# Gauge theories

**Gauge theories** refers to a quite general class of
quantum field theories used for the description of elementary
particles and their interactions. The theories are characterized by
the presence of vector fields, and as such are a generalization of
the older theory of Quantum Electrodynamics (QED) that is used to
describe the electromagnetic interactions of charged elementary
particles with spin 1/2. **Local gauge invariance** is a very
central issue. An important feature is that these theories are often
renormalizable when used in 3 space- and 1 time dimension.

## 1. Maxwell's equations and gauge invariance

The simplest example of a gauge theory is electrodynamics, as described by the Maxwell equations. The electric field strength \(\vec E(\vec x,t)\) and the magnetic field strength \(\vec B(\vec x,t)\) obey the homogeneous Maxwell equations (in SI units):

\[\tag{1} \vec\nabla\times\vec E+{\partial \vec B\over\partial t}=0\]

\[\tag{2}
\vec\nabla\cdot\vec B=0\ .\]

According to Poincaré's Lemma,
Eq. (2) implies that there exists another
vector field \(\vec A(\vec x,t)\) such that

\[\tag{3} \vec B=\vec\nabla\times \vec A \ .\]

Since Eq. (1) now reads

\[\tag{4} \vec\nabla\times(\vec E+{\partial\vec A\over\partial t})=0\ ,\]

we can also conclude that there is a potential field \(\Phi(\vec
x,t)\) such that

\[\tag{5} \vec E=-\vec\nabla\Phi-{\partial \vec A\over\partial t} \ .\]

The field \(\Phi\) is the **electric potential field**;
the vector field \(\vec A\) is called the **vector potential field**.
The strengths of these potential fields are
determined by the inhomogeneous Maxwell equations, which are the
equations that relate the strengths of the electromagnetic fields to
the electric charges and currents that generate these fields. The use
of potential fields often simplifies the problem of solving Maxwell's
equations.

What turns this theory into a gauge theory is the fact that the
values of these potential fields are not completely determined by
Maxwell's equations. Consider an electromagnetic field configuration
\((\vec E(\vec x,t),\,\vec B(\vec x,t))\ ,\) and suppose that
it is described by the potential fields \((\Phi(\vec x,t),\,\vec
A(\vec x,t))\ .\) Then, using any arbitrary scalar function
\(\Lambda(\vec x,t)\ ,\) one can find a *different* set of
potential fields describing the same electric and magnetic fields, by
writing

\[\tag{6} \Phi'=\Phi+{\partial\Lambda\over\partial t}\ ,\quad\vec A'=\vec A-\vec\nabla\Lambda \ .\]

Inspecting Equations (3) and (5), one
easily observes that \(\vec E=\vec E'\) and \(\vec
B=\vec B'\ .\) Thus, the set (\(\Phi',\,\vec A'\)) and
(\(\Phi,\,\vec A\)) describe the same physical situation.
Because of this, we call the transformation (6) a
**gauge transformation**. Since \(\Lambda\) may be chosen
to be an arbitrary function of the points \((\vec x,t)\) in
space-time, we speak of a *local* gauge transformation. The fact
that the electromagnetic fields are invariant under these local gauge
transformations turns Maxwell's theory into a **gauge theory**.

In relativistic quantum field theory, the field \(\psi(\vec x,t)\) of a non-interacting spinless particle would typically obey the equation

\[\tag{7} (\vec\nabla^2-{\partial^2\over\partial t^2})\psi=m^2\psi \ ,\]

where units where used such that the velocity of light \(c=1\ ,\) and Planck's
constant \(\hbar=1\ .\) This gives the dispersion relation between energy and momentum as
dictated by Special Relativity:

\[\tag{8} E=\sqrt{{\vec p}^{\,2}+m^2} \ ,\]

Suppose now that the particle in question carries an electric charge
\(q\ .\) How is its equation then affected by the presence of
electro-magnetic fields? It turns out that one cannot write the
correct equations using the fields \(\vec E\) and \(\vec
B\) directly. Here, one can only choose to add terms depending
on the (vector) potential fields instead:

\[\tag{9} (\vec\nabla-iq\vec A)^2\psi-({\partial\over\partial t}+iq\Phi)^2\psi=m^2\psi\ .\]

It can be verified that this equation correctly produces waves that
are deflected by the electro-magnetic forces in the way one expects.
For instance, the energy \(E\) is easily seen to be enhanced
by an amount \(q\,\Phi(\vec x,t)\ ,\) which is the potential
energy of a charged particle in an electric potential field.

However, what happens to this equation when performing a gauge transformation? It appears as if the equation changes, so that the solution for the field \(\psi\) should change as well. Indeed, \(\psi\) changes in the following way:

\[\tag{10} \psi'=e^{-iq\Lambda}\psi \ ,\quad {\partial\psi'\over\partial t}=e^{-iq\Lambda}({\partial\psi\over\partial t}-iq\psi {\partial\Lambda\over\partial t} ) \ .\]

Thus, the field \(\psi\) makes a rotation in the complex plane. This is closely related
to a 'scale transformation', which would result if one were to remove the 'i' from Eq. (10).
It was Hermann Weyl who noted that this symmetry transformation simply redefines the scale of the field
\(\psi\ ,\) and introduced the word 'gauge' to describe this feature.

The combinations

\[\tag{11} \vec D\psi=(\vec\nabla-iq\vec A)\psi \quad,\quad D_t\psi=({\partial\over\partial t}+iq\Phi)\psi \ ,\]

are called **covariant derivatives**, because they are chosen in
such a way that the derivatives of the function \(\Lambda(\vec
x,t)\) cancel out in a gauge transformation:

\[\tag{12} (\vec D\psi)'=e^{-iq\Lambda}(\vec\nabla-iq(\vec\nabla\Lambda)-iq(\vec A-\vec\nabla\Lambda)) \psi = e^{-iq\Lambda} (\vec D\psi)\ ,\]

\[\tag{13} ( D_t\psi)'=e^{-iq\Lambda}({\partial\over\partial t}-iq{\partial\Lambda\over\partial t}+iq( \Phi+{\partial\Lambda\over\partial t})) \psi = e^{-iq\Lambda} (D_t\psi)\ ,\]

and this makes it easy to see that Equation (10)
correctly describes the way \(\psi\) transforms under a
local gauge transformation, obeying the same field equation
(9) both before and after the transformation (all terms
in the equation are multiplied by the same exponential
\(e^{-iq\Lambda}\ ,\) so that that factor is immaterial).

The absolute value, \(|\psi(\vec x,t)|^2\) does not change at all under a gauge transformation, and indeed this is the quantity that corresponds to something that is physically observable: it is the probability that a particle can be found at \((\vec x,t)\ .\) A rule of thumb is that local gauge invariance requires all derivatives in our equations to be replaced by covariant derivatives.

## 2. Yang-Mills theory

In the 1950s, it was known that the field equations for the field of a proton, \(P(\vec x,t)\ ,\) and the field of a neutron, \(N(\vec x,t)\ ,\) are such that one can rotate these fields in a complex two-dimensional space:

\[\tag{14} \left({P'(\vec x,t)\atop N'(\vec x,t)}\right)=\left({a\quad b\atop c\quad d}\right)\left({P(\vec x,t)\atop N(\vec x,t)}\right) \ ,\]

where the matrix \( U=\left({a\quad b\atop c\quad
d}\right)\) may contain four arbitrary complex numbers, as long
as it is unitary (\(U\,U^\dagger=I\)), and usually, the
determinant of \(U\) is restricted to be 1. Since these
equations resemble the rotations one can perform in ordinary space,
to describe spin of a particle, the symmetry in question here was
called **isospin**.

In 1954, C.N. Yang and R.L. Mills published a very important idea.
Could one modify the equations in such a way that these isospin
rotations could be regarded as *local gauge rotations*? this would
mean that, unlike the case that was known, the matrices
\(U\) should be allowed to depend on space and time, just
like the gauge generator \(\Lambda(\vec x,t)\) in
electromagnetism. Yang and Mills were also inspired by the observation
that Einstein's theory of gravity, General Relativity, also
allows for transformations very similar to local gauge
transformations: the replacement of the coordinate frame by other
coordinates in an arbitrary, space-time dependent way.

To write down field equations for protons and neutrons, one needs the derivatives of these fields. The way these derivatives transform under a local gauge transformation implies that there will be terms containing the gradients \(\vec\nabla U\) of the matrices \(U\ .\) To make the theory gauge-invariant, these gradients would have to be cancelled out, and in order to do that, Yang and Mills replaced the derivatives \(\vec\nabla\) by covariant derivatives \(\vec D=\vec\nabla -ig\vec A(\vec x,t)\ ,\) as was done in electromagnetism, see Equation (11). Here, however, the fields \(\vec A\) had to be matrix-valued, just as the isospin \(U\) matrices:

\[\tag{15} \vec A=\left({\vec a_{11}\quad \vec a_{12}\atop \vec a_{21}\quad \vec a_{22}}\right)\ ,\]

\[ \hbox{Tr}\,\vec A=\vec a_{11}+\vec a_{22}=0\ ,\quad \vec a_{11}=\vec a_{11}^{\,*}\,,\quad \vec a_{21} =\vec a_{12}^{\,*}\ . \]

Since the \(U\) matrices contain four coefficients with one constraint (the determinant has to be 1), one ends up with a set of three new vector fields (there are 3 independent real vectors in the matrix (15)). At first sight, they appear to be the fields of a vector particle with isospin one. In practice, this should correspond to particles with one unit of spin (i.e., the particle rotates about its axis), and its electric charge could be neutral or one or minus one unit. Yang-Mills theory therefore predicts and describes a new type of particles with spin one that transmit a force not unlike the electro-magnetic force.

The fields that are equivalent to Maxwell's electric and magnetic
fields are obtained by considering the *commutator* of two
covariant derivatives:

\[\tag{16} [D_\mu,\,D_\nu]=D_\mu D_\nu-D_\nu D_\mu= -ig(\partial_\mu A_\nu-\partial_\nu A_\mu-ig[A_\mu,\,A_\nu]) = -ig F_{\mu\nu}\ ,\]

where the indices take the values \(\mu,\ \nu=0,1,2,3\ ,\)
with 0 referring to the time-component.

Since \( F_{\mu\nu}=-F_{\nu\mu}\ ,\) this tensor has 6 independent components, three forming an electric vector field, and three a magnetic field. Each of these components is also a matrix. The commutator, \([A_\mu,\,A_\nu]\) is a new, non-linear term, which makes the Yang-Mills equations a lot more complicated than the Maxwell system.

In other respects, the Yang-Mills particles, being the energy quanta of the Yang-Mills fields, are similar to photons, the quanta of light. Yang-Mills particles also carry no intrinsic mass, and travel with the speed of light. Indeed, these features were at first reasons to dismiss this theory, because massless particles of this sort should have been detected long ago, whereas they were conspicuously absent.

## 3. The Brout-Englert-Higgs mechanism

The theory was revived when it was combined with
**spontaneous breakdown of local gauge symmetry**, also known as the
**Brout-Englert-Higgs mechanism**. Consider a scalar (spinless)
particle described by a field \(\phi(\vec x,t)\ .\) This field
is assumed to be a vector field, in the sense that it undergoes some
rotation when a gauge transformation is performed. In practice this
means that the particle carries one or several kinds of charges that
make it sensitive to the Yang-Mills force, and often it has several
components, which means there are various species of this particle.
Such particles must obey **Bose-Einstein statistics**, which
implies that it can undergo **Bose-Einstein condensation**. In
terms of its field \(\phi\) this means the following:

*In the vacuum the field \(\phi\) takes a non-vanishing value \(F\ .\)*

This is usually written as

\[\tag{17} \langle\phi(\vec x,t)\rangle=F \ .\]

After a local gauge transformation, this would look like

\[\tag{18} \langle\phi'(\vec x,t)\rangle=U(\vec x,t)\,F \ ,\]

where \( U(\vec x,t) \) is a matrix field representing the
local gauge transformation.

It is often said that, therefore, the vacuum is not gauge-invariant,
but, strictly speaking, this is not correct. The situation described
by Equation (18) is the *same* vacuum as
(17); it is only described differently. However, this
property of the vacuum does have important consequences. Due to the
fact that the rotated field now describes the same situation as the
previous value, there is no different physical particle associated to
the rotated field. Only the *length* of the vector
\(\phi\) has physical significance. This length is
gauge-invariant. therefore, only the length of the vector \( \phi
\) is associated to one type of particle, which must be neutral
for the Yang-Mills forces. This particle is now called the
**Higgs particle**.

As the Higgs field is a constant source for the Yang-Mills field
strength, the Yang-Mills field equations are modified by it. Due to
the Higgs field, the Yang-Mills "photons" described by the Yang-Mills
field \(A_\mu(\vec x,t)\) get a *mass*. This can also be
explained as follows. Massless photons can only have two helicity
states, that is, they can spin only in two directions. This is
related to the fact that light can be polarized in exactly two
directions. Massive photons (particles with non-vanishing mass and
with one unit of spin), can always spin in *three* directions. This
third rotation mode is now provided by the Higgs field, which itself
loses several of its physical components. The total number of
physical field components stays the same before and after the Brout-Englert-Higgs
mechanism. A further consequence of this effect on the Yang-Mills
field is that the force transmitted by the massive photons is a
short-range one (the range of the force being inversely proportional
to the mass of the photon).

The **weak interactions** could now be successfully described
by a Yang-Mills theory. The set of local gauge transformations forms
the mathematical **group** \(SU(2)\times U(1)\ .\) This
group generates 4 species of photons (3 for \(SU(2)\) and 1
for \( U(1)\)). The Brout-Englert-Higgs mechanism breaks this group down
in such a way that a subgroup of the form \(U(1)\) remains.
This is the electromagnetic theory, with just one photon. The other
three photons become massive; they are responsible for the weak
interactions, which in practice appear to be weak just because these
forces have a very short range. With respect to electromagnetism, two
of these **intermediate vector bosons**, \(W^\pm\ ,\)
are electrically charged, and a third, \( Z^0\ ,\) is
electrically neutral. When the latter's existence was derived from
group theoretical arguments, this gave rise to the prediction of a
hitherto unnoticed form of the weak interaction: the
**neutral current interaction**. This theory, that combines electromagnetism
and the weak force into one, is called the **electro-weak theory**,
and it was the first fully renormalizable theory for the weak force
(see Chapter 5).

## 4. Quantum Chromodynamics

When it was understood that the weak interactions, together with the
electromagnetic ones, can be ascribed to a Yang-Mills gauge theory,
the question was asked how to address the strong force, a very strong
force with relatively short range of action, which controls the
behavior of the hadronic particles such as the **nucleons** and the
**pions**. It was understood since 1964 that these particles behave
as if built from subunits, called **quarks**. Three varieties
of quarks were known (**up**, **down**, and **strange**), and
three more would be discovered later (**charm**, **top**, and
**bottom**). These quarks have the peculiar property that they
permanently stick together either in triplets, or one quark sticks
together with one anti-quark. Yet when they approach one another very
closely, they begin to behave more freely as individuals.

These features we now understand as, again, being due to a Yang-Mills
gauge theory. Here, we have the mathematical group \(SU(3)\)
as local gauge group, while now the symmetry is *not* affected by
any Brout-Englert-Higgs mechanism. Due to the non-linear nature of
the Yang-Mills field, it self-interacts, which forces the fields to
come in patterns quite different from the electromagnetic case:
**vortex lines** are formed, which form unbreakable bonds
between quarks. At close distances, the Yang-Mills force becomes
weak, and this is a feature that can be derived in an elementary way
using perturbation expansions, but it is a property of the quantized
Yang-Mills system that hitherto had been thought to be impossible for
any quantum field theory, called **asymptotic freedom**. The
discovery of this feature has a complicated history.

\(SU(3)\) implies that every species of quark comes in three
types, referred to as *color*: they are "red", "green" or "blue".
The field of a quark is therefore a 3-component vector in an internal
'color' space. Yang-Mills gauge transformations rotate this vector in
color space. The Yang-Mills fields themselves form 3 by 3 matrices,
with one constraint (since the determinant of the Yang-Mills gauge
rotation matrices must be kept equal to one). Therefore, the
Yang-Mills field has 8 colored photon-like particles, called
**gluons**. Anti-quarks carry the conjugate colors ("cyan",
"magenta" or "yellow"). The theory is now called Quantum chromodynamics
(QCD). It is also a renormalizable theory.

The gluons effectively keep the quarks together in such a way that
their colors add up to a total that is color-neutral ("white" or a
"shade of gray"). This is why either three quarks or one quark and
one anti-quark can sit together to form a physically observable
particle (a **hadron**). This property of the theory is called
**permanent quark confinement**. Because of the strongly non-linear
nature of the fields, quark confinement is in fact quite difficult to
prove, whereas the property of asymptotic freedom can be demonstrated
exactly. Indeed, a mathematically air-tight demonstration of
confinement, with the associated phenomenon of a **mass gap** in the
theory (the absence of strictly massless hadronic objects) has not
yet been given, and is the subject of a
$1,000,000,- prize, issued by
the Clay Mathematics Institute of Cambridge, Massachusetts.

## 5. The Lagrangian

One cannot choose all field equations at will. They must obey conditions such as energy conservation. This implies that there is an action principle (action = reaction), and this principle is most conveniently expressed by writing the Lagrangian for the theory. The Lagrangian (more precisely, Lagrange density) \( \mathcal{L}(\vec x,t)\) is an expression in terms of the fields of the system. For a real scalar field \(\Phi\) it is

\[\tag{19} \mathcal{L}=-{1\over 2}\Big((\vec D\Phi)^2-(D_t\Phi)^2+m^2\Phi^2\Big)\ , \]

and for the Maxwell fields it is

\[\tag{20} \mathcal{L}={1\over 2}(\vec E^2-\vec B^2)=-{1\over 4}\sum_{\mu,\nu}F_{\mu\nu}F_{\mu\nu}\ , \]

where the summation is the Lorentz covariant summation over the Lorentz indices \(\mu,\ \nu\ .\) The field equations can all be derived from this expression by demanding that the action integral,

\[\tag{21} S=\int\mathrm{d}^3\vec x\mathrm{d}t\,\mathcal{L}(\vec x,t)\ , \]

where \(\mathcal{L}\) is the sum of the Lagrangians of all fields in the system, be stationary under all infinitesimal variations of these fields. This is called the Euler-Lagrange principle, and the equations are the Euler-Lagrange equations.

For gauge theories this generalizes directly: one writes

\[\tag{22} \mathcal{L}=-{1\over 4}\hbox{Tr}\sum_{\mu,\nu}F_{\mu\nu}F_{\mu\nu}+ ...\ ,\]

using the expression (16) for the gauge fields \(F_{\mu\nu}\ ,\) and adds all terms associated to the other fields that are introduced. All symmetries of the theory are the symmetries of the Lagrangian, and the dimensionality of all coupling strengths can easily be read off from the Lagrangian as well, which is of importance for the renormalization procedure (see next chapter).

## 6. Renormalization and Anomalies

According to the laws of quantum mechanics, the energy in a field consists of energy packets, and these energy packets are in fact the particles associated to the field. Quantum mechanics gives extremely precise prescriptions on how these particles interact, as soon as the field equations are known and can be given in the form of a Lagrangian. The theory is then called quantum field theory (QFT), and it explains not only how forces are transmitted by the exchange of particles, but it also states that multiple exchanges should occur. In many older theories, these multiple exchange gave rise to difficulties: their effects seem to be unbounded, or infinite. In a gauge theory, however, the small distance structure is very precisely prescribed by the requirement of gauge-invariance. In such a theory one can combine the infinite effects of the multiple exchanges with redefinitions of masses and charges of the particles involved. This procedure is called renormalization. In 3 space and 1 time dimension, most gauge theories are renormalizable. This allows us to compute the effects of multiple particle exchanges to high accuracy, thus allowing for detailed comparison with experimental data.

Renormalization requires that masses and coupling strengths of particles be defined very carefully. If all coupling parameters of a theory are given a mass-dimensionality that is zero or positive, the number of divergent expressions stays under control. Usually, requiring the theory to remain gauge invariant throughout the renormalization procedure leaves no ambiguity for the definitions. However, it is not obvious that unambiguous, gauge invariant definitions exist at all, since gauge invariance has to hold for all interactions, whereas only a few infinite expressions can be replaced by finite ones.

The proof that showed how and why unambiguous renormalized
expressions can be obtained, could be most elegantly obtained by
realizing that gauge theories can be formulated in any number of
space-time dimensions. It was even possible to define all Feynman
diagrams unambiguously for theories in spaces where the dimensions
are \(3-\epsilon\ ,\) where \(\epsilon\) is an
infinitesimal quantity. Taking the limit \(\epsilon\rightarrow
0\) requires the subtraction of poles of the form
\(C_n/\epsilon^n\) from the original, "bare" mass and coupling
parameters. The result is a set of unique, finite and gauge invariant
expressions. In practice, it was found that this procedure, called
**dimensional regularization** and **renormalization** is also
convenient for carrying out technically complicated calculations of
loop diagrams.

However, there is a special case where extension to dimensions
different from the canonical one is impossible. This is when
fermionic particles exhibit *chiral symmetry*. Chiral symmetry is a
symmetry that distinguishes left-rotating from right rotating
particles, and indeed it plays a crucial role in the Standard Model.
Chiral symmetry is only possible if space is 3 dimensional, and so
does not allow for dimensional renormalization. Indeed, sometimes chiral
symmetry cannot be preserved when renormalizing the theory.
An anomaly occurs, called **chiral anomaly**.
It was first discovered when a calculation of the
\(\pi_0\rightarrow\gamma\gamma\) decay amplitude gave answers that did not
follow the expected symmetry pattern.

Since the gauge symmetries of the Standard Model do distinguish left rotating from right rotating particles (in particular, only left-rotating neutrinos are produced in a weak interaction), anomalies were a big concern. It so happens, however, that all anomalous amplitudes that would jeopardize gauge invariance and hence the self consistency of our equations, all cancel out. This is related to the fact that certain "grand unified" extensions of the Standard Model are based on anomaly free gauge groups (see Chapter 7).

The anomaly has a direct physical implication. A topologically
twisted field configuration called the **instanton** (because
it represents an event at a given instant in time), represents
exactly the gauge field configuration where the anomaly is maximal. It causes a
violation of the conservation of some of the gauge charges. When
there is an anomaly, at least one of the charges involved *cannot*
be a gauge charge, but must be a charge to which no gauge field is coupled, like baryonic charge.
Indeed, in the electroweak theory, instantons trigger the violation of the conservation laws of
baryons. It is now believed that this might explain the imbalance between matter
and antimatter that must have arisen during early phases of the Universe.

## 7. Standard Model

Apart from the weak force, the electromagnetic force and the strong
force, there is the **gravitational force** acting upon elementary
particles. No other elementary forces are known. At the level of
individual particles, gravity is so weak that it can be ignored in
most cases. Suppose now that we take the \( SU(2)\times
U(1)\) Yang-Mills system, together with the Higgs field, to
describe electromagnetism and the weak force, and add to this the
\(SU(3)\) Yang-Mills theory for the strong force, and we
include all known elementary matter fields, being the *quarks* and
the *leptons*, with their appropriate transformation rules under a
gauge transformation; suppose we add to this all possible ways these
fields can *mix*, a feature observed experimentally, which can be
accounted for as a basic type of self-interaction of the fields. Then
we obtain what is called the **Standard Model**. It is one great
gauge theory that literally represents all our present understanding
of the subatomic particles and their interactions.

The Standard Model owes its strength to the fact that it is renormalizable. It has been subject of numerous experimental experiments and observations. It has withstood all these tests remarkably well. One important modification became inevitable around the early 1990's: in the leptonic sector, also the neutrinos carry a small amount of mass, and their fields mix. This was not totally unexpected, but highly successful neutrino experiments (in particular the Japanese Kamiokande experiment) now had made it clear that these effects are really there. They actually implied a further reinforcement of the Standard Model.

One ingredient has not yet been confirmed: the Higgs particle. Observation of this object is expected in the near future, notably by the Large Hadron Collider at CERN, Geneva. The simplest versions of the Standard model only require one single, electrically neutral Higgs particle, but the 'Higgs sector' could be more complicated: the Higgs could be much heavier than presently expected, or there could exist more than one variety, in which case also electrically charged scalar particles would be found.

The Standard Model is not perfect from a mathematical point of view.
At extremely high energies (energies much higher than what can be
attained today in the particle accelerators), the theory becomes
*unnatural*. In practice, this means that we do not believe anymore
that everything will happen exactly as prescribed in the theory; new
phenomena are to be expected. The most popular scenario is the
emergence of a new symmetry called **supersymmetry**, a symmetry
relating bosons with fermions (particles such as electrons and
quarks, which require Dirac fields for their description).

## 8. Grand Unified Theories

It is natural to suspect that the electroweak forces and the strong
forces should also be connected by gauge rotations. This would imply
that all forces among the subatomic particles are actually related by
gauge transformations. There is no direct evidence for this, but
there are several circumstances that appear to point in this
direction. In the present version of the Standard Model, the \(
SU(3)\) Yang-Mills fields, describing the strong force, indeed
exhibit very large coupling strengths, whereas the \(U(1)\)
sector, describing the electric (and part of the weak) sector, has a
tiny coupling strength. One can now use the mathematics of
renormalization, in particular the so-called **renormalization group**,
to calculate the effective strengths of these forces at
much higher energies. It is found that the \(SU(3)\) forces
decrease in strength, due to asymptotic freedom, but that the
\(U(1)\) coupling strength increases. The \(SU(2)\)
force varies more slowly. At extremely high energies, corresponding
to ultra short distance scales, around \(10^{-32}\) cm, the
three coupling strengths appear to approach one another, as if that
is the place where the forces unite.

It was found that \(SU(2)\times U(1)\) and
\(SU(3)\) fit quite nicely in a group called
\(SU(5)\ .\) They indeed form a subgroup of
\(SU(5)\ .\) One may then assume that a Brout-Englert-Higgs
mechanism breaks this group down to a \(SU(2)\times U(1)\times
SU(3)\) subgroup. One obtains a so-called **Grand Unified Field theory**.
In this theory, one assumes three **generations** of
fermions, each transforming in the same way under \(SU(5)\)
transformations (mathematically, they form a \(\mathbf{10}\)
and a \(\overline\mathbf{5}\) representation).

The \(SU(5)\) theory, however, predicts that the proton can decay, extremely slowly, into leptons and pions. The decay has been searched for but not found. Also, in this model, it is not easy to account for the neutrino mass and its mixings. A better theory was found where \( SU(5)\) is enlarged into \(SO(10)\ .\) The \(\mathbf{10}\) and the \(\overline\mathbf{5}\) representations of \(SU(5)\) together with a single right handed neutrino field, combine in to a \(\mathbf{16}\) representation of \(SO(10)\) (one for each of the three generations). This grand unified model puts the neutrinos at the same level as the charged leptons. Often, it is extended to a supersymmetric version.

## 9. Final remarks

Any gauge theory is constructed as follows. First, choose the gauge group. This can be the direct product of any number of irreducible, compact Lie groups, either of the series \(SU(N)\ ,\) \(SO(N)\) or \(Sp(2N)\ ,\) or the exceptional groups \(G_2,\ F_4,\ E_6, E_7,\) or \(E_8\ .\) Then, choose fermionic (spin 1/2) and scalar (spin 0) fields forming representations of this local gauge group. The left helicity and the right helicity components of the fermionic fields may be in different representations, provided that the anomalies cancel out. Besides the local gauge group, we may impose exact and/or approximate global symmetries as well. Finally, choose mass terms and interaction terms in the Lagrangian, described by freely adjustable coupling parameters. There will be only a finite number of such parameters, provided that all interactions are chosen to be of the renormalizable type (this can now be read off easily from the theory's Lagrangian).

There are infinitely many ways to construct gauge theories along these lines. However, it seems that the models that are most useful to describe observed elementary particles, are the relatively simple ones, based on fairly elementary mathematical groups and representations. One may wonder why Nature appears to be so simple, and whether it will stay that way when new particles and interactions are discovered. Conceivably, more elaborate gauge theories will be needed to describe interactions at energies that are not yet attainable in particle accelerators today.

Related subjects are **Supersymmetry** and **Superstring theory**.
They are newer ideas about particle structure and
particle symmetries, where gauge invariance also plays a very basic
role.

## References

- Yang, C N and Mills, R L (1954). Conservation of Isotopic Spin and Isotopic Gauge Invariance.
*Physical Review*96: 191-195. - Higgs, P W (1964). Broken symmetries, massless particlees and gauge fields.
*Phys. Lett.*12: 132. - Higgs, P W (1964). Broken Symmetries and the Masses of Gauge Bosons.
*Phys. Rev. Lett.*13: 508. - Higgs, P W (1966). Spontaneous Symmetry Breakdown without Massless Bosons.
*Phys. Rev.*145: 1156. - Englert, F and Brout, R (1964). Broken Symmetry and the Mass of Gauge Vector Mesons.
*Phys. Rev. Lett.*13: 321. - Weinberg, S (1967). A Model of Leptons.
*Phys. Rev. Lett.*19: 1264. - Faddeev, L D and Popov, V N (1967). Feynman diagrams for the Yang-Mills field.
*Phys. Lett.*25B: 29. - 't Hooft, G (1971). Renormalization of massless Yang-Mills fields.
*Nucl. Phys.*B33: 173. - 't Hooft, G (1971). Renormalizable Lagrangians for massive Yang-Mills fields.
*Nucl. Phys.*B35: 167. - Taylor, J C (1971). Ward identities and charge renormalization of the Yang-Mills field.
*Nucl. Phys.*B33: 436. - Slavnov, A (1972). Ward identities in gauge theories
*Theor. Math. Phys.*10: 153. - 't Hooft, G and Veltman, M (1972). Regularization and renormalization of gauge fields.
*Nucl. Phys.*B44: 189. - Adler, S L (1969). Axial-Vector Vertex in Spinor Electrodynamics
*Phys. Rev.*177: 2426. - Bell, J S and Jackiw, R (1969). A PCAC puzzle: π0→γγ in the σ-model
*Nuovo Cim.*60A: 47. - Adler, S L and Bardeen, W A (1969). Absence of Higher-Order Corrections in the Anomalous Axial-Vector Divergence Equation.
*Phys. Rev.*182: 1517. - Bardeen, W A (1969). Anomalous Ward Identities in Spinor Field Theories.
*Phys. Rev.*184: 1848. - Fritzsch, H; Gell-Mann , M and Leutwyler, H (1973). Advantages of the color octet gluon picture
*Phys. Lett.*47B: 365. - De Rujula, A; Georgi, H; Glashow, S L and Quinn, H (1974). Fact and fancy in neutrino physics.
*Rev. Mod. Phys.*46: 391.

## Further reading

- Crease, R P and Mann, C C (1986). The Second Creation: makers of the revolution in twentieth-century physics, Macmillan, New York. ISBN 0-02-521440-3.
- 't Hooft, G (1997). In Search of the Ultimate Building Blocks (English translation of: "Bouwstenen van de Schepping") Cambridge Univ. Press, Cambridge. ISBN 0521550831.
- 't Hooft, G (1994). Under the spell of the gauge principle. Advanced Series in Mathematical Physics 19. World Scientific, Singapore. ISBN 9810213093.
- 't Hooft, G (2005). 50 years of Yang-Mills theory World Scientific, Singapore. ISBN 978-981-256-007-0.
- de Wit, B and Smith, J (1986). Field Theory in Particle Physics North Holland, Amsterdam. ISBN 0444869999.
- Aitchison, I J R and Hey, A J G (1989). Gauge Theories in Particle Physics, a practical introduction Adam Hilger, Bristol and Philadelphia. ISBN 0-85274-329-7.
- Itzykson, C and Zuber, J B (2006). Quantum Field Theory Dover Publications, New York. ISBN 0486445682.
- Ryder, L H (1997). Quantum Field Theory Cambridge University Press, Cambridge. ISBN 0521478146.

## See Also

Becchi-Rouet-Stora-Tyutin symmetry, Englert-Brout-Higgs-Guralnik-Hagen-Kibble_mechanism, Gauge invariance, Slavnov-Taylor identities, Zinn-Justin equation