Standard Model

The Standard Model includes 12 elementary particles of spin 1⁄2, known as fermions. According to the spin–statistics theorem, fermions respect the Pauli exclusion principle. Each fermion has a corresponding antiparticle.

Fermions are classified according to how they interact (or equivalently, by what charges they carry). There are six quarks (up, down, charm, strange, top, bottom), and six leptons (electron, electron neutrino, muon, muon neutrino, tau, tau neutrino). Each class is divided into pairs of particles that exhibit a similar physical behavior called a generation (see the table).

The defining property of quarks is that they carry color charge, and hence interact via the strong interaction. The phenomenon of color confinement results in quarks being very strongly bound to one another, forming color-neutral composite particles called hadrons that contain either a quark and an antiquark (mesons) or three quarks (baryons). The lightest baryons are the proton and the neutron. Quarks also carry electric charge and weak isospin. Hence they interact with other fermions via electromagnetism and the weak interaction. The remaining six fermions do not carry color charge and are called leptons. The three neutrinos do not carry electric charge either, so their motion is directly influenced only by the weak nuclear force and gravity, which makes them notoriously difficult to detect. By contrast, by virtue of carrying an electric charge, the electron, muon, and tau all interact electromagnetically.

Each member of a generation has greater mass than the corresponding particle of any generation before it. The first-generation charged particles do not decay, hence all ordinary (baryonic) matter is made of such particles. Specifically, all atoms consist of electrons orbiting around atomic nuclei, ultimately constituted of up and down quarks. On the other hand, second- and third-generation charged particles decay with very short half-lives and are observed only in very high-energy environments. Neutrinos of all generations also do not decay, and pervade the universe, but rarely interact with baryonic matter.

Interactions in the Standard Model. All Feynman diagrams in the model are built from combinations of these vertices. q is any quark, g is a gluon, X is any charged particle, γ is a photon, f is any fermion, m is any particle with mass (with the possible exception of the neutrinos), m_B is any boson with mass. In diagrams with multiple particle labels separated by / one particle label is chosen. In diagrams with particle labels separated by | the labels must be chosen in the same order. For example, in the four boson electroweak case the valid diagrams are WWWW, WWZZ, WWγγ, WWZγ. The conjugate of each listed vertex (reversing the direction of arrows) is also allowed.[27]

In the Standard Model, gauge bosons are defined as force carriers that mediate the strong, weak, and electromagnetic fundamental interactions.

Interactions in physics are the ways that particles influence other particles. At a macroscopic level, electromagnetism allows particles to interact with one another via electric and magnetic fields, and gravitation allows particles with mass to attract one another in accordance with Einstein's theory of general relativity. The Standard Model explains such forces as resulting from matter particles exchanging other particles, generally referred to as force mediating particles. When a force-mediating particle is exchanged, the effect at a macroscopic level is equivalent to a force influencing both of them, and the particle is therefore said to have mediated (i.e., been the agent of) that force.[28] The Feynman diagram calculations, which are a graphical representation of the perturbation theory approximation, invoke "force mediating particles", and when applied to analyze high-energy scattering experiments are in reasonable agreement with the data. However, perturbation theory (and with it the concept of a "force-mediating particle") fails in other situations. These include low-energy quantum chromodynamics, bound states, and solitons.

The gauge bosons of the Standard Model all have spin (as do matter particles). The value of the spin is 1, making them bosons. As a result, they do not follow the Pauli exclusion principle that constrains fermions: thus bosons (e.g. photons) do not have a theoretical limit on their spatial density (number per volume). The types of gauge bosons are described below.

• Photons mediate the electromagnetic force between electrically charged particles. The photon is massless and is well-described by the theory of quantum electrodynamics.
• The
  W⁺
  ,
  W⁻
  , and
  Z
  gauge bosons mediate the weak interactions between particles of different flavours (all quarks and leptons). They are massive, with the
  Z
  being more massive than the
  W^±
  . The weak interactions involving the
  W^±
  act only on left-handed particles and right-handed antiparticles. The
  W^±
  carries an electric charge of +1 and −1 and couples to the electromagnetic interaction. The electrically neutral
  Z
  boson interacts with both left-handed particles and right-handed antiparticles. These three gauge bosons along with the photons are grouped together, as collectively mediating the electroweak interaction.
• The eight gluons mediate the strong interactions between color charged particles (the quarks). Gluons are massless. The eightfold multiplicity of gluons is labeled by a combination of color and anticolor charge (e.g. red–antigreen).[note 2] Because gluons have an effective color charge, they can also interact among themselves. Gluons and their interactions are described by the theory of quantum chromodynamics.

The interactions between all the particles described by the Standard Model are summarized by the diagrams on the right of this section.

The Higgs particle is a massive scalar elementary particle theorized by Peter Higgs in 1964, when he showed that Goldstone's 1962 theorem (generic continuous symmetry, which is spontaneously broken) provides a third polarisation of a massive vector field. Hence, Goldstone's original scalar doublet, the massive spin-zero particle, was proposed as the Higgs boson, and is a key building block in the Standard Model.[29] It has no intrinsic spin, and for that reason is classified as a boson (like the gauge bosons, which have integer spin).

The Higgs boson plays a unique role in the Standard Model, by explaining why the other elementary particles, except the photon and gluon, are massive. In particular, the Higgs boson explains why the photon has no mass, while the W and Z bosons are very heavy. Elementary-particle masses, and the differences between electromagnetism (mediated by the photon) and the weak force (mediated by the W and Z bosons), are critical to many aspects of the structure of microscopic (and hence macroscopic) matter. In electroweak theory, the Higgs boson generates the masses of the leptons (electron, muon, and tau) and quarks. As the Higgs boson is massive, it must interact with itself.

Because the Higgs boson is a very massive particle and also decays almost immediately when created, only a very high-energy particle accelerator can observe and record it. Experiments to confirm and determine the nature of the Higgs boson using the Large Hadron Collider (LHC) at CERN began in early 2010 and were performed at Fermilab's Tevatron until its closure in late 2011. Mathematical consistency of the Standard Model requires that any mechanism capable of generating the masses of elementary particles must become visible at energies above 1.4 TeV;[30] therefore, the LHC (designed to collide two 7 TeV proton beams) was built to answer the question of whether the Higgs boson actually exists.[31]

On 4 July 2012, two of the experiments at the LHC (ATLAS and CMS) both reported independently that they had found a new particle with a mass of about 125 GeV/c² (about 133 proton masses, on the order of 10×10⁻²⁵ kg), which is "consistent with the Higgs boson".[32][33] On 13 March 2013, it was confirmed to be the searched-for Higgs boson.[34][35]

Technically, quantum field theory provides the mathematical framework for the Standard Model, in which a Lagrangian controls the dynamics and kinematics of the theory. Each kind of particle is described in terms of a dynamical field that pervades space-time.[36] The construction of the Standard Model proceeds following the modern method of constructing most field theories: by first postulating a set of symmetries of the system, and then by writing down the most general renormalizable Lagrangian from its particle (field) content that observes these symmetries.

The global Poincaré symmetry is postulated for all relativistic quantum field theories. It consists of the familiar translational symmetry, rotational symmetry and the inertial reference frame invariance central to the theory of special relativity. The local SU(3)×SU(2)×U(1) gauge symmetry is an internal symmetry that essentially defines the Standard Model. Roughly, the three factors of the gauge symmetry give rise to the three fundamental interactions. The fields fall into different representations of the various symmetry groups of the Standard Model (see table). Upon writing the most general Lagrangian, one finds that the dynamics depends on 19 parameters, whose numerical values are established by experiment. The parameters are summarized in the table (made visible by clicking "show") above.

The quantum chromodynamics (QCD) sector defines the interactions between quarks and gluons, which is a Yang–Mills gauge theory with SU(3) symmetry, generated by T^a. Since leptons do not interact with gluons, they are not affected by this sector. The Dirac Lagrangian of the quarks coupled to the gluon fields is given by

$${\displaystyle {\mathcal {L}}_{\text{QCD}}=\sum _{\psi }{\overline {\psi }}_{i}\left(i\gamma ^{\mu }(\partial _{\mu }\delta _{ij}-ig_{s}G_{\mu }^{a}T_{ij}^{a})\right)\psi _{j}-{\frac {1}{4}}G_{\mu \nu }^{a}G_{a}^{\mu \nu },}$$

where

• ψ
  _i is the Dirac spinor of the quark field, where i = {r, g, b} represents color,
• γ^μ are the Dirac matrices,
• G^a
  _μ is the 8-component (${\displaystyle a=1,2,\dots ,8}$) SU(3) gauge field,
• T^a
  _ij are the 3 × 3 Gell-Mann matrices, generators of the SU(3) color group,
• G^a
  _μν represents the gluon field strength tensor,
• g_s is the strong coupling constant.

The electroweak sector is a Yang–Mills gauge theory with the symmetry group U(1) × SU(2)_L,

$${\displaystyle {\mathcal {L}}_{\text{EW}}=\sum _{\psi }{\bar {\psi }}\gamma ^{\mu }\left(i\partial _{\mu }-g'{\tfrac {1}{2}}Y_{\text{W}}B_{\mu }-g{\tfrac {1}{2}}{\vec {\tau }}_{\text{L}}{\vec {W}}_{\mu }\right)\psi -{\tfrac {1}{4}}W_{a}^{\mu \nu }W_{\mu \nu }^{a}-{\tfrac {1}{4}}B^{\mu \nu }B_{\mu \nu },}$$

where

• B_μ is the U(1) gauge field,
• Y_W is the weak hypercharge – the generator of the U(1) group,
• W→_μ is the 3-component SU(2) gauge field,
• τ_L→ are the Pauli matrices – infinitesimal generators of the SU(2) group – with subscript L to indicate that they only act on left-chiral fermions,
• g' and g are the U(1) and SU(2) coupling constants respectively,
• ${\displaystyle W^{a\mu \nu }}$ (${\displaystyle a=1,2,3}$) and ${\displaystyle B^{\mu \nu }}$ are the field strength tensors for the weak isospin and weak hypercharge fields.

Notice that the addition of fermion mass terms into the electroweak Lagrangian is forbidden, since terms of the form ${\displaystyle m{\overline {\psi }}\psi }$ do not respect U(1) × SU(2)_L gauge invariance. Neither is it possible to add explicit mass terms for the U(1) and SU(2) gauge fields. The Higgs mechanism is responsible for the generation of the gauge boson masses, and the fermion masses result from Yukawa-type interactions with the Higgs field.

In the Standard Model, the Higgs field is a complex scalar of the group SU(2)_L:

$${\displaystyle \varphi ={\frac {1}{\sqrt {2}}}{\begin{pmatrix}\varphi ^{+}\\\varphi ^{0}\end{pmatrix}},}$$

where the superscripts + and 0 indicate the electric charge (Q) of the components. The weak hypercharge (Y_W) of both components is 1.

Before symmetry breaking, the Higgs Lagrangian is

$${\displaystyle {\mathcal {L}}_{\text{H}}=\varphi ^{\dagger }\left(\partial ^{\mu }-{\frac {i}{2}}\left(g'Y_{\text{W}}B^{\mu }+g{\vec {\tau }}{\vec {W}}^{\mu }\right)\right)\left(\partial _{\mu }+{\frac {i}{2}}\left(g'Y_{\text{W}}B_{\mu }+g{\vec {\tau }}{\vec {W}}_{\mu }\right)\right)\varphi -{\frac {\lambda ^{2}}{4}}\left(\varphi ^{\dagger }\varphi -v^{2}\right)^{2},}$$

which up to a divergence term, (i.e., after partial integration) can also be written as

$${\displaystyle {\mathcal {L}}_{\text{H}}=\left|\left(\partial _{\mu }+{\frac {i}{2}}\left(g'Y_{\text{W}}B_{\mu }+g{\vec {\tau }}{\vec {W}}_{\mu }\right)\right)\varphi \right|^{2}-{\frac {\lambda ^{2}}{4}}\left(\varphi ^{\dagger }\varphi -v^{2}\right)^{2}.}$$

The Higgs self-coupling strength λ is approximately 1⁄8. This is not included in the table above because it can be derived from the mass (after symmetry breaking) and the vacuum expectation value.

The Yukawa interaction terms are

$${\displaystyle {\mathcal {L}}_{\text{Yukawa}}={\overline {U}}_{\rm {L}}G_{\rm {u}}U_{\rm {R}}\varphi ^{0}-{\overline {D}}_{\rm {L}}G_{\rm {u}}U_{\rm {R}}\varphi ^{-}+{\overline {U}}_{\rm {L}}G_{\rm {d}}D_{\rm {R}}\varphi ^{+}+{\overline {D}}_{\rm {L}}G_{\rm {d}}D_{\rm {R}}\varphi ^{0}+\mathrm {h.c.} ,}$$

where G_u,d are 3 × 3 matrices of Yukawa couplings, with the ij term giving the coupling of the generations i and j, and h.c. means Hermitian conjugate of preceding terms.

Despite being perhaps the most familiar fundamental interaction, gravity is not described by the Standard Model, due to contradictions that arise when combining general relativity, the modern theory of gravity, and quantum mechanics. However, gravity is so weak at microscopic scales, that it is essentially unmeasurable. The graviton is postulated as the mediating particle.

Electromagnetism is the only long-range force in the Standard Model. It is mediated by photons and couples to electric charge. Electromagnetism is responsible for a wide range of phenomena including atomic electron shell structure, chemical bonds, electric circuits and electronics. Electromagnetic interactions in the Standard Model are described by quantum electrodynamics.

The weak interaction is responsible for various forms of particle decay, such as beta decay. It is weak and short-range, due to the fact that the weak mediating particles, W and Z bosons, have mass. W bosons have electric charge and mediate interactions that change the particle type (referred to as flavour) and charge. Interactions mediated by W bosons are charged current interactions. Z bosons are neutral and mediate neutral current interactions, which do not change particle flavour. Thus Z bosons are similar to the photon, aside from them being massive and interacting with the neutrino. The weak interaction is also the only interaction to violate parity and CP. Parity violation is maximal for charged current interactions, since the W boson interacts exclusively with left-handed fermions and right-handed antifermions.

In the Standard Model, the weak force is understood in terms of the electroweak theory, which states that the weak and electromagnetic interactions become united into a single electroweak interaction at high energies.

The strong nuclear force is responsible for hadronic and nuclear binding. It is mediated by gluons, which couple to color charge. Since gluons themselves have color charge, the strong force exhibits confinement and asymptotic freedom. Confinement means that only color-neutral particles can exist in isolation, therefore quarks can only exist in hadrons and never in isolation, at low energies. Asymptotic freedom means that the strong force becomes weaker, as the energy scale increases. The strong force overpowers the electrostatic repulsion of protons and quarks in nuclei and hadrons respectively, at their respective scales.

While quarks are bound in hadrons by the fundamental strong interaction, which is mediated by gluons, nucleons are bound by an emergent phenomenon termed the residual strong force or nuclear force. This interaction is mediated by mesons, such as the pion. The color charges inside the nucleon cancel out, meaning most of the gluon and quark fields cancel out outside of the nucleon. However, some residue is "leaked", which appears as the exchange of virtual mesons, that causes the attractive force between nucleons. The (fundamental) strong interaction is described by quantum chromodynamics, which is a component of the Standard Model.

• I. Aitchison; A. Hey (2003). Gauge Theories in Particle Physics: A Practical Introduction. Institute of Physics. ISBN 978-0-585-44550-2.
• W. Greiner; B. Müller (2000). Gauge Theory of Weak Interactions. Springer. ISBN 978-3-540-67672-0.
• G.D. Coughlan; J.E. Dodd; B.M. Gripaios (2006). The Ideas of Particle Physics: An Introduction for Scientists. Cambridge University Press.
• D.J. Griffiths (1987). Introduction to Elementary Particles. John Wiley & Sons. ISBN 978-0-471-60386-3.
• G.L. Kane (1987). Modern Elementary Particle Physics. Perseus Books. ISBN 978-0-201-11749-3.

• T.P. Cheng; L.F. Li (2006). Gauge theory of elementary particle physics. Oxford University Press. ISBN 978-0-19-851961-4. Highlights the gauge theory aspects of the Standard Model.
• J.F. Donoghue; E. Golowich; B.R. Holstein (1994). Dynamics of the Standard Model. Cambridge University Press. ISBN 978-0-521-47652-2. Highlights dynamical and phenomenological aspects of the Standard Model.
• L. O'Raifeartaigh (1988). Group structure of gauge theories. Cambridge University Press. ISBN 978-0-521-34785-3.
• Nagashima, Yorikiyo (2013). Elementary Particle Physics: Foundations of the Standard Model, Volume 2. Wiley. ISBN 978-3-527-64890-0. 920 pages.
• Schwartz, Matthew D. (2014). Quantum Field Theory and the Standard Model. Cambridge University. ISBN 978-1-107-03473-0. 952 pages.
• Langacker, Paul (2009). The Standard Model and Beyond. CRC Press. ISBN 978-1-4200-7907-4. 670 pages. Highlights group-theoretical aspects of the Standard Model.

• E.S. Abers; B.W. Lee (1973). "Gauge theories". Physics Reports. 9 (1): 1–141. Bibcode:1973PhR.....9....1A. doi:10.1016/0370-1573(73)90027-6.
• M. Baak; et al. (2012). "The Electroweak Fit of the Standard Model after the Discovery of a New Boson at the LHC". The European Physical Journal C. 72 (11): 2205. arXiv:1209.2716. Bibcode:2012EPJC...72.2205B. doi:10.1140/epjc/s10052-012-2205-9. S2CID 15052448.
• Y. Hayato; et al. (1999). "Search for Proton Decay through p → νK⁺ in a Large Water Cherenkov Detector". Physical Review Letters. 83 (8): 1529–1533. arXiv:hep-ex/9904020. Bibcode:1999PhRvL..83.1529H. doi:10.1103/PhysRevLett.83.1529. S2CID 118326409.
• S.F. Novaes (2000). "Standard Model: An Introduction". arXiv:hep-ph/0001283.
• D.P. Roy (1999). "Basic Constituents of Matter and their Interactions – A Progress Report". arXiv:hep-ph/9912523.
• F. Wilczek (2004). "The Universe Is A Strange Place". Nuclear Physics B: Proceedings Supplements. 134: 3. arXiv:astro-ph/0401347. Bibcode:2004NuPhS.134....3W. doi:10.1016/j.nuclphysbps.2004.08.001. S2CID 28234516.