In particle and condensed matter physics,
Goldstone bosons or Nambu–Goldstone bosons
(NGBs) are bosons that appear necessarily
in models exhibiting spontaneous breakdown
of continuous symmetries. They were discovered
by Yoichiro Nambu in the context of the BCS
superconductivity mechanism, and subsequently
elucidated by Jeffrey Goldstone, and systematically
generalized in the context of quantum field
theory.These spinless bosons correspond to
the spontaneously broken internal symmetry
generators, and are characterized by the quantum
numbers of these.
They transform nonlinearly (shift) under the
action of these generators, and can thus be
excited out of the asymmetric vacuum by these
generators. Thus, they can be thought of as
the excitations of the field in the broken
symmetry directions in group space—and are
massless if the spontaneously broken symmetry
is not also broken explicitly.
If, instead, the symmetry is not exact, i.e.
if it is explicitly broken as well as spontaneously
broken, then the Nambu–Goldstone bosons
are not massless, though they typically remain
relatively light; they are then called pseudo-Goldstone
bosons or pseudo-Nambu–Goldstone bosons
(abbreviated PNGBs).
== Goldstone's theorem ==
Goldstone's theorem examines a generic continuous
symmetry which is spontaneously broken; i.e.,
its currents are conserved, but the ground
state is not invariant under the action of
the corresponding charges. Then, necessarily,
new massless (or light, if the symmetry is
not exact) scalar particles appear in the
spectrum of possible excitations. There is
one scalar particle—called a Nambu–Goldstone
boson—for each generator of the symmetry
that is broken, i.e., that does not preserve
the ground state. The Nambu–Goldstone mode
is a long-wavelength fluctuation of the corresponding
order parameter.
By virtue of their special properties in coupling
to the vacuum of the respective symmetry-broken
theory, vanishing momentum ("soft") Goldstone
bosons involved in field-theoretic amplitudes
make such amplitudes vanish ("Adler zeros").
In theories with gauge symmetry, the Goldstone
bosons are "eaten" by the gauge bosons. The
latter become massive and their new, longitudinal
polarization is provided by the Goldstone
boson.
== Examples ==
=== Natural ===
In fluids, the phonon is longitudinal and
it is the Goldstone boson of the spontaneously
broken Galilean symmetry. In solids, the situation
is more complicated; the Goldstone bosons
are the longitudinal and transverse phonons
and they happen to be the Goldstone bosons
of spontaneously broken Galilean, translational,
and rotational symmetry with no simple one-to-one
correspondence between the Goldstone modes
and the broken symmetries.
In magnets, the original rotational symmetry
(present in the absence of an external magnetic
field) is spontaneously broken such that the
magnetization points into a specific direction.
The Goldstone bosons then are the magnons,
i.e., spin waves in which the local magnetization
direction oscillates.
The pions are the pseudo-Goldstone bosons
that result from the spontaneous breakdown
of the chiral-flavor symmetries of QCD effected
by quark condensation due to the strong interaction.
These symmetries are further explicitly broken
by the masses of the quarks, so that the pions
are not massless, but their mass is significantly
smaller than typical hadron masses.
The longitudinal polarization components of
the W and Z bosons correspond to the Goldstone
bosons of the spontaneously broken part of
the electroweak symmetry SU(2)⊗U(1), which,
however, are not observable. Because this
symmetry is gauged, the three would-be Goldstone
bosons are "eaten" by the three gauge bosons
corresponding to the three broken generators;
this gives these three gauge bosons a mass,
and the associated necessary third polarization
degree of freedom. This is described in the
Standard Model through the Higgs mechanism.
An analogous phenomenon occurs in superconductivity,
which served as the original source of inspiration
for Nambu, namely, the photon develops a dynamical
mass (expressed as magnetic flux exclusion
from a superconductor), cf. the Ginzburg–Landau
theory.
=== Theory ===
Consider a complex scalar field ϕ, with the
constraint that ϕ*ϕ = v², a constant. One
way to impose a constraint of this sort is
by including a potential interaction term
in its Lagrangian density,
λ
(
ϕ
∗
ϕ
−
v
2
)
2
,
{\displaystyle \lambda (\phi ^{*}\phi -v^{2})^{2}~,}
and taking the limit as λ → ∞ (this is
called the "Abelian nonlinear σ-model". It
corresponds to the Goldstone sombrero potential
where the tip and the sides shoot to infinity,
preserving the location of the minimum at
its base).
The constraint, and the action, below, are
invariant under a U(1) phase transformation,
δϕ=iεϕ. The field can be redefined to
give a real scalar field (i.e., a spin-zero
particle) θ without any constraint by
ϕ
=
v
e
i
θ
{\displaystyle \phi =ve^{i\theta }}
where θ is the Nambu–Goldstone boson (actually
vθ is), and the U(1) symmetry transformation
effects a shift on θ, namely
δ
θ
=
ϵ
,
{\displaystyle \delta \theta =\epsilon ~,}
but does not preserve the ground state |0〉
(i.e. the above infinitesimal transformation
does not annihilate it—the hallmark of invariance),
as evident in the charge of the current below.
Thus, the vacuum is degenerate and noninvariant
under the action of the spontaneously broken
symmetry.
The corresponding Lagrangian density is given
by
L
=
−
1
2
(
∂
μ
ϕ
∗
)
∂
μ
ϕ
+
m
2
ϕ
∗
ϕ
=
−
1
2
(
−
i
v
e
−
i
θ
∂
μ
θ
)
(
i
v
e
i
θ
∂
μ
θ
)
+
m
2
v
2
,
{\displaystyle {\mathcal {L}}=-{\frac {1}{2}}(\partial
^{\mu }\phi ^{*})\partial _{\mu }\phi +m^{2}\phi
^{*}\phi =-{\frac {1}{2}}(-ive^{-i\theta }\partial
^{\mu }\theta )(ive^{i\theta }\partial _{\mu
}\theta )+m^{2}v^{2},}
and thus
=
−
v
2
2
(
∂
μ
θ
)
(
∂
μ
θ
)
+
m
2
v
2
.
{\displaystyle =-{\frac {v^{2}}{2}}(\partial
^{\mu }\theta )(\partial _{\mu }\theta )+m^{2}v^{2}~.}
Note that the constant term m²v² in the
Lagrangian density has no physical significance,
and the other term in it is simply the kinetic
term for a massless scalar.
The symmetry-induced conserved U(1) current
is
J
μ
=
−
v
2
∂
μ
θ
.
{\displaystyle J_{\mu }=-v^{2}\partial _{\mu
}\theta ~.}
The charge, Q, resulting from this current
shifts θ and the ground state to a new, degenerate,
ground state. Thus, a vacuum with 〈θ〉
= 0 will shift to a different vacuum with
〈θ〉 = −ε. The current connects the
original vacuum with the Nambu–Goldstone
boson state, 〈0|J0(0)|θ〉≠ 0.
In general, in a theory with several scalar
fields, ϕj, the Nambu–Goldstone mode ϕg
is massless, and parameterises the curve of
possible (degenerate) vacuum states. Its hallmark
under the broken symmetry transformation is
nonvanishing vacuum expectation 〈δϕg〉,
an order parameter, for vanishing 〈ϕg〉
= 0, at some ground state |0〉 chosen at
the minimum of the potential, 〈∂V/∂ϕi〉
= 0. Symmetry dictates that all variations
of the potential with respect to the fields
in all symmetry directions vanish. The vacuum
value of the first order variation in any
direction vanishes as just seen; while the
vacuum value of the second order variation
must also vanish, as follows. Vanishing vacuum
values of field symmetry transformation increments
add no new information.
By contrast, however, nonvanishing vacuum
expectations of transformation increments,
〈δϕg〉, specify the relevant (Goldstone)
null eigenvectors of the mass matrix,
and hence the corresponding zero-mass eigenvalues.
== Goldstone's argument ==
The principle behind Goldstone's argument
is that the ground state is not unique. Normally,
by current conservation, the charge operator
for any symmetry current is time-independent,
d
d
t
Q
=
d
d
t
∫
x
J
0
(
x
)
=
0.
{\displaystyle {d \over dt}Q={d \over dt}\int
_{x}J^{0}(x)=0.}
Acting with the charge operator on the vacuum
either annihilates the vacuum, if that is
symmetric; else, if not, as is the case in
spontaneous symmetry breaking, it produces
a zero-frequency state out of it, through
its shift transformation feature illustrated
above. Actually, here, the charge itself is
ill-defined, cf. the Fabri–Picasso argument
below.
But its better behaved commutators with fields,
that is, the nonvanishing transformation shifts
〈δϕg〉, are, nevertheless, time-invariant,
d
⟨
δ
ϕ
g
⟩
d
t
=
0
,
{\displaystyle {\frac {d\langle \delta \phi
_{g}\rangle }{dt}}=0,}
thus generating a δ(k0) in its Fourier transform.
(This ensures that, inserting a complete set
of intermediate states in a nonvanishing current
commutator can lead to vanishing time-evolution
only when one or more of these states is massless.)
Thus, if the vacuum is not invariant under
the symmetry, action of the charge operator
produces a state which is different from the
vacuum chosen, but which has zero frequency.
This is a long-wavelength oscillation of a
field which is nearly stationary: there are
physical states with zero frequency, k0, so
that the theory cannot have a mass gap.
This argument is further clarified by taking
the limit carefully. If an approximate charge
operator acting in a huge but finite region
A is applied to the vacuum,
d
d
t
Q
A
=
d
d
t
∫
x
e
−
x
2
2
A
2
J
0
(
x
)
=
−
∫
x
e
−
x
2
2
A
2
∇
⋅
J
=
∫
x
∇
(
e
−
x
2
2
A
2
)
⋅
J
,
{\displaystyle {d \over dt}Q_{A}={d \over
dt}\int _{x}e^{-{\frac {x^{2}}{2A^{2}}}}J^{0}(x)=-\int
_{x}e^{-{\frac {x^{2}}{2A^{2}}}}\nabla \cdot
J=\int _{x}\nabla \left(e^{-{\frac {x^{2}}{2A^{2}}}}\right)\cdot
J,}
a state with approximately vanishing time
derivative is produced,
‖
d
d
t
Q
A
|
0
⟩
‖
≈
1
A
‖
Q
A
|
0
⟩
‖
.
{\displaystyle \left\|{d \over dt}Q_{A}|0\rangle
\right\|\approx {\frac {1}{A}}\left\|Q_{A}|0\rangle
\right\|.}
Assuming a nonvanishing mass gap m0, the frequency
of any state like the above, which is orthogonal
to the vacuum, is at least m0,
‖
d
d
t
|
θ
⟩
‖
=
‖
H
|
θ
⟩
‖
≥
m
0
‖
|
θ
⟩
‖
.
{\displaystyle \left\|{\frac {d}{dt}}|\theta
\rangle \right\|=\|H|\theta \rangle \|\geq
m_{0}\||\theta \rangle \|.}
Letting A become large leads to a contradiction.
Consequently m0 = 0. However this argument
fails when the symmetry is gauged, because
then the symmetry generator is only performing
a gauge transformation. A gauge transformed
state is the same exact state, so that acting
with a symmetry generator does not get one
out of the vacuum.
Fabri–Picasso Theorem. Q does not properly
exist in the Hilbert space, unless Q|0〉
= 0.The argument requires both 
the vacuum and the charge Q 
to be translationally invariant, P|0〉 = 0,
[P,Q]= 0.
Consider the correlation function of the charge
with itself,
⟨
0
|
Q
Q
|
0
⟩
=
∫
d
3
x
⟨
0
|
j
0
(
x
)
Q
|
0
⟩
=
∫
d
3
x
⟨
0
|
e
i
P
x
j
0
(
0
)
e
−
i
P
x
Q
|
0
⟩
=
∫
d
3
x
⟨
0
|
e
i
P
x
j
0
(
0
)
e
−
i
P
x
Q
e
i
P
x
e
−
i
P
x
|
0
⟩
=
∫
d
3
x
⟨
0
|
j
0
(
0
)
Q
|
0
⟩
{\displaystyle {\begin{aligned}\langle 0|QQ|0\rangle
&=\int d^{3}x\langle 0|j_{0}(x)Q|0\rangle
\\&=\int d^{3}x\left\langle 0\left|e^{iPx}j_{0}(0)e^{-iPx}Q\right|0\right\rangle
\\&=\int d^{3}x\left\langle 0\left|e^{iPx}j_{0}(0)e^{-iPx}Qe^{iPx}e^{-iPx}\right|0\right\rangle
\\&=\int d^{3}x\left\langle 0\left|j_{0}(0)Q\right|0\right\rangle
\end{aligned}}}
so the integrand in the right hand side does
not depend on the position.
Thus, its value is proportional to the total
space volume,
‖
Q
|
0
⟩
‖
2
=
∞
{\displaystyle \|Q|0\rangle \|^{2}=\infty
}
— unless the symmetry is unbroken, Q|0〉
= 0. Consequently, Q does not properly exist
in the Hilbert space.
== Infraparticles ==
There is an arguable loophole in the theorem.
If one reads the theorem carefully, it only
states that there exist non-vacuum states
with arbitrarily small energies. Take for
example a chiral N = 1 super QCD model with
a nonzero squark VEV which is conformal in
the IR. The chiral symmetry is a global symmetry
which is (partially) spontaneously broken.
Some of the "Goldstone bosons" associated
with this spontaneous symmetry breaking are
charged under the unbroken gauge group and
hence, these composite bosons have a continuous
mass spectrum with arbitrarily small masses
but yet there is no Goldstone boson with exactly
zero mass. In other words, the Goldstone bosons
are infraparticles.
== Nonrelativistic theories ==
A version of Goldstone's theorem also applies
to nonrelativistic theories (and also relativistic
theories with spontaneously broken spacetime
symmetries, such as Lorentz symmetry or conformal
symmetry, rotational, or translational invariance).
It essentially states that, for each spontaneously
broken symmetry, there corresponds some quasiparticle
with no energy gap—the nonrelativistic version
of the mass gap. (Note that the energy here
is really H−μN−α→⋅P→ and not H.)
However, two different spontaneously broken
generators may now give rise to the same Nambu–Goldstone
boson. For example, in a superfluid, both
the U(1) particle number symmetry and Galilean
symmetry are spontaneously broken. However,
the phonon is the Goldstone boson for both.
In general, the phonon is effectively the
Nambu–Goldstone boson for spontaneously
broken Galilean/Lorentz symmetry. However,
in contrast to the case of internal symmetry
breaking, when spacetime symmetries are broken,
the order parameter need not be a scalar field,
but may be a tensor field, and the corresponding
independent massless modes may now be fewer
than the number of spontaneously broken generators,
because the
Goldstone modes may now be linearly dependent
among themselves: e.g., the Goldstone modes
for some generators might be expressed as
gradients of Goldstone modes for other broken
generators.
== Nambu–Goldstone fermions ==
Spontaneously broken global fermionic symmetries,
which occur in some supersymmetric models,
lead to Nambu–Goldstone fermions, or goldstinos.
These have spin ½, instead of 0, and carry
all quantum numbers of the respective supersymmetry
generators broken spontaneously.
Spontaneous supersymmetry breaking smashes
up ("reduces") supermultiplet structures into
the characteristic nonlinear realizations
of broken supersymmetry, so that goldstinos
are superpartners of all particles in the
theory, of any spin, and the only superpartners,
at that. That is, to say, two non-goldstino
particles
are connected to only goldstinos through supersymmetry
transformations, and not to each other, even
if they were so connected before the breaking
of supersymmetry. As a result, the masses
and spin multiplicities of such particles
are then arbitrary.
== See also ==
Pseudo-Goldstone boson
Majoron
Higgs mechanism
Mermin–Wagner theorem
Vacuum expectation value
Noether's theorem
