# Glashow-Iliopoulos-Maiani mechanism

Jean Iliopoulos (2010), Scholarpedia, 5(5):7125. | doi:10.4249/scholarpedia.7125 | revision #91319 [link to/cite this article] |

The **Glashow-Iliopoulos-Maiani mechanism** was discovered in 1970 by Sheldon Lee Glashow, John Iliopoulos, and Luciano Maiani (Glashow et al., 1970). It describes a mechanism to naturally suppress Flavour Changing Neutral Currents (FCNC), as well as \(\Delta S =2\) transitions, in weak interactions. Its discovery involved the introduction of a fourth quark flavour, *charm*, which was unknown at that time.

## Contents |

## Introduction

The quest for discovering the "elementary" constituents of matter has always been a central problem for physics. In the last hundred years we went through the steps:

*molecules \(\rightarrow\) atoms \(\rightarrow\) nuclei + electrons \(\rightarrow\) protons + neutrons + electrons \(\rightarrow\) quarks + electrons \(\rightarrow\ ?\) ??*

Obviously, there is no reason to believe that we have reached the innermost layer, if such a thing exists at all. Our present observational tools allow us to explore distances down to \(10^{-16}\) cm and, at that scale, we have obtained a quite coherent picture of the structure of matter. Table 1 contains the elementary particles known today.

TABLE OF ELEMENTARY PARTICLES
| |||

QUANTA OF RADIATION | |||

Strong Interactions | Eight gluons | ||

Electromagnetic Interactions | Photon (\(\gamma\)) | ||

Weak Interaction | Bosons \(W^+\ ,\) \(W^-\ ,\) \(Z^0\) | ||

Gravitational Interactions | Graviton (?) | ||

MATTER PARTICLES | |||

Leptons | Quarks | ||

1st Family | \({\nu}_e\ ,\) \(e^-\) | \(u_a\ ,\) \(d_a\ ,\) \(a=1,2,3\) | |

2nd Family | \({\nu}_{\mu}\ ,\) \({\mu}^-\) | \(c_a\ ,\) \(s_a\ ,\) \(a=1,2,3\) | |

3rd Family | \({\nu}_{\tau}\ ,\) \({\tau}^-\) | \(t_a\ ,\) \(b_a\ ,\) \(a=1,2,3\) | |

HIGGS BOSON (?) |

A closer look at Table 1 reveals some remarkable regularities.

- (i) All interactions are produced by the exchange of virtual quanta. They are called "radiation quanta" in the Table. They are the eight gluons for the
*strong*interactions, which are responsible for the nuclear forces, the photon for the*electromagnetic*ones, responsible for the structure of atoms and molecules, the bosons \(W\) and \(Z\) for the*weak*interactions, responsible for \(\beta\)-decay, as well as the decays of many unstable particles. Similarly, we believe that the*gravitational*interactions result from the exchange of virtual gravitons. The radiation quanta for the first three are vector (spin-one) fields, while the graviton is assumed to be a tensor, spin-two field.

- (ii) The constituents of matter appear to be all spin one-half particles. They are divided into quarks, which are subject to strong interactions, and "leptons" which are not.

- (iii) At present we know the existence of six quark species, called "flavours". Each one appears under three forms, often called "colours" (no relation with the ordinary sense of these words). The \(u\ ,\) \(c\) and \(t\) quarks have electric charge equal to 2/3 times the electric charge of the proton, while the other three \(d\ ,\) \(s\) and \(b\) have charge equal to -1/3.

- (iv) Quarks and gluons do not appear as free particles. They form a large number of bound states, the hadrons.

- (v) Quarks and leptons seem to fall into three distinct groups, or "families". This family structure is one of the great puzzles in elementary particle physics. We believe we understand the importance of the first family. It is composed by the electron and its associated neutrino as well as the up and down quarks. These quarks are the constituents of protons and neutrons. The role of each member of this family in the structure of matter is obvious. In contrast, the role of the other two remains obscure. The muon and the tau leptons seem to be heavier versions of the electron but they cannot be viewed as excited states of it because they seem to carry their own quantum numbers. The associated quarks with exotic names such as charm, strange, top and bottom, form new, unstable hadrons which are not present in ordinary matter. They decay to each other by weak interactions. Why does Nature produce three similar copies of apparently the same structure?

- (vi) The sum of all electric charges inside any family is equal to zero.

## GIM mechanism

A firmly established experimental fact is that flavour changing weak processes obey certain selection rules: One of them, known as the \(\Delta S =1\) rule, states that the flavour number, in this case strangeness \(S\ ,\) changes by at most one unit. A second rule is that the allowed \(\Delta\)Flavour = 1 processes involve only charged currents. It follows that \(\Delta S =2\) transitions, as well as FCNC processes, must occur only at second order in the weak interactions. The best experimental evidence for the first is the measured \(K_L-K_S\) mass difference which equals 3.48 \(10^{-12}\) MeV and, for the second, the branching ratio \(B_{\mu^+ \mu^-}= \Gamma (K_L\rightarrow \mu^+ \mu^-)/\Gamma (K_L\rightarrow all)\) which equals 6.87 \(10^{-9}\) (PDG). The GIM mechanism offers a natural explanation for both. It is based on two ingredients:

The first is a generalisation of the Cabibbo universality principle (Cabibbo, 1963) for the charged weak current. With only three quark flavours, \(u\ ,\) \(d\) and \(s\ ,\) Cabibbo postulated that the charged weak current is given by

\[\tag{1} J_{\mu}(x)=\bar{u}(x)\gamma_{\mu}(1+\gamma_5)[\cos \theta d(x)+\sin \theta s(x)]\]

where \(\theta\) denotes the mismatch between the flavour symmetry breaking directions chosen by the strong and weak interactions, known as *the Cabibbo angle.* The expression (1) can be interpreted as saying that the \(u\) quark is coupled to a certain linear combination of the \(d\) and \(s\) quarks, \(d_C=\cos \theta~d+\sin \theta~s\ .\) The orthogonal combination, namely \(s_C=-\sin \theta~d+\cos \theta~s\) remains uncoupled. With the addition of a fourth quark \(c\) with electric charge 2/3, GIM conjectured that the full charged weak current is given by

\[\tag{2} J_{\mu}(x)=\bar{u}(x)\gamma_{\mu}(1+\gamma_5)d_C(x)+\bar{c}(x)\gamma_{\mu}(1+\gamma_5)s_C(x)\]

or, in a matrix notation,

\[\tag{3} J_{\mu}(x)=\bar{U}(x)\gamma_{\mu}(1+\gamma_5)CD(x)\]

with

\[\tag{4} U=\left( \begin{array}{c} u \\ c \end{array} \right); D=\left( \begin{array}{c} d \\ s \end{array} \right); C=\left( \begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array} \right) \]

The important point is that, now, the current \(J_3\ ,\) given by the commutator of \(J\) and \(J^{\dagger}\ ,\) is diagonal in flavour space. As a result in a gauge theory the neutral current, which is a linear superposition of \(J_3\) and the electromagnetic current, will also be diagonal. This solves the first part of the problem, namely it ensures that FCNC processes will not be generated in the tree approximation.

However, this is not enough to explain the observed rates. For example, the \(K_L\rightarrow \mu^+ \mu^-\) decay can be generated by the box diagram of Figure 1 which, in a renormalisable gauge theory, is expected to give a branching ratio of order \(g^4 \sim \alpha^2 \sim 10^{-4}\ ,\) with \(\alpha\) the fine structure constant.

Here comes the second ingredient of the mechanism. GIM observed that, with a fourth quark, there is a second diagram, with \(c\) replacing \(u\ ,\) Figure 2.

In the limit of exact flavour symmetry the two diagrams cancel. The breaking of flavour symmetry induces a mass difference between the quarks, so the sum of the two diagrams is of order \(g^4 (m_c^2-m_u^2)/m_W^2 \sim \alpha^2 m_c^2/m_W^2\ .\) With the measured charm quark mass \(m_c \sim 1.27\)GeV (PDG), the predicted rates are in agreement with observation. Before the experimental discovery of the charm particles, this mechanism was used to put upper limits on their masses (see the history section). The same mechanism applies to the present theory with six quark flavours with the Cabibbo-Kobayashi-Maskawa matrix (Kobayashi and Maskawa, 1973) replacing the \(C\) matrix of equation (4).

## GIM history

Until the late 1960's weak interactions were assumed to be described by the Fermi current x current theory. The existence of an intermediate vector boson was accepted as a possibility, although, of course, there was no evidence for it and it was assumed to be only charged. This theory is non-renormalisable, which means that higher orders are not computable, but, at least, it has a very simple and elegant structure. There was a vague hope that strong interactions, if properly understood, would provide for a cut-off for the weak ones, but, through the success of Current Algebra, people began realising that such could not be the case. The argument is rather simple. At one loop a weak interaction amplitude can be written as:

\[\tag{5} A(i\rightarrow f) \sim g^2\int d^4x e^{-ikx} \frac{d^4k}{(2\pi)^4} G^{\mu \nu}(k)<f|T(J_{\mu}(x) J^{\dagger}_{\nu}(0)|i> \]

where \(|i>\) and \(|f>\) are two hadronic states, \(G^{\mu\nu}(k)=[g^{\mu\nu}-k^{\mu}k^{\nu}/m_W^2]/(k^2-m_W^2)\) is the massive, intermediate vector boson propagator, \(J\) is the weak current and \(g\) denotes the strength of the vector boson-current coupling. The Fermi coupling constant is given by \(G_F/\sqrt{2}\sim g^2/m_W^2\ .\) The dangerous piece is the \(k^{\mu}k^{\nu}\) part of the propagator. By partial integration we relate this piece to the equal time commutators of the currents. Since the weak currents, which contain both vector and axial parts, are not exactly conserved, it follows that the coefficient of the leading divergence, which is quadratic at one loop, is non-zero. And this is true for any theory of strong interactions, provided it satisfies the algebra of currents. Notice that, for exactly conserved currents, the divergence is logarithmic. In order to define the theory at all orders let us introduce a cut-off \(\Lambda\) which has the dimensions of a mass. It can be viewed as determining the scale up to which the theory can be trusted. Up to logarithmic corrections, an \(n\)-th order diagram behaves as \(C_0(G_F\Lambda^2)^n+C_1G_F(G_F\Lambda^2)^{n-1}+...+C_nG_F^n\) where the \(C\)'s are numerical constants. It follows that the effective, dimensionless, coupling constant is \(G_F\Lambda^2\ ,\) so we expect, näively, the theory to be reliable as long as this coupling constant stays smaller than one, which means \(\Lambda \le\) 300 GeV. However, B.L. Joffe and E.P. Shabalin (Joffe and Shabalin, 1967a; Joffe and Shabalin, 1968; Joffe and Shabalin, 1967b; Joffe and Shabalin, 1967c; Lee, 1969; Mohapatra et al., 1968; Low 1968; Mohapatra and Olesen, 1969) remarked that the bound on \(\Lambda\) is in fact much stronger. The simplest way to show their argument is to rewrite the perturbation expansion in terms of the effective coupling constant:

\[\tag{6} \begin{array}{lllll} A & \sim & C^1_0(G_F\Lambda^2) & +C^1_1G_FM^2 & \\ & + & C^2_0(G_F\Lambda^2)^2 & +C^2_1G_FM^2(G_F\Lambda^2) &+C^2_2(G_FM^2)^2 \\ & + &.....& & \\ & + & C^n_0(G_F\Lambda^2)^n &+C^n_1G_FM^2(G_F\Lambda^2)^{n-1} &+.... \\ & + &.....& & \end{array} \]

where \(M\) is some typical mass. When \(G_F\Lambda^2 \sim 1\ ,\) the leading divergent terms become, effectively, of zero order in the weak interactions, the next-to-leading ones of first order etc. But
weak interactions violate strangeness by one unit as well as parity. Since both are known to be conserved by strong interactions, the available precision measurements allowed Joffe and Shabalin to obtain a bound on \(G_F\Lambda^2\) corresponding to a very low value of the cut-off \(\Lambda\) of the order of a few GeV. This was an astonishing result. It meant that the Fermi theory of weak interactions should break down at energies already available in running accelerators. It was the solution of this paradox that prompted the work on the divergence structure of weak interactions. The purpose was to obtain a theory whose divergences are *well ordered*, namely, the leading divergent terms respect the symmetries of strong interactions, and the next-to-leading respect those of first order weak interactions.

Progress went in two steps: First was the problem of the leading divergences which, as we said, raised the spectre of having parity and strangeness violations in strong interactions. The solution was found by C. Bouchiat, J. Iliopoulos and J. Prentki (Bouchiat, Iliopoulos, and Prentki, 1968; Iliopoulos, 1969) who remarked that this term contains an operator, proportional to the divergence of the currents, whose properties depend crucially on the symmetries of the strong interaction Hamiltonian. In particular they proved that, under the assumption that the chiral \(SU(3)\times SU(3)\) symmetry breaking term belongs to a \((3,\bar{3})\oplus (\bar{3},3)\) representation, this operator is diagonal, *i.e.* it does not connect states with different quantum numbers, strangeness and/or parity. Therefore, all its effects could be absorbed in a redefinition of the parameters of the strong interactions and no observable effects would be induced. Notice that this particular form of the symmetry breaking is the simplest possible and, in the language of the quark model, it corresponds to an explicit mass term. It was the favourite symmetry breaking term to most theorists, so this was considered a welcome result.

The second step dealt with the next-to-leading divergences which appeared to be of first order in the weak interactions. The problem there was that these terms appeared to induce Strangeness Changing Neutral Currents (SCNC) and \(\Delta S =2\) transitions. It is this step that led to the GIM mechanism.

Let me open a parenthesis here and notice that during these years the subject of the divergences of weak interactions attracted considerable attention and many interesting suggestions were made (Gatto et al., 1968; Gatto et al., 1969; Cabibo and Maiani, 1968; Cabibo and Maiani, 1970; [[Gell-Mann et al., 1969; Lee and Wick, 1969; Fronsdal, 1964; Kummer and Segrè, 1965; Segrè, 1969; Li and Segrè, 1969; Christ, 1968). The final solution was obtained by M. Veltman and G. 't Hooft (Veltman, 1968; ’t Hooft 1971; ’t Hooft 1971; Veltman and ’t Hooft 1972) who proved that a Yang-Mills theory with spontaneous breaking of the gauge symmetry is renormalisable. However, even with this solution, the physical application required the ingredients of the well ordering we discuss here.

Coming back to the GIM mechanism we remark that the solution of the problem of the leading divergences was found in the framework of the commonly accepted theory at that time. On the contrary, the next to leading divergences required a drastic modification, although, in retrospect, it is quite simple: At the limit of exact flavour symmetry, quark quantum numbers, such as strangeness, are not well defined. Any basis in quark space is as good as any other. By breaking this symmetry medium strong interactions choose a particular basis, which becomes the privileged one. Weak interactions, however, define a different direction, which forms an angle \(\theta\) with respect to the first one. Having only three quarks to play with, one can form only one doublet of weak \(SU(2)\ .\) It will contain the Cabibbo rotated combination \(d_C\) we introduced before. The orthogonal combination \(s_C\) will be necessarily a singlet. The neutral component of the current contains \(\bar{d}_Cd_C\) and, therefore, has flavour changing pieces. The only way out is to add the \(\bar{s}_Cs_C\) term in order to form a flavour invariant. This implies that \(s_C\) should also belong to a doublet, that is, one needs a second up-type quark. In this case \(\bar{d}_Cd_C+ \bar{s}_Cs_C=\bar{d}d+ \bar{s}s\) for all values of \(\theta\) and the Cabibbo angle remains undetermined. Notice that, because of this form of the coupling, the charm quark is expected to decay predominantly into a strange quark. All the above argument is exact in the limit of flavour symmetry. In the real world one expects corrections proportional to the quark mass differences. Therefore, Joffe and Shabalin's estimations can be translated into a limit for the new quark mass and yield an upper bound of a few GeV for the masses of the new hadrons. This fact is very important. A prediction for the existence of new particles is interesting only if they cannot be arbitrarily heavy.

The end of the story is well known. When, in 1971, the renormalisability of gauge theories was proven (Veltman, 1968; ’t Hooft 1971; ’t Hooft 1971; Veltman and ’t Hooft 1972) and the electroweak theory was constructed (Glashow, 1961; Weinberg, 1967; Salam, 1968), the GIM mechanism was an essential ingredient. The weak neutral currents, without flavour changing pieces, were discovered by the Gargamelle collaboration between 1973 and 1974 (PDG). Charmed particles were discovered between 1974 and 1976, first as \(\bar{c}c\) bound states (PDG) and next as charmed hadrons decaying into strange particles , as predicted by the theory.

## Further developments

The GIM mechanism implies that the quarks should form doublets of the weak \(SU(2)\ .\)
The Kobayashi-Maskawa suggestion (Kobayashi and Maskawa, 1973) respects this structure.
When later the \(b\) quark was discovered in FermiLab (see PDG) this was interpreted as a prediction for the existence of its partner, the \(t\) quark, prediction brilliantly verified by experiment (PDG).
However, the mechanism does not tie together leptons and quarks, in spite of the fact that the title of the original GIM paper was *Weak interactions with lepton-hadron symmetry*.
Such a symmetry is not implied by the requirement for the suppression of processes with FCNC.
It is remarkable that such a symmetry is imposed by the mere mathematical consistency of the theory and it was discovered immediately after the renormalisability of general Yang-Mills theories, with or without spontaneous symmetry breaking, was proven.

In order to explain how the argument goes we have to present a strange phenomenon which is manifest in many models of quantum field theory.
It is related to an essential ingredient of most physical theories, the concept of symmetry.
Our understanding of the fundamental laws of Nature has always gone in parallel with that of their symmetry properties.
A symmetry is the invariance of the classical equations of motion under a group of transformations.
For continuous transformations this implies the existence of conserved quantities.
For example, the conservation of energy and momentum follows from the invariance of the equations under time and space translations.
However, there are cases where Quantum Mechanics brings an important complication.
Going from the classical equations to the quantum theory involves a series of steps which often include a limiting procedure, for example the limit of some parameter going to infinity.
This limit, although well defined, may not respect some of the symmetries of the classical equations.
We call such symmetries *anomalous*^{1}, which means, in fact, that their consequences will not be satisfied in the quantum theory.
The simplest example we shall use is that of quantum electrodynamics.
In order to simplify the discussion we consider the case of a massless electron moving in an external electromagnetic field.
It is easy to check, using the Dirac equation, that the theory admits two conserved currents:

\(
J_{\mu}(x)=\bar{\Psi}(x)\gamma_{\mu}\Psi(x) ~~~ ;~~~ \frac{\partial}{\partial
x_{\mu}}J_{\mu}(x)=0
\)

\[\tag{7}
J_{\mu}^5(x)=\bar{\Psi}(x)\gamma_{\mu}\gamma^5\Psi(x) ~~~ ;~~~ \frac{\partial}{\partial x_{\mu}}J_{\mu}^5(x)=0
\]

The notation is the standard one in the Dirac theory. \(\Psi(x)\) is the field of the electron, the index \(\mu\) runs from 0 to 3 and a summation over repeated indices is understood. \(\gamma_{\mu}\ ,\) (\(\mu=0,1,2,3\)) and \(\gamma^5\) represent a set of five 4x4 numerical matrices. \(\gamma^5\) anti-commutes with all of the \(\gamma_{\mu}\)'s.

The two conserved currents of (7) correspond to the invariance of the massless Dirac equation under two sets of phase transformations, called *chiral transformations*.
We shall not need their explicit form but we can prove that the two currents cannot be simultaneously conserved in the quantum theory (Adler, 1969; Bell and Jackiw, 1969).
The simplest way to obtain this result is to compute explicitly the triangle diagram of Figure 3.
If we enforce the conservation of the vector current \(J_{\mu}(x)\ ,\) the equation for the axial one becomes:

\[\tag{8} \frac{\partial}{\partial x_{\mu}}J_{\mu}^5(x)=\frac{e^2}{8\pi^2}\epsilon_{\nu \rho \sigma \tau}F^{\nu \rho}(x)F^{\sigma \tau}(x) \]

where \(e\) is the charge of the electron, \(\epsilon_{\nu \rho \sigma \tau}\) is the completely anti-symmetric four index tensor which equals 1 if the indices form an even permutation of (0,1,2,3) and \(F^{\nu \rho}\) is the electromagnetic field strength given by

\[F^{\nu \rho}(x)=\frac{\partial A^{\rho}(x)}{\partial x_{\nu}}-\frac{\partial A^{\nu}(x)}{\partial x_{\rho}}\]

with \(A^{\nu}(x)\) the electromagnetic vector potential.
The r.h.s. of equation (8) is called *the axial anomaly*, which is a fancy way to say that the axial current of massless quantum electrodynamics is not conserved, contrary to what the classical equations of motion indicate.

This result has important physical consequences in particle physics but here we shall present only its implications for the electro-weak theory. For quantum electrodynamics the non-conservation of the axial current can be considered as a curiosity because this current does not play any direct physical role. However, in the electro-weak theory both vector and axial currents are important because of the non-conservation of parity in weak interactions. In proving the renormalisability of the gauge theory we need the full power of all its symmetries and, therefore, the conservation of all its currents. The axial anomaly breaks this conservation and the entire program collapses. As a result, the purely leptonic model, the one which was first constructed, is mathematically inconsistent.

The solution was first found by C. Bouchiat, J. Iliopoulos and Philippe Meyer in 1972. The important observation is that the anomaly is independent of the fermion mass. Every fermion of the theory, light or heavy, contributes the same amount and we must add all contributions in order to get the right answer. For the electroweak theory this means that we need both the leptons and the quarks. Each one will go around the triangle loop of Figure 3 and its contribution will depend only on the coefficients in the three vertices. A simple calculation shows that the total anomaly produced by the fermions of each family will be proportional to \({\mathcal A}\) given by:

\[\tag{9} {\mathcal A}=\sum_i Q_i \]

where the sum extends over all fermions in a given family and \(Q_i\) is the electric charge of the \(i\)th fermion. Since \({\mathcal A}=0\) is a necessary condition for the mathematical consistency of the theory, we conclude that each family must contain the right amount of leptons and quarks to make the anomaly vanish. This condition is satisfied by the three colour model with charges 2/3 and -1/3, but also by other models such as the Han-Nambu model which assumes three quark doublets with integer charges given by (1,0), (1,0) and (0,-1). In fact, the anomaly cancellation condition (9) has a wider application. The Standard Model could have been invented after the Yang-Mills theory was written, much before the discovery of the quarks. At that time the "elementary" particles were thought to be the electron and its neutrino, the proton and the neutron, so we would have used one lepton and one hadron doublet. The condition (9) is satisfied. When quarks were discovered we changed from nucleons to quarks. The condition is again satisfied. If tomorrow we find that our known leptons and/or quarks are composite, the new building blocks will be required to satisfy this condition again. Since the contribution of a chiral fermion to the anomaly is independent of its mass, it must be the same no matter which mass scale we are using to compute it.

The moral of the story is that families must be complete. The title of the GIM paper was correct. Thus, the discovery of a new lepton, the tau, implied the existence of two new quarks, the \(b\) and the \(t\ ,\) prediction which was again verified experimentally.

The above discussion was confined to the \(SU(2)\times U(1)\) gauge theory but the principle of anomaly cancellation should be imposed in any gauge theory in order to ensure mathematical consistency. This includes models of strong interactions and grand-unified theories. H. Georgi and S.L. Glashow found the generalisation of the anomaly equation (9) for a gauge theory based on any Lie algebra. It takes a surprisingly simple form:

\[\tag{10} {\mathcal A}_{abc}={\mathrm Tr}\left ( \gamma^5 \{\Gamma_a,\Gamma_b\}\Gamma_c\right ) \]

where \(\Gamma_a\) denotes the Hermitian matrix which determines the coupling of the gauge field \(W^{\mu}_a\) to the fermions through the interaction \(\bar{\Psi}\gamma_{\mu}\Gamma_a\Psi W^{\mu}_a\ .\) As we see, \(\Gamma_a\) may include a \(\gamma^5\ .\) Georgi and Glashow showed that the anomaly is always a positive multiplet of \({\mathcal A}_{abc}\ ,\) so this quantity should vanish identically for all values of the Lie algebra indices \(a\ ,\) \(b\) and \(c\ .\)

Since gauge theories are believed to describe all fundamental interactions, the anomaly cancellation condition plays an important role not only in the framework of the Standard Model, but also in all modern attempts to go beyond, from grand unified theories to superstrings. It is remarkable that this seemingly obscure higher order effect dictates to a certain extent the structure of the world.

^{1} The term may be misleading, as it may give the impression that something contrary to common sense has happened.

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**Internal references**

- Roman W. Jackiw (2008) Axial anomaly. Scholarpedia, 3(10):7302.

- Gerard ′t Hooft (2008) Gauge theories. Scholarpedia, 3(12):7443.

## Further reading

- None

## External links

## See also

Cabibbo-Kobayashi-Maskawa matrix, Gauge theories, Gluon, Higgs boson, Quark, Weak interaction