 (2016) 29-59 (spire:1497884, pdf)