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Quantum Flavordynamics - SU(3)
------------------------------
SU(3) is a 3 x 3 matrix operating on ψ_{i}, ψ_{j} and ψ_{k} quark field operators.
This representation is called the '3' and is referred to as the FUNDAMENTAL
representation. Likewise, we can identify complex conjugate matrices
operating on the anti quarks field operators, ψ_{i}*, ψ_{j}* and ψ_{k}*. This
_{ } _
representation is called the '3' (≡ 3*) and is referred to as the CONJUGATE
_
representation of SU(3). Thus, ψ_{i,j,k} ≡ ψ_{i,j,k}*. The i, j and k notation can
represent u, d and s quarks or R, G and B colored quarks - the math is the
same.
Mesons
------
We can construct a state composed of 2 quarks by taking the tensor
product:
3 ⊗ 3
This has 9 states. If you work this out you get the curious result
that,
^{ } _
3 ⊗ 3 = 3 ⊕ 6
_
The 3 represents the antisymmetric states, the 6 represents the
symmetric states.
Note: The analogy for spin 1/2 particles (i.e. electrons) would be
2 ⊗ 2 = 1 ⊕ 3 where the 1 is antisymmetric and the 3 are the symmetric
states.
Using the same approach we we can also construct a state consisting
of a quark and an anti quark.
^{ } _
3 ⊗ 3 = 1 ⊕ 8
1 is an antisymmetric SINGLET that is completely invariant (transforms
into itself) under SU(3) rotation. The 8 respresents the symmetric
states that transform in the same way as the group generators. The
1 is referred to as the TRIVIAL representation and the 8 is referred
to as the ADJOINT representation.
Baryons
-------
Continuing from before, we can also construct a 3 quark state as
follows:
3 ⊗ 3 ⊗ 3
_
= (3 ⊕ 6) ⊗ 3
_
= (3 ⊗ 3) ⊕ (6 ⊗ 3)
= 1 ⊕ 8 ⊕ 8 ⊕ 10
Again, 1 is an antisymmetric singlet that is completely invariant under
SU(3) rotation. The 8s are 'mixed' symmetry states and the 10 represents
the symmetric states that transform in the same way as the group
generators.
The lower energy 8 is the BARYON OCTET containing the protons and
neutrons. The 10 corresponds to the BARYON DECUPLET. This can be
summarized in terms of the Gell-Mann's EIGHTFOLD WAY.
The Eightfold Way
------------------
The Eightfold Way is a theory organizing baryons and mesons into octets
based on combinations of u, d and s quarks. The overall wavefunctions
are constructed as follows:
|ψ> = |ψ_{Flavor}> ⊗ |ψ_{Spin}> ⊗ |ψ_{Space}>
The flavor, spin and space parts are symmetric under interchange of any
2 quarks.
The total momentum, J, is given by:
J = L + S
Physicists are generally only interested in studying the ground
states of particles where L = 0.
A single quark can have S_{z} = ± 1/2
2 quarks can have their spins aligned as S_{z} = -1, 0, 1
3 quarks can have their spins aligned as S_{z} = -3/2, -1/2, 1/2, 3/2
The Pseudoscalar Meson Octet (1 ⊕ 8)
-------------------------------------
_
Pseudoscalar mesons have the qq spins antialigned. Therefore,
J = 0.
_ _ _
|η^{o}> = (1/√6)(|uu> + |dd> - 2|ss>)
_ _ _
|η'> = (1/√3)(|uu> + |dd> + |ss>)
_ _
|π^{o}> = (1/√2)(|uu> - |dd>)
The Vector Meson Octet (1 ⊕ 8)
-------------------------------
_
Vector mesons have 0 the qq spins aligned. Therefore, J = 1.
_ _
|ρ^{o}> = (1/√2)(|uu> - |dd>)
_
|φ^{o}> = |ss>)
_ _
|ω^{o}> = (1/√2)(|uu> + |dd>)
The Baryon Octet (1 ⊕ 8 ⊕ 8 ⊕ 10)
----------------------------------
For the baryon octet we choose J = 1/2.
The principles of the Eightfold Way also applied to the isospin 3/2
baryons, forming a decuplet.
The Baryon Decuplet (1 ⊕ 8 ⊕ 8 ⊕ 10)
-------------------------------------
For the baryon decuplet we choose J = 3/2.
The charges (green) are computed using the Gell-Mann - Nishijimi
formula:
Q = I_{3} + (1/2)(B + S)
where B is the baryon number:
B = (1/3)(n_{q} - n_{q}) = (# of quarks - # of antiquarks)
and S is the strangeness quantum number.
The masses (blue) follow the GELL-MANN - OKUBO mass formula that
describes an approximate relation between meson and baryon masses.
Baryons:
(N + Ξ)/2 = (3Λ + Σ)/4
(N + Ξ)/2 ~ 1129 MeV
(3Λ + Σ)/4 ~ 1135 MeV
Mesons:
_
[((K^{-} + K^{0})/2)^{2} + ((K^{+} + K^{0})/2)^{2}]/2 = (3η^{2} + π^{2})/4
_
[((K^{-} + K^{0})/2)^{2} + ((K^{+} + K^{0})/2)^{2}]/2 ~ 246 x 10^{3} MeV
(3η^{2} + π^{2})/4 ~ 230 x 10^{3} MeV
SU(3) FLAVOR transformations move around an 8/10 dimensional vector space
which correspond to the 8/10 particles in the baryon octet/decuplet. All
particles in the octet would have equal mass if we were to only consider
SU(3). However, SU(3) flavor symmetry contains SU(2) isospin as a subgroup.
We can see this by looking at the Gell-Mann matrices which include the Pauli
spin matrices as indicated in orange.
- -
_{ } | 0 1 0 |
λ_{1} = | 1 0 0 |
_{ } | 0 0 0 |
- -
- -
_{ } | 0 -i 0 |
λ_{2} = | i 0 0 |
_{ } | 0 0 0 |
- -
- -
_{ } | 1 0 0 |
λ_{3} = | 0 -1 0 |
_{ } | 0 0 0 |
- -
- -
_{ } | 0 0 1 |
λ_{4} = | 0 0 0 |
_{ } | 1 0 0 |
- -
- -
_{ } | 0 0 -i |
λ_{5} = | 0 0 0 |
_{ } | i 0 0 |
- -
- -
_{ } | 0 0 0 |
λ_{6} = | 0 0 1 |
_{ } | 0 1 0 |
- -
- -
_{ } | 0 0 0 |
λ_{7} = | 0 0 -i |
_{ } | 0 i 0 |
- -
- -
_{ } | 1 0 0 |
λ_{8} = 1/√3 | 0 1 0 |
_{ } | 0 0 -2 |
- -
The SU(3) flavor symmetry is not exact because the quarks have different
rest masses and different electroweak interactions. Thus, interchanging
quarks results in changes to the overall mass*.
* The masses for the different particle species is not just the cumulative
mass of their quarks. The vast majority of the mass is due to the kinetic
energy of the quarks and to the energy of the gluon fields that bind the
quarks together.
Let look in detail at some examples of the isospin (flavor) multiplets that
make up the above octets and decuplets.
Isospin Multiplets
------------------
Recall the allowed spin/isospin states are as follows:
| | | + +3/2
| | + +1 |
| | | |
| + +1/2 | + +1/2
| | | |
+ 0 | + 0 |
| | | |
| + -1/2 | + -1/2
| | | |
| | | |
| | + -1 + -3/2
and,
n_{states} = (2l + 1) where l is the spin/isospin
+---+------+------+------+------+
| | Spin | I_{3} | Q | J |
+---+------+------+------+------+
| u | 1/2 | 1/2 | 2/3 | 1/2 |
+---+------+------+------+------+
| d | 1/2 | -1/2 | -1/3 | 1/2 |
+---+------+------+------+------+
| s | 1/2 | 0 | -1/3 | 1/2 |
+---+------+------+------+------+
| _ | | | | |
| u | 1/2 | -1/2 | -2/3 | 1/2 |
+---+------+------+------+------+
| _ | | | | |
| d | 1/2 | 1/2 | 1/3 | 1/2 |
+---+------+------+------+------+
| _ | | | | |
| s | 1/2 | 0 | 1/3 | 1/2 |
+---+------+------+------+------+
The idea of isospin multiplets is best illustrated using the following
example. Consider isospin 1. There are 3 states +1, -1 and 0. These
states can be constructed in the following way:
1. |uu> with I_{3} = 1
2. |dd> with I_{3} = -1
3. |ud + du> with I_{3} = 0 and I_{1}, I_{2} ≠ 0
4. |ud - du> with I_{3} = I_{1} = I_{2} = 0
1, 2 and 3 form a multiplet. 4 is a singlet state. In the multiplet,
the states transform into each other under a SU(2) transformation.
In the singlet case, the state just transforms into itself.
Physically isospin multiplets are families of hadrons with approximately
equal masses. All particles within a multiplet, have the same spin angular
momentum, J, but differ in their electric charges. An isospin projection
(denoted by I_{3}) is associated with each charged state.
Δ Particles
-----------
Δ particles consist of 3 quarks therefore there are 8 possible
combinations. They form an isospin, I, 3/2 multiplet. Their spins
are aligned to create J = ±3/2. According to the rule, n = (2l + 1),
they are allowed to have 4 states. These are:
Δ^{++} has I_{3} = 3/2, J = 3/2 and Q = 2
Δ^{+} has I_{3} = 1/2, J = 3/2 and Q = 1
Δ^{0} has I_{3} = -1/2, J = -3/2 and Q = 0
Δ^{-} has I_{3} = -3/2, J = -3/2 and Q = -1
The Δs have similar mass to each other but they are much heavier
than either the proton or the neutron. Δs decay. Consider Δ^{++}.
It decays in the following manner:
Pions are very short lived ~ 10^{-23}s
Historically, the discovery of the Δ_{3/2} was very important because
it led to the question "how do 3 fermions with spin 1/2 occupy the
same state without violating the Pauli Exclusion principle?". This
also applies to protons and neutrons which are comprised of 2
identical quarks. The answer lies in the emergence of 'COLOR CHARGE'
and QCD. The color charge is assigned as follows:
+---------+----+---+----+----+----+---+
| | L_{ } | R || L_{ } | R || L _{ }| R |
+---------+----+---+----+----+----+---+
| Red | u_{ } | d || c_{ } | s || t _{ }| b |
+---------+----+---+----+----+----+---+
| Blue | u_{ } | d || c_{ } | s || t _{ }| b |
+---------+----+---+----+----+----+---+
| Green | u_{ } | d || c_{ } | s || t_{ } | b |
+---------+----+---+----+----+----+---+
+---------+----+---+----+----+----+---+
| Leptons | ν_{e} | e || ν_{μ} | μ | ν_{τ} || τ_{ }|
+---------+----+---+----+----+----+---+
Weak interaction symmetries (SU(2)) act horizontally and never
vertically. All particles in a row are transformed at the same
time.
Transitions from R column to L column emits a W^{-} gauge boson.
Transitions from L column to R column emits a W^{+} gauge boson.
Strong interaction symmetries (SU(3)) act vertically and never
horizontally. All particles in a column are transformed at the
same time.
This can be summarized as follows:
The fact that SU(3) and SU(2) and U(1) don't mix is expressed as:
SU(3)⊗SU(2)⊗U(1)
Note: Leptons are part of SU(2) and do not participate in the color
symmetry.
The net color in hadrons must vanish. Since color is an unobserved
property, all hadrons must be colorless objects. Therefore, protons
and neutrons must consist of red, blue and green quarks in
superpositions that result in no color.
Protons and Neutrons
--------------------
Protons and neutrons consist of 3 quarks therefore there are 8
possible combinations. They form an isospin 1/2 multiplet. Their
spins are aligned to create J = ±1/2. According to the rule,
n_{states} = (2l + 1), they are allowed to have 2 states. These are:
p has I_{3} = 1/2, J = 1/2 and Q = 1
n has I_{3} = -1/2, J = -1/2 and Q = 0
NOTE: Technically speaking, we should use the full description to
describe the various combinations of quarks that make up the
various particles. For brevity, however, it is conventional to use
a simpler shorthand notation. Under this notation a protons would
be written as 'uud' and a neutron as 'ddu' with the above
constructions implied.
Pions
-----
Pions are combination of quark anti-quark pairs. They form an
isospin 1 multiplet. Their spins are aligned to create J = 0.
According to the rule, n_{states} = (2l + 1), they are allowed to have
3 states. These are:
π^{+} has I_{3} = 1, J = 0 and Q = 1
π^{0} has I_{3} = 0, J = 0 and Q = 0
π^{-} has I_{3} = -1, J = 0 and Q = -1
Quantum Chromodynamics - SU(3)
------------------------------
Historically, the idea of color as a quantum number came about to
overcome the violation of the Pauli Exclusion principle that was apparent
with combining identical quarks into particles like the proton and neutron.
As mentioned previously, color states also be described by the 3 and 3
representations (i, j, k ≡ R, G, B). In this case, the singlets are
referred to as COLOR SINGLETS.
We can construct the overall wavefunction for a particle as follows:
|ψ> = |ψ_{Flavor}> ⊗ |ψ_{Spin}> ⊗ |ψ_{Space}> ⊗ |ψ_{Color}>
All observable particles in nature are color singlets. Furthermore, the
overall wavefunction must be antisymmetric under the exchange of any
2 quarks. The flavor, space and spin parts are symmetric under interchange
of any pair of labels. This implies that the color part must be asymmetric
and colorless.
In the case of mesons we can construct a colorless antisymmetric state as
follows:
_{ } _ _ _
ψ_{Color}^{M} = (1/√3){RR + GG + BB}
In the case of baryons we can construct a colorless antisymmetric state via
a Slater determinant as follows.
_{ } | R_{1} G_{1} B_{1} |
ψ_{Color}^{B} = (1/√6)| R_{2} G_{2} B_{2} |
_{ } | R_{3} G_{3} B_{3} |
_{ } = (1/√6){R_{1}G_{2}B_{3} - R_{1}B_{2}G_{3} + G_{1}B_{3}R_{2} - G_{1}R_{3}B_{2} + B_{1}R_{2}G_{3} - B_{1}G_{2}R_{3}}
The antisymmetric colour combination is the only combination of 3 quark
states that has no net color.
Gluons
------
Force between quarks: The exchange of a gluon between 2 quarks.
This represents the force between quarks.
Force between gluons: Exchange of gluon between 2 gluons. This
represent a forces between gluons. THUS, GLUONS ARE DIFFERENT
FROM PHOTONS IN THAT THEY CAN INTERACT WITH EACH OTHER.
Gluons BEHAVE like quark anti-quark pairs with respect to their
color symmetry. They are NOT mesons but we can use the same
representations. Thus we can write:
^{ } _
3 ⊗ 3 = 1 ⊕ 8
There are many ways of presenting the 8. One commonly used list is:
_ _ _ _ _ _
(1/√2)(RB + BR) (i/√2)(RB - BR) (1/√2)(RR - GG)
_ _ _ _
(1/√2)(BG + GB) (i/√2)(BG - GB)
_ _ _ _ _ _ _
(1/√2)(GR + RG) (i/√2)(GR - RG) (1/√6)(RR + GG - 2BB)
_ _ _
(1/√3)(RR + GG + BB) which is colorless.
These are the combined states that are constructed according to
the principle of superposition. If one were somehow able to make
a direct measurement of the color of a gluon in one of these
states, there would be a 50% chance of it having a particular
color charge.
We can construct LADDER OPERATORS that map these eigenstates into
the gluon octet. Ladder operators were introduced in the context
of angular momentum and rotation so it is also appropriate to use
them in the context of rotations associated with SU(3).
T± = T ± iT'
Thus, for example:
_ _ _ _ _
(1/√2)(RB + BR) + i(i/√2)(RB - BR) => BR
and,
_ _ _ _ _
(1/√2)(RB + BR) - i(i/√2)(RB - BR) => BR
Not surprisingly, the gluon colour wavefunctions have a similar form
to the mesons.
The singlet does not appear in nature and we can disregard it for now.
We are left with thr 8 gluons that constitute the strong force.
We can combine 2 gluons as follows:
8 ⊗ 8 = 1 ⊕ 63
The color singlet state of the 2 gluon combinations corresponds
to a GLUEBALL.
The Nature of the Strong Interaction
------------------------------------
Gluons carry a color charge. The color force resulting from this charge
does not follow an inverse square force like the electromagnetic force.
In fact it behaves in the opposite way and increases linearly with distance.
This behavior is responsible for QUARK CONFINEMENT. As a quark-antiquark
pair separates, the chromoelectric flux lines associated with the gluon field
between them attract each other to form flux tubes rather like stretched
chewing gum. As the distance increases, more and more energy is required
to pull them apart. At some point, it becomes energetically favorable for the
tube to 'snap' rather than extend further, and a new quarkâ€“antiquark pair
spontaneously appears. In so doing, energy is conserved because the energy
of the color-force field is converted into the mass of the new quarks, and the
color-force field can 'relax' back to an unstretched state. In reality, in high
energy particle accelerators, the tube actually snaps in multiple places
forming a spray of hadrionic particles referred to as a JET. This process of
jet formation is referred to a HADRONIZATION. Quark confinement is the
reason individual quarks are never observed in nature.
The color force appears to exert little force at short distances so that the
quarks are like free particles within the confining boundary of the color
force and only experience the strong confining force when they begin to get
too far apart. The behavior is referred to as term ASYMPTOTIC FREEDOM.
Gauge symmetry prohibits gluons to have mass and as far as we know the
SU(3) gauge symmetry is not broken (see the Higgs mechanism). Massless
particles are mediators for long range forces such as electromagnetism.
However, the range of the strong interaction is limited by confinement.
Like photons, gluons are spin 1 particles with 0 conventional charge.
Nuclear Force
--------------
The nuclear force is the force that hold the nucleons together. In 1935
(long before QCD) Hideki Yukawa modelled the nuclear force in terms
of the YUKAWA POTENTIAL given by:
V(r) = -g^{2}exp(-kmr)/r
where g and k are constants, m is the mass and r is the radial distance.
The nuclear force is maximally attractive between nucleons at distances
of about 1 fermi (10^{-15} metres) between their centers, but exponentially
decreases at distances beyond this. At about 2.5 fermi the force becomes
negligible. At distances less than 0.7 fermi, the nuclear force becomes
repulsive. The repulsive component is responsible for the size of the
nucleus. The nuclear force is stronger than the Coulomb force between
protons at distance of less than 1.7 fermi. Nucleons have a radius of
about 0.8 fermi.
Based on the range of the force, Yukawa calculated that the virtual
particles associated with the nuclear force should have a mass in the
neighborhood of 100 MeV. This led to the eventual discovery of the
pion.
With the development of QCD, the nuclear force is considered to be a
residual effect of the more fundamental strong force. As previously
stated quarks and gluons are confined within nucleons. However, some
combinations of quarks and gluons can 'leak' away from nucleons, in
the form of pions. As a result, the nuclear force is sometimes called
the residual strong force.
Pions are different from gluons in that gluons are massless and are
confined within the nucleons. In contrast, pions have a mass of about
140 MeV and are not confined.
The 'mechanism' of the nuclear force is illustrated in the following
graphic (courtesy of Wikipedia)
The Origin of Mass in QCD
-------------------------
When we talk about combinations of quarks in the hadrons we are
referring to VALENCE quarks. In reality the picture is much more
complicated. Virtual quark-antiquark pairs known as SEA quarks form
when gluons split. This process also occurs in reverse so that the
annihilation of 2 sea quarks produces a gluon. The result is a constant
flux of gluon splits and creations inside the hadron. This flux is
referred to as the quark/gluon condensate and characterizes the QCD
VACUUM state. The VACUUM EXPECTATION LEVEL of the QCD vacuum is given
by:
_
VEV = <ψ_{i}ψ_{i}> where ψ_{i} is the quark field operator summed over i (flavor).
_ _ _
= <uu + dd + ss>
It is possible to imagine a valence quark as a CURRENT quark that is
'dressed' by their interactions with this condensate. Valence quarks
are often referred to as CONSTITUENT quarks with the following
approximate masses:
Constituent Mass
----------------------
Up quark 336 MeV
Down quark 340 MeV
Strange quark 486 MeV
Charm quark 1550 MeV
Bottom quark 4730 MeV
Top quark 177000 MeV
The bulk of the mass of the hadrons comes from these high energy
interactions with the QCD vacuum. The current quarks themselves
are very light and get their mass via the HIGGS MECHANISM.
The process of aquiring mass is related to CHIRAL SYMMETRY breaking.
Chiral Symmetry Breaking
------------------------
We have said before that the strong force behaves differently to
other forces in that it becomes weaker as the distance decreases.
This is equivalent to saying that the coupling to the strong field,
α_{S}, gets smaller as the distance becomes smaller. This is referred
to as RUNNING coupling. The running coupling introduces a dimensional
parameter, Λ_{QCD}, which sets the scale at which the coupling constant
becomes large and the physics becomes highly non linear and NON-
PERTURBATIVE requiring extremely intense numerical computations such
as LATTICE QCD. Lattice QCD is similar to lattice QED in that fields
representing quarks are defined by lattice cells, while the gluon
fields are defined by the propagators connecting neighboring cells.
This approximation approaches continuum QCD as the spacing between
lattice cells is reduced to zero. Below Λ_{QCD} the physics becomes
PERTURBATIVE and perturbation methods can be used to solve complex
problems. In fact, Λ_{QCD} simply sets the scale for strong interaction
physics. Experimentally it is determined that Λ_{QCD} ~ 200 MeV. With
this it is possible to introduce the notion of light and heavy quarks.
Briefly speaking, light quarks are ones having masses much smaller
than Λ_{QCD}, and heavy quarks having masses much larger than Λ_{QCD}.
Therefore, under this regime u, d and s quarks qualify as light
quarks whereas the c, b and t quarks may be regarded as heavy.
To understand the physics of the u and d quarks, it is convenient
to consider a theoretical limit in which their masses are exactly
zero. The quarks can be right-handed or left-handed. The chirality
can be selected using 5th Dirac matrix:
γ^{5} = γ^{0}γ^{1}γ^{2}γ^{3}
ψ_{Lf} = (1/2)(1 - γ^{5})ψ_{f}
ψ_{Rf} = (1/2)(1 + γ^{5})ψ_{f}
where f labels the flavor (u or d). The total quark field is simply
a linear combination of the two,
ψ = ψ_{L} + ψ_{R}
Without mass there is no coupling between the L and R quarks and they
live in 2 separate worlds (see the Dirac equation). Because the u
and d quarks are both massless, they may be regarded as two independent
states of the same object forming a two component spinor in 'isospin
space' in the same way that the proton and neutron can be regarded
as 2 states of the same particle if charge is ignored.
- - - -
| u'_{L,R} | = U_{L,R}| u_{L,R} |
| d'_{L,R} | _{ } | d_{L,R} |
- - - -
U_{L,R} are 2 x 2 unitary matrices:
U(2)_{L}⊗U(2)_{R}
The quark part of the QCD Lagrangian is:
_
L = ψ(iγ^{μ}D_{μ} - m)ψ
when m = 0 this reduces to:
_
L = ψiγ^{μ}D_{μ}ψ
It is easy to see that the Lagrangian is invariant under under these
transformations.
U(2)_{L}⊗U(2)_{R} can be decomposed into:
SU(2)_{L}⊗SU(2)_{R}⊗U(1)_{V}⊗U(1)_{A}
Where V and A represent the VECTOR and AXIAL symmetries. Vector means
that L and R transform in the same way. Axial means L and R transform
in the opposite way. Therefore,
U(1)_{V} acts as:
ψ_{L} -> exp(iθ)ψ_{L} and ψ_{R} -> exp(iθ)ψ_{R}
and U(1)_{A} acts as:
ψ_{L} -> exp(iθ)ψ_{L} and ψ_{R} -> exp(-iθ)ψ_{R}
Applying Noether's theorem to the Lagrangian shows that the conserved
quantity for U(1) is electric charge. Charge is related to the baryon
number through the Gell-Mann-Nishijima formula (Q = I_{3} + (1/2)(B + S))
and therefore the baryon number can be interpreted as the conserved
quantity. It turns out that this is true for U(1)_{V}. However, U(1)_{A}
does not correspond to a conserved quantity because the symmetry is
explicitly violated due to a quantum anomaly. This will not be
explained here since it is not crucial to the ensuing discussion.
If we just focus on the two SU(2) symmetries we can write:
SU(2)_{L}⊗SU(2)_{R}
This corresponds to 6 generators: 3 for left, T_{L}^{a} and 3 for right, T_{R}^{a}.
These generators can be combined as follows:
T_{L}^{a} + T_{R}^{a} corresponding to SU(2)_{V}
and
T_{L}^{a} - T_{R}^{a} corresponding to SU(2)_{A}
In addition there are 6 conserved Noether currents that can be derived
from the Lagrangian:
^{ } _
J^{μ}_{L,R,a} = ψ_{L,R}γ^{μ}(T_{a}/2)ψ_{L,R} with ∂_{μ}J^{μ}_{R,L} = 0
These can be also be combined as follows to form the vector and axial
currents:
^{ } _
V^{μ}_{a} = J^{μ}_{L,a} + J^{μ}_{R,a} = ψγ^{μ}(T_{a}/2)ψ
and
^{ } _
A^{μ}_{a} = J^{μ}_{L,a} - J^{μ}_{R,a} = ψγ^{5}γ^{μ}(T_{a}/2)ψ
With corresponding charges:
Q^{V}_{a} = ∫d^{3}xψ^{†}(T_{a}/2)ψ
and
Q^{A}_{a} = ∫d^{3}xψ^{†}γ^{5}(T_{a}/2)ψ
that are, likewise, generators of SU(2)_{L}⊗SU(2)_{R}
In QCD, it is the vacuum that SPONTANEOUSLY breaks the chiral symmetry.
In particular, it is the axial generators that get broken. Thus, we
can write:
SU(2)_{L}⊗SU(2)_{R}⊗U(1)_{V}⊗U(1)_{A} -> SU(2)_{V}⊗U(1)_{V}
Where SU(2)_{V} is the diagonal vector subgroup describing the isospin.
Isospin is conserved in the strong interaction but not in the weak
interaction because the strong interaction is agnostic to charge
differences. Also, the implication from the Gell-Mann - Nishijimi
equation is that if charge is conserved and baryon number is
conserved, then I_{3} must also be conserved.
Spontaneous chiral symmetry breaking by itself results in 3 massless
NAMBU GOLDSTONE BOSONS (NGBs) - one for each generator of the SU(2)_{A}
symmetry that is broken. SU(2)_{V} and U(1)_{V} are unbroken.
Likewise, we can write the following for SU(3) flavor symmetry involving
u, d and s quarks if we assume that the s quark is very similar to the
u and d quarks. The s quark is heavier than the u and d quarks but
if we set the scale correctly (see renormalization) we can treat them
in the same way.
U(3)_{L}⊗U(3)_{R} decomposes to:
SU(3)_{L}⊗SU(3)_{R}⊗U(1)_{V}⊗U(1)_{A}
Again, if we just focus on the two SU(3) symmetries we can write:
SU(3)_{L}⊗SU(3)_{R}
In this case there are 16 generators. Again, the 8 axial generators
get spontaneosly broken by the QCD vacuum to produce 8 NGBs. As before,
SU(3)_{V} and U(1)_{V} are unbroken. Thus,
SU(3)_{L}⊗SU(3)_{R}⊗U(1)_{V}⊗U(1)_{A} -> SU(3)_{V}⊗U(1)_{V}
If the symmetry is EXPLICITLY broken as well as spontaneously broken,
the NGBs are not massless and are termed PSEUDO NAMBU GOLDSTONE BOSONS
(PNGBs). It is the quark masses that explicitly break the symmetry.
The observed pion masses are obtained using CHIRAL PERTURBATION THEORY.
This leads to the GELL-MANN - OAKES - RENNER relationship:
^{ } _ _
m_{π}^{2} = -(1/2f_{π}^{2})(m_{u} + m_{d})<uu + dd> + O(m_{u,d}^{2})
\ \
(explicit) (spontaneous)
Where f_{π} is the experimentally determined PION DECAY CONSTANT (~ 93 MeV).
Similarly, for the remaining members of the pseudoscalar octet:
m_{K}^{2} ∝ (m_{s} + m_{u,d})
m_{η}^{2} ∝(m_{u} + m_{d} + 4m_{s})
Vector Mesons
--------------
The quark color charges lead to a spin-spin interaction between quarks
and antiquarks called the CHROMOMAGNETIC INTERACTION. The chromomagnetic
energy depends on the relative spin orientations of the quark and the
antiquarks. This is analagous to the electromagnetic interaction that
results in the spliiting of energy levels. The equations are very similar.
However, the spin-spin interaction is much much stronger than the
electromagnetic case due to the much larger coupling constant for the
strong force.