A chiral phenomenon is one that is not identical
to its mirror image (see the article on mathematical
chirality). The spin of a particle may be
used to define a handedness, or helicity,
for that particle, which, in the case of a
massless particle, is the same as chirality.
A symmetry transformation between the two
is called parity transformation. Invariance
under parity transformation by a Dirac fermion
is called chiral symmetry.
An experiment on the weak decay of cobalt-60
nuclei carried out by Chien-Shiung Wu and
collaborators in 1957 demonstrated that parity
is not a symmetry of the universe.
== Chirality and helicity ==
The helicity of a particle is right-handed
if the direction of its spin is the same as
the direction of its motion. It is left-handed
if the directions of spin and motion are opposite.
So a standard clock, with its spin vector
defined by the rotation of its hands, tossed
with its face directed forwards, has left-handed
helicity.
Mathematically, helicity is the sign of the
projection of the spin vector onto the momentum
vector: “left” is negative, “right”
is positive.
The chirality of a particle is more abstract:
It is determined by whether the particle transforms
in a right- or left-handed representation
of the Poincaré group.For massless particles
– photons, gluons, and (hypothetical) gravitons
– chirality is the same as helicity; a given
massless particle appears to spin in the same
direction along its axis of motion regardless
of point of view of the observer.
For massive particles – such as electrons,
quarks, and neutrinos – chirality and helicity
must be distinguished: In the case of these
particles, it is possible for an observer
to change to a reference frame moving faster
than the spinning particle, in which case
the particle will then appear to move backwards,
and its helicity (which may be thought of
as “apparent chirality”) will be reversed.
A massless particle moves with the speed of
light, so no real observer (who must always
travel at less than the speed of light) can
be in any reference frame where the particle
appears to reverse its relative direction
of spin, meaning that all real observers see
the same helicity. Because of this, the direction
of spin of massless particles is not affected
by a change of viewpoint (Lorentz boost) in
the direction of motion of the particle, and
the sign of the projection (helicity) is fixed
for all reference frames: The helicity of
massless particles is a “relativistic invariant”
(a quantity whose value is the same in all
inertial reference frames) which always matches
the massless particles' chirality.
The discovery of neutrino oscillation implies
that neutrinos have mass, so the only observed
massless particle is the photon. The gluon
is also expected to be massless, although
the assumption that it is has not been conclusively
tested. Hence, these are the only two particles
now known for which helicity could be identical
to chirality, and only one of them has been
confirmed by measurement. All other observed
particles have mass and thus may have different
helicities in different reference frames.
It is still possible that as-yet unobserved
particles, like the graviton, might be massless,
and hence have invariant helicity that matches
their chirality, like the photon.
== Chiral theories ==
Only left-handed fermions and right-handed
antifermions interact with the weak interaction.
In most circumstances, two left-handed fermions
interact more strongly than right-handed or
opposite-handed fermions, implying that the
universe has a preference for left-handed
chirality, which violates a symmetry of the
other forces of nature.
Chirality for a Dirac fermion ψ is defined
through the operator γ5, which has eigenvalues
±1.
Any Dirac field can thus be projected into
its left- or right-handed component by acting
with the projection operators ½(1−γ5)
or ½(1+γ5) on ψ.
The coupling of the charged weak interaction
to fermions is proportional to the first projection
operator, which is responsible for this interaction's
parity symmetry violation.
A common source of confusion is due to conflating
this operator with the helicity operator.
Since the helicity of massive particles is
frame-dependent, it might seem that the same
particle would interact with the weak force
according to one frame of reference, but not
another. The resolution to this false paradox
is that the chirality operator is equivalent
to helicity for massless fields only, for
which helicity is not frame-dependent. By
contrast, for massive particles, chirality
is not the same as helicity, so there is no
frame dependence of the weak interaction:
a particle that couples the weak force in
one frame, does so in every frame.
A theory that is asymmetric with respect to
chiralities is called a chiral theory, while
a non-chiral (i.e., parity-symmetric) theory
is sometimes called a vector theory. Many
pieces of the Standard Model of physics are
non-chiral, which is traceable to anomaly
cancellation in chiral theories. Quantum chromodynamics
is an example of a vector theory, since both
chiralities of all quarks appear in the theory,
and couple to gluons in the same way.
The electroweak theory, developed in the mid
20th century, is an example of a chiral theory.
Originally, it assumed that neutrinos were
massless, and only assumed the existence of
left-handed neutrinos (along with their complementary
right-handed antineutrinos). After the observation
of neutrino oscillations, which imply that
neutrinos are massive like all other fermions,
the revised theories of the electroweak interaction
now include both right- and left-handed neutrinos.
However, it is still a chiral theory, as it
does not respect parity symmetry.
The exact nature of the neutrino is still
unsettled and so the electroweak theories
that have been proposed are somewhat different,
but most accommodate the chirality of neutrinos
in the same way as was already done for all
other fermions.
== Chiral symmetry ==
Vector gauge theories with massless Dirac
fermion fields ψ exhibit chiral symmetry,
i.e., rotating the left-handed and the right-handed
components independently makes no difference
to the theory. We can write this as the action
of rotation on the fields:
ψ
L
→
e
i
θ
L
ψ
L
{\displaystyle \psi _{L}\rightarrow e^{i\theta
_{L}}\psi _{L}}
and
ψ
R
→
ψ
R
{\displaystyle \psi _{R}\rightarrow \psi _{R}}
or
ψ
L
→
ψ
L
{\displaystyle \psi _{L}\rightarrow \psi _{L}}
and
ψ
R
→
e
i
θ
R
ψ
R
.
{\displaystyle \psi _{R}\rightarrow e^{i\theta
_{R}}\psi _{R}.}
With N flavors, we have unitary rotations
instead: U(N)L×U(N)R.
More generally, we write the right-handed
and left-handed states as a projection operator
acting on a spinor. The right-handed and left-handed
projection operators are
P
R
=
1
+
γ
5
2
{\displaystyle P_{R}={\frac {1+\gamma ^{5}}{2}}}
and
P
L
=
1
−
γ
5
2
{\displaystyle P_{L}={\frac {1-\gamma ^{5}}{2}}}
Massive fermions do not exhibit chiral symmetry,
as the mass term in the Lagrangian, m—ψψ,
breaks chiral symmetry explicitly.
Spontaneous chiral symmetry breaking may also
occur in some theories, as it most notably
does in quantum chromodynamics.
The chiral symmetry transformation can be
divided into a component that treats the left-handed
and the right-handed parts equally, known
as vector symmetry, and a component that actually
treats them differently, known as axial symmetry.
(cf. Current algebra.) A scalar field model
encoding chiral symmetry and its breaking
is the chiral model.
The most common application is expressed as
equal treatment of clockwise and counter-clockwise
rotations from a fixed frame of reference.
The general principle is often referred to
by the name chiral symmetry. The rule is absolutely
valid in the classical mechanics of Newton
and Einstein, but results from quantum mechanical
experiments show a difference in the behavior
of left-chiral versus right-chiral subatomic
particles.
=== Example: u and d quarks in QCD ===
Consider quantum chromodynamics (QCD) with
two massless quarks u and d (massive fermions
do not exhibit chiral symmetry). The Lagrangian
reads
L
=
u
¯
i
⧸
D
u
+
d
¯
i
⧸
D
d
+
L
g
l
u
o
n
s
.
{\displaystyle {\mathcal {L}}={\overline {u}}\,i\displaystyle
{\not }D\,u+{\overline {d}}\,i\displaystyle
{\not }D\,d+{\mathcal {L}}_{\mathrm {gluons}
}~.}
In terms of left-handed and right-handed spinors,
it reads
L
=
u
¯
L
i
⧸
D
u
L
+
u
¯
R
i
⧸
D
u
R
+
d
¯
L
i
⧸
D
d
L
+
d
¯
R
i
⧸
D
d
R
+
L
g
l
u
o
n
s
.
{\displaystyle {\mathcal {L}}={\overline {u}}_{L}\,i\displaystyle
{\not }D\,u_{L}+{\overline {u}}_{R}\,i\displaystyle
{\not }D\,u_{R}+{\overline {d}}_{L}\,i\displaystyle
{\not }D\,d_{L}+{\overline {d}}_{R}\,i\displaystyle
{\not }D\,d_{R}+{\mathcal {L}}_{\mathrm {gluons}
}~.}
(Here, i is the imaginary unit and
⧸
D
{\displaystyle \displaystyle {\not }D}
the Dirac operator.)
Defining
q
=
[
u
d
]
,
{\displaystyle q={\begin{bmatrix}u\\d\end{bmatrix}},}
it can be written as
L
=
q
¯
L
i
⧸
D
q
L
+
q
¯
R
i
⧸
D
q
R
+
L
g
l
u
o
n
s
.
{\displaystyle {\mathcal {L}}={\overline {q}}_{L}\,i\displaystyle
{\not }D\,q_{L}+{\overline {q}}_{R}\,i\displaystyle
{\not }D\,q_{R}+{\mathcal {L}}_{\mathrm {gluons}
}~.}
The Lagrangian is unchanged under a rotation
of qL by any 2×2 unitary matrix L, and qR
by any 2×2 unitary matrix R.
This symmetry of the Lagrangian is called
flavor chiral symmetry, and denoted as U(2)L×U(2)R.
It decomposes into
S
U
(
2
)
L
×
S
U
(
2
)
R
×
U
(
1
)
V
×
U
(
1
)
A
.
{\displaystyle SU(2)_{L}\times SU(2)_{R}\times
U(1)_{V}\times U(1)_{A}~.}
The singlet vector symmetry, U(1)V, acts as
q
L
→
e
i
θ
q
L
q
R
→
e
i
θ
q
R
,
{\displaystyle q_{L}\rightarrow e^{i\theta
}q_{L}\qquad q_{R}\rightarrow e^{i\theta }q_{R}~,}
and corresponds to baryon number conservation.
The singlet axial group U(1)A acts as
q
L
→
e
i
θ
q
L
q
R
→
e
−
i
θ
q
R
,
{\displaystyle q_{L}\rightarrow e^{i\theta
}q_{L}\qquad q_{R}\rightarrow e^{-i\theta
}q_{R}~,}
and it does not correspond to a conserved
quantity, because it is explicitly violated
by a quantum anomaly.
The remaining chiral symmetry SU(2)L×SU(2)R
turns out to be spontaneously broken by a
quark condensate
⟨
q
¯
R
a
q
L
b
⟩
=
v
δ
a
b
{\displaystyle \textstyle \langle {\bar {q}}_{R}^{a}q_{L}^{b}\rangle
=v\delta ^{ab}}
formed through nonperturbative action of QCD
gluons,
into the diagonal vector subgroup SU(2)V known
as isospin. The Goldstone bosons corresponding
to the three broken generators are the three
pions.
As a consequence, the effective theory of
QCD bound states like the baryons, must now
include mass terms for them, ostensibly disallowed
by unbroken chiral symmetry. Thus, this chiral
symmetry breaking induces the bulk of hadron
masses, such as those for the nucleons — in
effect, the bulk of the mass of all visible
matter.
In the real world, because of the nonvanishing
and differing masses of the quarks, SU(2)L×SU(2)R
is only an approximate symmetry to begin with,
and therefore the pions are not massless,
but have small masses: they are pseudo-Goldstone
bosons.
=== More Flavors ===
For more "light" quark species, N flavors
in general, the corresponding chiral symmetries
are U(N)L×U(N)R, decomposing into
S
U
(
N
)
L
×
S
U
(
N
)
R
×
U
(
1
)
V
×
U
(
1
)
A
,
{\displaystyle SU(N)_{L}\times SU(N)_{R}\times
U(1)_{V}\times U(1)_{A}~,}
and exhibiting a very analogous chiral symmetry
breaking pattern.
Most usually, N = 3 is taken, the u, d, and
s quarks taken to be light (the Eightfold
way (physics)), so then approximately massless
for the symmetry to be meaningful to a lowest
order, while the other three quarks are sufficiently
heavy to barely have a residual chiral symmetry
be visible for practical purposes.
=== An application in Particle Physics ===
In theoretical physics, the electroweak model
breaks parity maximally. All its fermions
are chiral Weyl fermions, which means that
the charged weak gauge bosons only couple
to left-handed quarks and leptons. (Note that
the neutral electroweak Z boson couples to
left and right-handed fermions.)
Some theorists found this objectionable, and
so conjectured a GUT extension of the weak
force which has new, high energy W' and Z'
bosons, which now couple with right handed
quarks and leptons:
[
S
U
(
2
)
W
×
U
(
1
)
Y
]
Z
2
{\displaystyle {[SU(2)_{W}\times U(1)_{Y}]
\over \mathbb {Z} _{2}}}
to
S
U
(
2
)
L
×
S
U
(
2
)
R
×
U
(
1
)
B
−
L
Z
2
.
{\displaystyle {SU(2)_{L}\times SU(2)_{R}\times
U(1)_{B-L} \over \mathbb {Z} _{2}}.}
Here, SU(2)L (pronounced SU(2) left) is none
other than the above SU(2)W, while B−L is
the baryon number minus the lepton number.
The electric charge formula in this model
is given by
Q
=
I
3
L
+
I
3
R
+
B
−
L
2
{\displaystyle Q=I_{3L}+I_{3R}+{\frac {B-L}{2}}}
;where
I
3
L
,
R
{\displaystyle \!I_{3L,R}}
are the weak isospin values of the fields
in the theory.
There is also the chromodynamic SU(3)C. The
idea was to restore parity by introducing
a left-right symmetry. This is a group extension
of Z2 (the left-right symmetry) by
S
U
(
3
)
C
×
S
U
(
2
)
L
×
S
U
(
2
)
R
×
U
(
1
)
B
−
L
Z
6
{\displaystyle {SU(3)_{C}\times SU(2)_{L}\times
SU(2)_{R}\times U(1)_{B-L} \over \mathbb {Z}
_{6}}}
to the semidirect product
S
U
(
3
)
C
×
S
U
(
2
)
L
×
S
U
(
2
)
R
×
U
(
1
)
B
−
L
Z
6
⋊
Z
2
.
{\displaystyle {SU(3)_{C}\times SU(2)_{L}\times
SU(2)_{R}\times U(1)_{B-L} \over \mathbb {Z}
_{6}}\rtimes \mathbb {Z} _{2}.}
This has two connected components where Z2
acts as an automorphism, which is the composition
of an involutive outer automorphism of SU(3)C
with the interchange of the left and right
copies of SU(2) with the reversal of U(1)B−L.
It was shown by Rabindra N. Mohapatra and
Goran Senjanovic in 1975 that left-right symmetry
can be spontaneously broken to give a chiral
low energy theory, which is the Standard Model
of Glashow, Weinberg and Salam and it also
connects the small observed neutrino masses
to the breaking of left-right symmetry via
the seesaw mechanism.
In this setting, the chiral quarks
(
3
,
2
,
1
)
1
3
{\displaystyle (3,2,1)_{1 \over 3}}
and
(
3
¯
,
1
,
2
)
−
1
3
{\displaystyle ({\bar {3}},1,2)_{-{1 \over
3}}}
are unified into an irreducible representation
(“irrep”)
(
3
,
2
,
1
)
1
3
⊕
(
3
¯
,
1
,
2
)
−
1
3
.
{\displaystyle (3,2,1)_{1 \over 3}\oplus ({\bar
{3}},1,2)_{-{1 \over 3}}.}
The leptons are also unified into an irreducible
representation
(
1
,
2
,
1
)
−
1
⊕
(
1
,
1
,
2
)
1
.
{\displaystyle (1,2,1)_{-1}\oplus (1,1,2)_{1}.}
The Higgs bosons needed to implement the breaking
of left-right symmetry down to the Standard
Model
are
(
1
,
3
,
1
)
2
⊕
(
1
,
1
,
3
)
2
.
{\displaystyle (1,3,1)_{2}\oplus (1,1,3)_{2}.}
This then predicts three sterile neutrinos,
which is perfectly consistent with current
neutrino oscillation data. Within the seesaw
mechanism, the sterile neutrinos become superheavy
without affecting physics at low energies.
Because the left-right symmetry is spontaneously
broken, left-right models predict domain walls.
This left-right symmetry idea first appeared
in the Pati–Salam model (1974), Mohapatra–Pati
models (1975).
== See also ==
Electroweak theory
Chirality (chemistry)
Chirality (mathematics)
Chiral symmetry breaking
Handedness
Spinors and Dirac fields
Sigma model
Chiral model
== Notes
