The Quantum Chromodynamic Vacuum or QCD vacuum
is the vacuum state of quantum chromodynamics
(QCD). It is an example of a non-perturbative
vacuum state, characterized by non-vanishing
condensates such as the gluon condensate and
the quark condensate in the complete theory
which includes quarks. The presence of these
condensates characterizes the confined phase
of quark matter.
== Symmetries and symmetry breaking ==
=== 
Symmetries of the QCD Lagrangian ===
Like any relativistic quantum field theory,
QCD enjoys Poincaré symmetry including the
discrete symmetries CPT (each of which is
realized). Apart from these space-time symmetries,
it also has internal symmetries. Since QCD
is an SU(3) gauge theory, it has local SU(3)
gauge symmetry.
Since it has many flavours of quarks, it has
approximate flavour and chiral symmetry. This
approximation is said to involve the chiral
limit of QCD. Of these chiral symmetries,
the baryon number symmetry is exact. Some
of the broken symmetries include the axial
U(1) symmetry of the flavour group. This is
broken by the chiral anomaly. The presence
of instantons implied by this anomaly also
breaks CP symmetry.
In summary, the QCD Lagrangian has the following
symmetries:
Poincaré symmetry and CPT invariance
SU(3) local gauge symmetry
approximate global SU(Nf) × SU(Nf) flavour
chiral symmetry and the U(1) baryon number
symmetryThe following classical symmetries
are broken in the QCD Lagrangian:
scale, i.e., conformal symmetry (through the
scale anomaly), giving rise to asymptotic
freedom
the axial part of the U(1) flavour chiral
symmetry (through the chiral anomaly), giving
rise to the strong CP problem.
=== Spontaneous symmetry breaking ===
When the Hamiltonian of a system (or the Lagrangian)
has a certain symmetry, but the vacuum does
not, then one says that spontaneous symmetry
breaking (SSB) has taken place.
A familiar example of SSB is in ferromagnetic
materials. Microscopically, the material consists
of atoms with a non-vanishing spin, each of
which acts like a tiny bar magnet, i.e., a
magnetic dipole. The Hamiltonian of the material,
describing the interaction of neighbouring
dipoles, is invariant under rotations. At
high temperature, there is no magnetization
of a large sample of the material. Then one
says that the symmetry of the Hamiltonian
is realized by the system. However, at low
temperature, there could be an overall magnetization.
This magnetization has a preferred direction,
since one can tell the north magnetic pole
of the sample from the south magnetic pole.
In this case, there is spontaneous symmetry
breaking of the rotational symmetry of the
Hamiltonian.
When a continuous symmetry is spontaneously
broken, massless bosons appear, corresponding
to the remaining symmetry. This is called
the Goldstone phenomenon and the bosons are
called Goldstone bosons.
=== Symmetries of the QCD vacuum ===
The SU(Nf) × SU(Nf) chiral flavour symmetry
of the QCD Lagrangian is broken in the vacuum
state of the theory. The symmetry of the vacuum
state is the diagonal SU(Nf) part of the chiral
group. The diagnostic for this is the formation
of a non-vanishing chiral condensate ⟨ψiψi⟩,
where ψi is the quark field operator, and
the flavour index i is summed. The Goldstone
bosons of the symmetry breaking are the pseudoscalar
mesons.
When Nf=2, i.e., only the up and down quarks
are treated as massless, the three pions are
the Goldstone bosons. When the strange quark
is also treated as massless, i.e., Nf = 3,
all eight pseudoscalar mesons of the quark
model become Goldstone bosons. The actual
masses of these mesons are obtained in chiral
perturbation theory through an expansion in
the (small) actual masses of the quarks.
In other phases of quark matter the full chiral
flavour symmetry may be recovered, or broken
in completely different ways.
== Experimental evidence ==
The evidence for QCD condensates comes from
two eras, the pre-QCD era 1950–1973 and
the post-QCD era, after 1974. The pre-QCD
results established that the strong interactions
vacuum contains a quark chiral condensate,
while the post-QCD results established that
the vacuum also contains a gluon condensate.
=== Motivating results ===
==== 
Gradient coupling ====
In the 1950s, there were many attempts to
produce a field theory to describe the interactions
of pions and nucleons. The obvious renormalizable
interaction between the two objects is the
Yukawa coupling to a pseudoscalar:
L
I
=
N
¯
γ
5
π
N
{\displaystyle L_{I}={\bar {N}}\gamma _{5}\pi
N}
And this is clearly theoretically correct,
since it is leading order and it takes all
the symmetries into account. But it doesn't
match experiment. The interaction that does
couples the nucleons to the gradient of the
pion field.
g
N
¯
γ
μ
γ
5
∂
μ
π
N
{\displaystyle g{\bar {N}}\gamma ^{\mu }\gamma
_{5}\partial _{\mu }\pi N}
This is the gradient-coupling model. This
interaction has a very different dependence
on the energy of the pion—it vanishes at
zero momentum.
This type of coupling means that a coherent
state of low momentum pions barely interacts
at all. This is a manifestation of an approximate
symmetry, a shift symmetry of the pion field.
The replacement
π
→
π
+
C
{\displaystyle \pi \rightarrow \pi +C}
leaves the gradient coupling alone, but not
the pseudoscalar coupling.
The modern explanation for the shift symmetry
was first proposed by Yoichiro Nambu. The
pion field is a Goldstone boson, and the shift
symmetry is the lowest order approximation
to moving along the flat directions.
==== Goldberger–Treiman relation ====
There is a mysterious relationship between
the strong interaction coupling of the pions
to the nucleons, the coefficient g in the
gradient coupling model, and the axial vector
current coefficient of the nucleon which determines
the weak decay rate of the neutron. The relation
is
g
π
N
N
F
π
=
G
A
M
N
{\displaystyle g_{\pi NN}F_{\pi }=G_{A}M_{N}}
and it is obeyed to 10% accuracy.
The constant GA is the coefficient that determines
the neutron decay rate. It gives the normalization
of the weak interaction matrix elements for
the nucleon. On the other hand, the pion-nucleon
coupling is a phenomenological constant describing
the scattering of bound states of quarks and
gluons.
The weak interactions are current-current
interactions ultimately because they come
from a non-Abelian gauge theory. The Goldberger–Treiman
relation suggests that the pions for some
reason interact as if they are related to
the same symmetry current.
===== Partially conserved axial current =====
The phenomenon which gives rise to the Goldberger–Treiman
relation was called the partially conserved
axial current (PCAC) hypothesis. Partially
conserved is an archaic term for spontaneously
broken, and the axial current is now called
the chiral symmetry current.
The idea is that the symmetry current which
performs axial rotations on the fundamental
fields does not preserve the vacuum. This
means that the current J applied to the vacuum
produces particles. The particles must be
spinless, otherwise the vacuum wouldn't be
Lorentz invariant. By index matching, the
matrix element is:
J
μ
|
0
⟩
=
k
μ
|
π
⟩
,
{\displaystyle J_{\mu }|0\rangle =k_{\mu }|\pi
\rangle \,,}
where kμ is the momentum carried by the created
pion. Since the divergence of the axial current
operator is zero, we must have
∂
μ
J
μ
|
0
⟩
=
k
μ
k
μ
|
π
⟩
=
m
π
2
|
π
⟩
=
0
.
{\displaystyle \partial _{\mu }J^{\mu }|0\rangle
=k^{\mu }k_{\mu }|\pi \rangle =m_{\pi }^{2}|\pi
\rangle =0\,.}
Hence the pions are massless, m2π = 0, in
accordance with Goldstone's theorem.
Now if the scattering matrix element is considered,
we have
k
μ
⟨
N
(
p
)
|
π
(
k
)
N
(
p
′
)
⟩
=
⟨
N
(
p
)
|
J
μ
|
N
(
p
′
)
⟩
.
{\displaystyle k_{\mu }\langle N(p)|\pi (k)N(p')\rangle
=\langle N(p)|J_{\mu }|N(p')\rangle \,.}
Up to a momentum factor, which is the gradient
in the coupling, it takes the same form as
the axial current turning a neutron into a
proton in the current-current form of the
weak interaction.
⟨
N
|
J
μ
|
N
⟩
⟨
e
|
J
μ
|
ν
⟩
{\displaystyle \langle N|J^{\mu }|N\rangle
\langle e|J_{\mu }|\nu \rangle \,}
==== Soft pion emission ====
Extensions of the PCAC ideas allowed Steven
Weinberg to calculate the amplitudes for collisions
which emit low energy pions from the amplitude
for the same process with no pions. The amplitudes
are those given by acting with symmetry currents
on the external particles of the collision.
These successes established the basic properties
of the strong interaction vacuum well before
QCD.
=== Pseudo-Goldstone bosons ===
Experimentally it is seen that the masses
of the octet of pseudoscalar mesons is very
much lighter than the next lightest states;
i.e., the octet of vector mesons (such as
the rho meson). The most convincing evidence
for SSB of the chiral flavour symmetry of
QCD is the appearance of these pseudo-Goldstone
bosons. These would have been strictly massless
in the chiral limit. There is convincing demonstration
that the observed masses are compatible with
chiral perturbation theory. The internal consistency
of this argument is further checked by lattice
QCD computations which allow one to vary the
quark mass and check that the variation of
the pseudoscalar masses with the quark mass
is as required by chiral perturbation theory.
=== Eta prime meson ===
This pattern of SSB solves one of the earlier
"mysteries" of the quark model, where all
the pseudoscalar mesons should have been of
nearly the same mass. Since Nf = 3, there
should have been nine of these. However, one
(the SU(3) singlet η′ meson) has quite
a larger mass than the SU(3) octet. In the
quark model, this has no natural explanation
– a mystery named the η−η′ mass splitting
(the η is one member of the octet, which
should have been degenerate in mass with the
η′).
In QCD, one realizes that the η′ is associated
with the axial UA(1) which is explicitly broken
through the chiral anomaly, and thus its mass
is not "protected" to be small, like that
of the η. The η–η′ mass splitting can
be explained through the 't Hooft instanton
mechanism, whose 1/N realization is also known
as Witten–Veneziano mechanism.
=== Current algebra and QCD sum rules ===
PCAC and current algebra also provide evidence
for this pattern of SSB. Direct estimates
of the chiral condensate also come from such
analysis.
Another method of analysis of correlation
functions in QCD is through an operator product
expansion (OPE). This writes the vacuum expectation
value of a non-local operator as a sum over
VEVs of local operators, i.e., condensates.
The value of the correlation function then
dictates the values of the condensates. Analysis
of many separate correlation functions gives
consistent results for several condensates,
including the gluon condensate, the quark
condensate, and many mixed and higher order
condensates. In particular one obtains
⟨
(
g
G
)
2
⟩
=
d
e
f
⟨
g
2
G
μ
ν
G
μ
ν
⟩
≈
0.5
GeV
4
⟨
ψ
¯
ψ
⟩
≈
(
−
0.23
)
3
GeV
3
⟨
(
g
G
)
4
⟩
≈
5
:
10
⟨
(
g
G
)
2
⟩
2
{\displaystyle {\begin{aligned}\left\langle
(gG)^{2}\right\rangle \ {\stackrel {\mathrm
{def} }{=}}\ \left\langle g^{2}G_{\mu \nu
}G^{\mu \nu }\right\rangle &\approx 0.5\;{\text{GeV}}^{4}\\\left\langle
{\overline {\psi }}\psi \right\rangle &\approx
(-0.23)^{3}\;{\text{GeV}}^{3}\\\left\langle
(gG)^{4}\right\rangle &\approx 5:10\left\langle
(gG)^{2}\right\rangle ^{2}\end{aligned}}}
Here G refers to the gluon field tensor, ψ
to the quark field, and g to the QCD coupling.
These analyses are being refined further through
improved sum rule estimates and direct estimates
in lattice QCD. They provide the raw data
which must be explained by models of the QCD
vacuum.
== Models of the QCD vacuum ==
A full solution of QCD should give a full
description of the vacuum, confinement and
the hadron spectrum. Lattice QCD is making
rapid progress towards providing the solution
as a systematically improvable numerical computation.
However, approximate models of the QCD vacuum
remain useful in more restricted domains.
The purpose of these models is to make quantitative
sense of some set of condensates and hadron
properties such as masses and form factors.
This section is devoted to models. Opposed
to these are systematically improvable computational
procedures such as large N QCD and lattice
QCD, which are described in their own articles.
=== The Savvidy vacuum, instabilities and
structure ===
The Savvidy vacuum is a model of the QCD vacuum
which at a basic level is a statement that
it cannot be the conventional Fock vacuum
empty of particles and fields. In 1977, George
Savvidy showed that the QCD vacuum with zero
field strength is unstable, and decays into
a state with a calculable non vanishing value
of the field. Since condensates are scalar,
it seems like a good first approximation that
the vacuum contains some non-zero but homogeneous
field which gives rise to these condensates.
However, Stanley Mandelstam showed that a
homogeneous vacuum field is also unstable.
The instability of a homogeneous gluon field
was argued by Niels Kjær Nielsen and Poul
Olesen in their 1978 paper. These arguments
suggest that the scalar condensates are an
effective long-distance description of the
vacuum, and at short distances, below the
QCD scale, the vacuum may have structure.
=== The dual superconducting model ===
In a type II superconductor, electric charges
condense into Cooper pairs. As a result, magnetic
flux is squeezed into tubes. In the dual superconductor
picture of the QCD vacuum, chromomagnetic
monopoles condense into dual Cooper pairs,
causing chromoelectric flux to be squeezed
into tubes. As a result, confinement and the
string picture of hadrons follows. This dual
superconductor picture is due to Gerard 't
Hooft and Stanley Mandelstam. 't Hooft showed
further that an Abelian projection of a non-Abelian
gauge theory contains magnetic monopoles.
While the vortices in a type II superconductor
are neatly arranged into a hexagonal or occasionally
square lattice, as is reviewed in Olesen's
1980 seminar one may expect a much more complicated
and possibly dynamical structure in QCD. For
example, nonabelian Abrikosov-Nielsen-Olesen
vortices may vibrate wildly or be knotted.
=== String models ===
String models of confinement and hadrons have
a long history. They were first invented to
explain certain aspects of crossing symmetry
in the scattering of two mesons. They were
also found to be useful in the description
of certain properties of the Regge trajectory
of the hadrons. These early developments took
on a life of their own called the dual resonance
model (later renamed string theory). However,
even after the development of QCD string models
continued to play a role in the physics of
strong interactions. These models are called
non-fundamental strings or QCD strings, since
they should be derived from QCD, as they are,
in certain approximations such as the strong
coupling limit of lattice QCD.
The model states that the colour electric
flux between a quark and an antiquark collapses
into a string, rather than spreading out into
a Coulomb field as the normal electric flux
does. This string also obeys a different force
law. It behaves as if the string had constant
tension, so that separating out the ends (quarks)
would give a potential energy increasing linearly
with the separation. When the energy is higher
than that of a meson, the string breaks and
the two new ends become a quark-antiquark
pair, thus describing the creation of a meson.
Thus confinement is incorporated naturally
into the model.
In the form of the Lund model Monte Carlo
program, this picture has had remarkable success
in explaining experimental data collected
in electron-electron and hadron-hadron collisions.
=== Bag models ===
Strictly, these models are not models of the
QCD vacuum, but of physical single particle
quantum states — the hadrons. The model
proposed originally in 1974 by A. Chodos et
al.
consists of inserting a quark model in a perturbative
vacuum inside a volume of space called a bag.
Outside this bag is the real QCD vacuum, whose
effect is taken into account through the difference
between energy density of the true QCD vacuum
and the perturbative vacuum (bag constant
B) and boundary conditions imposed on the
quark wave functions and the gluon field.
The hadron spectrum is obtained by solving
the Dirac equation for quarks and the Yang–Mills
equations for gluons. The wave functions of
the quarks satisfy the boundary conditions
of a fermion in an infinitely deep potential
well of scalar type with respect to the Lorentz
group. The boundary conditions for the gluon
field are those of the dual color superconductor.
The role of such a superconductor is attributed
to the physical vacuum of QCD. Bag models
strictly prohibit the existence of open color
(free quarks, free gluons, etc.) and lead
in particular to string models of hadrons.
The chiral bag model couples the axial vector
current ψγ5γμψ of the quarks at the bag
boundary to a pionic field outside of the
bag. In the most common formulation, the chiral
bag model basically replaces the interior
of the skyrmion with the bag of quarks. Very
curiously, most physical properties of the
nucleon become mostly insensitive to the bag
radius. Prototypically, the baryon number
of the chiral bag remains an integer, independent
of bag radius: the exterior baryon number
is identified with the topological winding
number density of the Skyrme soliton, while
the interior baryon number consists of the
valence quarks (totaling to one) plus the
spectral asymmetry of the quark eigenstates
in the bag. The spectral asymmetry is just
the vacuum expectation value ⟨ψγ0ψ⟩
summed over all of the quark eigenstates in
the bag. Other values, such as the total mass
and the axial coupling constant gA, are not
precisely invariant like the baryon number,
but are mostly insensitive to the bag radius,
as long as the bag radius is kept below the
nucleon diameter. Because the quarks are treated
as free quarks inside the bag, the radius-independence
in a sense validates the idea of asymptotic
freedom.
=== Instanton ensemble ===
Another view states that BPST-like instantons
play an important role in the vacuum structure
of QCD. These instantons were discovered in
1975 by Alexander Belavin, Alexander Markovich
Polyakov, Albert S. Schwarz and Yu. S. Tyupkin
as topologically stable solutions to the Yang-Mills
field equations. They represent tunneling
transitions from one vacuum state to another.
These instantons are indeed found in lattice
calculations. The first computations performed
with instantons used the dilute gas approximation.
The results obtained did not solve the infrared
problem of QCD, making many physicists turn
away from instanton physics. Later, though,
an instanton liquid model was proposed, turning
out to be more promising an approach.The dilute
instanton gas model departs from the supposition
that the QCD vacuum consists of a gas of BPST-like
instantons. Although only the solutions with
one or few instantons (or anti-instantons)
are known exactly, a dilute gas of instantons
and anti-instantons can be approximated by
considering a superposition of one-instanton
solutions at great distances from one another.
Gerard 't Hooft calculated the effective action
for such an ensemble, and he found an infrared
divergence for big instantons, meaning that
an infinite amount of infinitely big instantons
would populate the vacuum.
Later, an instanton liquid model was studied.
This model starts from the assumption that
an ensemble of instantons cannot be described
by a mere sum of separate instantons. Various
models have been proposed, introducing interactions
between instantons or using variational methods
(like the "valley approximation") endeavoring
to approximate the exact multi-instanton solution
as closely as possible. Many phenomenological
successes have been reached. Whether an instanton
liquid can explain confinement in 3+1 dimensional
QCD is not known, but many physicists think
that it is unlikely.
=== Center vortex picture ===
A more recent picture of the QCD vacuum is
one in which center vortices play an important
role. These vortices are topological defects
carrying a center element as charge. These
vortices are usually studied using lattice
simulations, and it has been found that the
behavior of the vortices is closely linked
with the confinement–deconfinement phase
transition: in the confinement phase vortices
percolate and fill the spacetime volume, in
the deconfinement phase they are much suppressed.
Also it has been shown that the string tension
vanished upon removal of center vortices from
the simulations, hinting at an important role
for center vortices.
== See also ==
Vacuum state and vacuum
QED vacuum of quantum electrodynamics
flavour (particle physics)
Top quark condensate
Goldstone boson
Higgs mechanism
