In the Standard Model of particle physics,
the Higgs mechanism is essential to explain
the generation mechanism of the property "mass"
for gauge bosons. Without the Higgs mechanism,
all bosons (one of the two classes of particles,
the other being fermions) would be considered
massless, but measurements show that the W+,
W−, and Z bosons actually have relatively
large masses of around 80 GeV/c2. The Higgs
field resolves this conundrum. The simplest
description of the mechanism adds a quantum
field (the Higgs field) that permeates all
space to the Standard Model. Below some extremely
high temperature, the field causes spontaneous
symmetry breaking during interactions. The
breaking of symmetry triggers the Higgs mechanism,
causing the bosons it interacts with to have
mass. In the Standard Model, the phrase "Higgs
mechanism" refers specifically to the generation
of masses for the W±, and Z weak gauge bosons
through electroweak symmetry breaking. The
Large Hadron Collider at CERN announced results
consistent with the Higgs particle on 14 March
2013, making it extremely likely that the
field, or one like it, exists, and explaining
how the Higgs mechanism takes place in nature.
The mechanism was proposed in 1962 by Philip
Warren Anderson, following work in the late
1950s on symmetry breaking in superconductivity
and a 1960 paper by Yoichiro Nambu that discussed
its application within particle physics.
A theory able to finally explain mass generation
without "breaking" gauge theory was published
almost simultaneously by three independent
groups in 1964: by Robert Brout and François
Englert; by Peter Higgs; and by Gerald Guralnik,
C. R. Hagen, and Tom Kibble. The Higgs mechanism
is therefore also called the Brout-Englert-Higgs
mechanism, or Englert-Brout-Higgs-Guralnik-Hagen-Kibble
mechanism, Anderson-Higgs mechanism, Anderson-Higgs-Kibble
mechanism, Higgs-Kibble mechanism by Abdus
Salam and ABEGHHK'tH mechanism [for Anderson,
Brout, Englert, Guralnik, Hagen, Higgs, Kibble,
and 't Hooft] by Peter Higgs.On 8 October
2013, following the discovery at CERN's Large
Hadron Collider of a new particle that appeared
to be the long-sought Higgs boson predicted
by the theory, it was announced that Peter
Higgs and François Englert had been awarded
the 2013 Nobel Prize in Physics.
== Standard model ==
The Higgs mechanism was incorporated into
modern particle physics by Steven Weinberg
and Abdus Salam, and is an essential part
of the standard model.
In the standard model, at temperatures high
enough that electroweak symmetry is unbroken,
all elementary particles are massless. At
a critical temperature, the Higgs field becomes
tachyonic; the symmetry is spontaneously broken
by condensation, and the W and Z bosons acquire
masses (also called “electroweak symmetry
breaking”, or EWSB.)
Fermions, such as the leptons and quarks in
the Standard Model, can also acquire mass
as a result of their interaction with the
Higgs field, but not in the same way as the
gauge bosons.
=== Structure of the Higgs field ===
In the standard model, the Higgs field is
an SU(2) doublet (i.e. the standard representation
with two complex components called isospin),
which is a scalar under Lorentz transformations.
Its electric charge is zero; its weak isospin
is ​1⁄2; its weak hypercharge (the charge
for the U(1) gauge group) is 1 . Under U(1)
rotations, it is multiplied by a phase, which
thus mixes the real and imaginary parts of
the complex spinor into each other, combining
to the standard two-component complex representation
of the group U(2).
The Higgs field, through the interactions
specified (summarized, represented, or even
simulated) by its potential, induces spontaneous
breaking of three out of the four generators
("directions") of the gauge group U(2). This
is often written as SU(2) × U(1), (which
is strictly speaking only the same on the
level of infinitesimal symmetries) because
the diagonal phase factor also acts on other
fields – quarks in particular. Three out
of its four components would ordinarily resolve
as Goldstone bosons, if they were not coupled
to gauge fields.
However, after symmetry breaking, these three
of the four degrees of freedom in the Higgs
field mix with the three W and Z bosons (W+,
W− and Z), and are only observable as components
of these weak bosons, which are made massive
by their inclusion; only the single remaining
degree of freedom becomes a new scalar particle:
the Higgs boson.
=== The photon as the part that remains massless
===
The gauge group of the electroweak part of
the standard model is SU(2) × U(1). The group
SU(2) is the group of all 2-by-2 unitary matrices
with unit determinant; all the orthonormal
changes of coordinates in a complex two dimensional
vector space.
Rotating the coordinates so that the second
basis vector points in the direction of the
Higgs boson makes the vacuum expectation value
of H the spinor (0, v). The generators for
rotations about the x, y, and z axes are by
half the Pauli matrices σx, σy, and σz,
so that a rotation of angle θ about the z-axis
takes the vacuum to
(
0
,
v
e
−
1
2
i
θ
)
.
{\displaystyle \left(0,ve^{-{\frac {1}{2}}i\theta
}\right).}
While the Tx and Ty generators mix up the
top and bottom components of the spinor, the
Tz rotations only multiply each by opposite
phases. This phase can be undone by a U(1)
rotation of angle 1/2θ. Consequently, under
both an SU(2) Tz-rotation and a U(1) rotation
by an amount 1/2θ, the vacuum is invariant.
This combination of generators
Q
=
T
z
+
1
2
Y
{\displaystyle Q=T_{z}+{\frac {1}{2}}Y}
defines the unbroken part of the gauge group,
where Q is the electric charge, Tz is the
generator of rotations around the z-axis in
the SU(2) and Y is the hypercharge generator
of the U(1). This combination of generators
(a z rotation in the SU(2) and a simultaneous
U(1) rotation by half the angle) preserves
the vacuum, and defines the unbroken gauge
group in the standard model, namely the electric
charge group. The part of the gauge field
in this direction stays massless, and amounts
to the physical photon.
=== Consequences for fermions ===
In spite of the introduction of spontaneous
symmetry breaking, the mass terms preclude
chiral gauge invariance. For these fields,
the mass terms should always be replaced by
a gauge-invariant "Higgs" mechanism. One possibility
is some kind of Yukawa coupling (see below)
between the fermion field ψ and the Higgs
field Φ, with unknown couplings Gψ, which
after symmetry breaking (more precisely: after
expansion of the Lagrange density around a
suitable ground state) again results in the
original mass terms, which are now, however
(i.e., by introduction of the Higgs field)
written in a gauge-invariant way. The Lagrange
density for the Yukawa interaction of a fermion
field ψ and the Higgs field Φ is
L
F
e
r
m
i
o
n
(
ϕ
,
A
,
ψ
)
=
ψ
¯
γ
μ
D
μ
ψ
+
G
ψ
ψ
¯
ϕ
ψ
,
{\displaystyle {\mathcal {L}}_{\mathrm {Fermion}
}(\phi ,A,\psi )={\overline {\psi }}\gamma
^{\mu }D_{\mu }\psi +G_{\psi }{\overline {\psi
}}\phi \psi ,}
where again the gauge field A only enters
Dμ (i.e., it is only indirectly visible).
The quantities γμ are the Dirac matrices,
and Gψ is the already-mentioned Yukawa coupling
parameter. Now the mass-generation follows
the same principle as above, namely from the
existence of a finite expectation value
|
⟨
ϕ
⟩
|
{\displaystyle |\langle \phi \rangle |}
. Again, this is crucial for the existence
of the property mass.
== History of research ==
=== Background ===
Spontaneous symmetry breaking offered a framework
to introduce bosons into relativistic quantum
field theories. However, according to Goldstone's
theorem, these bosons should be massless.
The only observed particles which could be
approximately interpreted as Goldstone bosons
were the pions, which Yoichiro Nambu related
to chiral symmetry breaking.
A similar problem arises with Yang–Mills
theory (also known as non-Abelian gauge theory),
which predicts massless spin-1 gauge bosons.
Massless weakly-interacting gauge bosons lead
to long-range forces, which are only observed
for electromagnetism and the corresponding
massless photon. Gauge theories of the weak
force needed a way to describe massive gauge
bosons in order to be consistent.
=== Discovery ===
That breaking gauge symmetries did not lead
to massless particles was observed in 1961
by Julian Schwinger, but he did not demonstrate
massive particles would eventuate. This was
done in Philip Warren Anderson's 1962 paper
but only in non-relativistic field theory;
it also discussed consequences for particle
physics but did not work out an explicit relativistic
model. The relativistic model was developed
in 1964 by three independent groups:
Robert Brout and François Englert
Peter Higgs
Gerald Guralnik, Carl Richard Hagen, and Tom
Kibble.Slightly later, in 1965, but independently
from the other publications the mechanism
was also proposed by Alexander Migdal and
Alexander Polyakov, at that time Soviet undergraduate
students. However, their paper was delayed
by the editorial office of JETP, and was published
late, in 1966.
The mechanism is closely analogous to phenomena
previously discovered by Yoichiro Nambu involving
the "vacuum structure" of quantum fields in
superconductivity. A similar but distinct
effect (involving an affine realization of
what is now recognized as the Higgs field),
known as the Stueckelberg mechanism, had previously
been studied by Ernst Stueckelberg.
These physicists discovered that when a gauge
theory is combined with an additional field
that spontaneously breaks the symmetry group,
the gauge bosons can consistently acquire
a nonzero mass. In spite of the large values
involved (see below) this permits a gauge
theory description of the weak force, which
was independently developed by Steven Weinberg
and Abdus Salam in 1967. Higgs's original
article presenting the model was rejected
by Physics Letters. When revising the article
before resubmitting it to Physical Review
Letters, he added a sentence at the end, mentioning
that it implies the existence of one or more
new, massive scalar bosons, which do not form
complete representations of the symmetry group;
these are the Higgs bosons.
The three papers by Brout and Englert; Higgs;
and Guralnik, Hagen, and Kibble were each
recognized as "milestone letters" by Physical
Review Letters in 2008. While each of these
seminal papers took similar approaches, the
contributions and differences among the 1964
PRL symmetry breaking papers are noteworthy.
All six physicists were jointly awarded the
2010 J. J. Sakurai Prize for Theoretical Particle
Physics for this work.Benjamin W. Lee is often
credited with first naming the "Higgs-like"
mechanism, although there is debate around
when this first occurred. One of the first
times the Higgs name appeared in print was
in 1972 when Gerardus 't Hooft and Martinus
J. G. Veltman referred to it as the "Higgs–Kibble
mechanism" in their Nobel winning paper.
== Examples ==
The Higgs mechanism occurs whenever a charged
field has a vacuum expectation value. In the
nonrelativistic context, this is the Landau
model of a charged Bose–Einstein condensate,
also known as a superconductor. In the relativistic
condensate, the condensate is a scalar field,
and is relativistically invariant.
=== Landau model ===
The Higgs mechanism is a type of superconductivity
which occurs in the vacuum. It occurs when
all of space is filled with a sea of particles
which are charged, or, in field language,
when a charged field has a nonzero vacuum
expectation value. Interaction with the quantum
fluid filling the space prevents certain forces
from propagating over long distances (as it
does in a superconducting medium; e.g., in
the Ginzburg–Landau theory).
A superconductor expels all magnetic fields
from its interior, a phenomenon known as the
Meissner effect. This was mysterious for a
long time, because it implies that electromagnetic
forces somehow become short-range inside the
superconductor. Contrast this with the behavior
of an ordinary metal. In a metal, the conductivity
shields electric fields by rearranging charges
on the surface until the total field cancels
in the interior. But magnetic fields can penetrate
to any distance, and if a magnetic monopole
(an isolated magnetic pole) is surrounded
by a metal the field can escape without collimating
into a string. In a superconductor, however,
electric charges move with no dissipation,
and this allows for permanent surface currents,
not just surface charges. When magnetic fields
are introduced at the boundary of a superconductor,
they produce surface currents which exactly
neutralize them. The Meissner effect is due
to currents in a thin surface layer, whose
thickness, the London penetration depth, can
be calculated from a simple model
(the Ginzburg–Landau theory).
This simple model treats superconductivity
as a charged Bose–Einstein condensate. Suppose
that a superconductor contains bosons with
charge q. The wavefunction of the bosons can
be described by introducing a quantum field,
ψ, which obeys the Schrödinger equation
as a field equation (in units where the reduced
Planck constant, ħ, is set 
to 1):
i
∂
∂
t
ψ
=
(
∇
−
i
q
A
)
2
2
m
ψ
.
{\displaystyle i{\partial \over \partial t}\psi
={(\nabla -iqA)^{2} \over 2m}\psi .}
The operator ψ(x) annihilates a boson at
the point x, while its adjoint ψ† creates
a new boson at the same point. The wavefunction
of the Bose–Einstein condensate is then
the expectation value ψ of ψ(x), which is
a classical function that obeys the same equation.
The interpretation of the expectation value
is that it is the phase that one should give
to a newly created boson so that it will coherently
superpose with all the other bosons already
in the condensate.
When there is a charged condensate, the electromagnetic
interactions are screened. To see this, consider
the effect of a gauge transformation on the
field. A gauge transformation rotates the
phase of the condensate by an amount which
changes from point to point, and shifts the
vector potential by a gradient:
ψ
→
e
i
q
ϕ
(
x
)
ψ
A
→
A
+
∇
ϕ
.
{\displaystyle {\begin{aligned}\psi &\rightarrow
e^{iq\phi (x)}\psi \\A&\rightarrow A+\nabla
\phi .\end{aligned}}}
When there is no condensate, this transformation
only changes the definition of the phase of
ψ at every point. But when there is a condensate,
the phase of the condensate defines a preferred
choice of phase.
The condensate wave function can be written
as
ψ
(
x
)
=
ρ
(
x
)
e
i
θ
(
x
)
,
{\displaystyle \psi (x)=\rho (x)\,e^{i\theta
(x)},}
where ρ is real amplitude, which determines
the local density of the condensate. If the
condensate were neutral, the flow would be
along the gradients of θ, the direction in
which the phase of the Schrödinger field
changes. If the phase θ changes slowly, the
flow is slow and has very little energy. But
now θ can be made equal to zero just by making
a gauge transformation to rotate the phase
of the field.
The energy of slow changes of phase can be
calculated from the Schrödinger kinetic energy,
H
=
1
2
m
|
(
q
A
+
∇
)
ψ
|
2
,
{\displaystyle H={1 \over 2m}\left|(qA+\nabla
)\psi \right|^{2},}
and taking the density of the condensate ρ
to be constant,
H
≈
ρ
2
2
m
(
q
A
+
∇
θ
)
2
.
{\displaystyle H\approx {\rho ^{2} \over 2m}(qA+\nabla
\theta )^{2}.}
Fixing the choice of gauge so that the condensate
has the same phase everywhere, the electromagnetic
field energy has an extra term,
q
2
ρ
2
2
m
A
2
.
{\displaystyle {q^{2}\rho ^{2} \over 2m}A^{2}.}
When this term is present, electromagnetic
interactions become short-ranged. Every field
mode, no matter how long the wavelength, oscillates
with a nonzero frequency. The lowest frequency
can be read off from the energy of a long
wavelength A mode,
E
≈
A
˙
2
2
+
q
2
ρ
2
2
m
A
2
.
{\displaystyle E\approx {{\dot {A}}^{2} \over
2}+{q^{2}\rho ^{2} \over 2m}A^{2}.}
This is a harmonic oscillator with frequency
1
m
q
2
ρ
2
.
{\displaystyle {\sqrt {{\frac {1}{m}}q^{2}\rho
^{2}}}.}
The quantity |ψ|2 (= ρ2) is the density
of the condensate of superconducting particles.
In an actual superconductor, the charged particles
are electrons, which are fermions not bosons.
So in order to have superconductivity, the
electrons need to somehow bind into Cooper
pairs. The charge of the condensate q is therefore
twice the electron charge −e. The pairing
in a normal superconductor is due to lattice
vibrations, and is in fact very weak; this
means that the pairs are very loosely bound.
The description of a Bose–Einstein condensate
of loosely bound pairs is actually more difficult
than the description of a condensate of elementary
particles, and was only worked out in 1957
by Bardeen, Cooper and Schrieffer in the famous
BCS theory.
=== Abelian Higgs mechanism ===
Gauge invariance means that certain transformations
of the gauge field do not change the energy
at all. If an arbitrary gradient is added
to A, the energy of the field is exactly the
same. This makes it difficult to add a mass
term, because a mass term tends to push the
field toward the value zero. But the zero
value of the vector potential is not a gauge
invariant idea. What is zero in one gauge
is nonzero in another.
So in order to give mass to a gauge theory,
the gauge invariance must be broken by a condensate.
The condensate will then define a preferred
phase, and the phase of the condensate will
define the zero value of the field in a gauge-invariant
way. The gauge-invariant definition is that
a gauge field is zero when the phase change
along any path from parallel transport is
equal to the phase difference in the condensate
wavefunction.
The condensate value is described by a quantum
field with an expectation value, just as in
the Ginzburg-Landau model.
In order for the phase of the vacuum to define
a gauge, the field must have a phase (also
referred to as 'to be charged'). In order
for a scalar field Φ to have a phase, it
must be complex, or (equivalently) it should
contain two fields with a symmetry which rotates
them into each other. The vector potential
changes the phase of the quanta produced by
the field when they move from point to point.
In terms of fields, it defines how much to
rotate the real and imaginary parts of the
fields into each other when comparing field
values at nearby points.
The only renormalizable model where a complex
scalar field Φ acquires a nonzero value is
the Mexican-hat model, where the field energy
has a minimum away from zero. The action for
this model is
S
(
ϕ
)
=
∫
1
2
|
∂
ϕ
|
2
−
λ
(
|
ϕ
|
2
−
Φ
2
)
2
,
{\displaystyle S(\phi )=\int {\frac {1}{2}}\left|\partial
\phi \right|^{2}-\lambda \left(\left|\phi
\right|^{2}-\Phi ^{2}\right)^{2},}
which results in the Hamiltonian
H
(
ϕ
)
=
1
2
|
ϕ
˙
|
2
+
|
∇
ϕ
|
2
+
V
(
|
ϕ
|
)
.
{\displaystyle H(\phi )={\frac {1}{2}}\left|{\dot
{\phi }}\right|^{2}+\left|\nabla \phi \right|^{2}+V(\left|\phi
\right|).}
The first term is the kinetic energy of the
field. The second term is the extra potential
energy when the field varies from point to
point. The third term is the potential energy
when the field has any given magnitude.
This potential energy, the Higgs potential,
V(z, Φ) = λ(|z|2 − Φ2)2, has a graph
which looks like a Mexican hat, which gives
the model its name. In particular, the minimum
energy value is not at z = 0, but on the circle
of points where the magnitude of z is Φ.
When the field Φ(x) is not coupled to electromagnetism,
the Mexican-hat potential has flat directions.
Starting in any one of the circle of vacua
and changing the phase of the field from point
to point costs very little energy. Mathematically,
if
ϕ
(
x
)
=
Φ
e
i
θ
(
x
)
{\displaystyle \phi (x)=\Phi e^{i\theta (x)}}
with a constant prefactor, then the action
for the field θ(x), i.e., the "phase" of
the Higgs field Φ(x), has only derivative
terms. This is not a surprise. Adding a constant
to θ(x) is a symmetry of the original theory,
so different values of θ(x) cannot have different
energies. This is an example of Goldstone's
theorem: spontaneously broken continuous symmetries
normally produce massless excitations.
The Abelian Higgs model is the Mexican-hat
model coupled 
to electromagnetism:
S
(
ϕ
,
A
)
=
∫
−
1
4
F
μ
ν
F
μ
ν
+
|
(
∂
−
i
q
A
)
ϕ
|
2
−
λ
(
|
ϕ
|
2
−
Φ
2
)
2
.
{\displaystyle S(\phi ,A)=\int -{\frac {1}{4}}F^{\mu
\nu }F_{\mu \nu }+\left|\left(\partial -iqA\right)\phi
\right|^{2}-\lambda \left(\left|\phi \right|^{2}-\Phi
^{2}\right)^{2}.}
The classical vacuum is again at the minimum
of the potential, where the magnitude of the
complex field φ is equal to Φ. But now the
phase of the field is arbitrary, because gauge
transformations change it. This means that
the field θ(x) can be set to zero by a gauge
transformation, and does not represent any
actual degrees of freedom at all.
Furthermore, choosing a gauge where the phase
of the vacuum is fixed, the potential energy
for fluctuations of the vector field is nonzero.
So in the Abelian Higgs model, the gauge field
acquires a mass. To calculate the magnitude
of the mass, consider a constant value of
the vector potential A in the x-direction
in the gauge where the condensate has constant
phase. This is the same as a sinusoidally
varying condensate in the gauge where the
vector potential is zero. In the gauge where
A is zero, the potential energy density in
the condensate is the scalar gradient energy:
E
=
1
2
|
∂
(
Φ
e
i
q
A
x
)
|
2
=
1
2
q
2
Φ
2
A
2
.
{\displaystyle E={\frac {1}{2}}\left|\partial
\left(\Phi e^{iqAx}\right)\right|^{2}={\frac
{1}{2}}q^{2}\Phi ^{2}A^{2}.}
This energy is the same as a mass term 1/2m2A2
where m = qΦ.
=== Non-Abelian Higgs mechanism ===
The Non-Abelian Higgs model has the following
action
S
(
ϕ
,
A
)
=
∫
1
4
g
2
tr
⁡
(
F
μ
ν
F
μ
ν
)
+
|
D
ϕ
|
2
+
V
(
|
ϕ
|
)
{\displaystyle S(\phi ,\mathbf {A} )=\int
{1 \over 4g^{2}}\mathop {\textrm {tr}} (F^{\mu
\nu }F_{\mu \nu })+|D\phi |^{2}+V(|\phi |)}
where now the 
non-Abelian field A is contained in the covariant
derivative D and in the tensor components
F
μ
ν
{\displaystyle F^{\mu \nu }}
and
F
μ
ν
{\displaystyle F_{\mu \nu }}
(the relation between A and those components
is well-known from the Yang–Mills theory).
It is exactly analogous to the Abelian Higgs
model. Now the field φ is in a representation
of the gauge group, and the gauge covariant
derivative is defined by the rate of change
of the field minus the rate of change from
parallel transport using the gauge field A
as a connection.
D
ϕ
=
∂
ϕ
−
i
A
k
t
k
ϕ
{\displaystyle D\phi =\partial \phi -iA^{k}t_{k}\phi
}
Again, the expectation value of Φ defines
a preferred gauge where the vacuum is constant,
and fixing this gauge, fluctuations in the
gauge field A come with a nonzero energy cost.
Depending on the representation of the scalar
field, not every gauge field acquires a mass.
A simple example is in the renormalizable
version of an early electroweak model due
to Julian Schwinger. In this model, the gauge
group is SO(3) (or SU(2) − there are no
spinor representations in the model), and
the gauge invariance is broken down to U(1)
or SO(2) at long distances. To make a consistent
renormalizable version using the Higgs mechanism,
introduce a scalar field φa which transforms
as a vector (a triplet) of SO(3). If this
field has a vacuum expectation value, it points
in some direction in field space. Without
loss of generality, one can choose the z-axis
in field space to be the direction that φ
is pointing, and then the vacuum expectation
value of φ is (0, 0, A), where A is a constant
with dimensions of mass (
c
=
ℏ
=
1
{\displaystyle c=\hbar =1}
).
Rotations around the z-axis form a U(1) subgroup
of SO(3) which preserves the vacuum expectation
value of φ, and this is the unbroken gauge
group. Rotations around the x and y-axis do
not preserve the vacuum, and the components
of the SO(3) gauge field which generate these
rotations become massive vector mesons. There
are two massive W mesons in the Schwinger
model, with a mass set by the mass scale A,
and one massless U(1) gauge boson, similar
to the photon.
The Schwinger model predicts magnetic monopoles
at the electroweak unification scale, and
does not predict the Z meson. It doesn't break
electroweak symmetry properly as in nature.
But historically, a model similar to this
(but not using the Higgs mechanism) was the
first in which the weak force and the electromagnetic
force were unified.
=== Affine Higgs mechanism ===
Ernst Stueckelberg discovered a version of
the Higgs mechanism by analyzing the theory
of quantum electrodynamics with a massive
photon. Effectively, Stueckelberg's model
is a limit of the regular Mexican hat Abelian
Higgs model, where the vacuum expectation
value H goes to infinity and the charge of
the Higgs field goes to zero in such a way
that their product stays fixed. The mass of
the Higgs boson is proportional to H, so the
Higgs boson becomes infinitely massive and
decouples, so is not present in the discussion.
The vector meson mass, however, equals to
the product eH, and stays finite.
The interpretation is that when a U(1) gauge
field does not require quantized charges,
it is possible to keep only the angular part
of the Higgs oscillations, and discard the
radial part. The angular part of the Higgs
field θ has the following gauge transformation
law:
θ
→
θ
+
e
α
A
→
A
+
∂
α
.
{\displaystyle {\begin{aligned}\theta &\rightarrow
\theta +e\alpha \,\\A&\rightarrow A+\partial
\alpha .\end{aligned}}}
The gauge covariant derivative for the angle
(which is actually gauge invariant) is:
D
θ
=
∂
θ
−
e
A
H
.
{\displaystyle D\theta =\partial \theta -eAH.\,}
In order to keep θ fluctuations finite and
nonzero in this limit, θ should be rescaled
by H, so that its kinetic term in the action
stays normalized. The action for the theta
field is read off from the Mexican hat action
by substituting
ϕ
=
H
e
i
θ
/
H
{\displaystyle \phi =He^{i\theta /H}}
.
S
=
∫
1
4
F
2
+
1
2
(
D
θ
)
2
=
∫
1
4
F
2
+
1
2
(
∂
θ
−
H
e
A
)
2
=
∫
1
4
F
2
+
1
2
(
∂
θ
−
m
A
)
2
{\displaystyle S=\int {\tfrac {1}{4}}F^{2}+{\tfrac
{1}{2}}(D\theta )^{2}=\int {\tfrac {1}{4}}F^{2}+{\tfrac
{1}{2}}(\partial \theta -HeA)^{2}=\int {\tfrac
{1}{4}}F^{2}+{\tfrac {1}{2}}(\partial \theta
-mA)^{2}}
since eH is the gauge boson mass. By making
a gauge transformation to set θ = 0, the
gauge freedom in the action is eliminated,
and the action becomes that of a massive vector
field:
S
=
∫
1
4
F
2
+
1
2
m
2
A
2
.
{\displaystyle S=\int {\tfrac {1}{4}}F^{2}+{\tfrac
{1}{2}}m^{2}A^{2}.\,}
To have arbitrarily small charges requires
that the U(1) is not the circle of unit complex
numbers under multiplication, but the real
numbers R under addition, which is only different
in the global topology. Such a U(1) group
is non-compact. The field θ transforms as
an affine representation of the gauge group.
Among the allowed gauge groups, only non-compact
U(1) admits affine representations, and the
U(1) of electromagnetism is experimentally
known to be compact, since charge quantization
holds to extremely high accuracy.
The Higgs condensate in this model has infinitesimal
charge, so interactions with the Higgs boson
do not violate charge conservation. The theory
of quantum electrodynamics with a massive
photon is still a renormalizable theory, one
in which electric charge is still conserved,
but magnetic monopoles are not allowed. For
non-Abelian gauge theory, there is no affine
limit, and the Higgs oscillations cannot be
too much more massive than the vectors.
== See also ==
Electroweak interaction
Electromagnetic mass
Higgs bundle
Mass generation
Quantum triviality
Yang–Mills–Higgs equations
== Notes
