Download PX430: Gauge Theories for Particle Physics

December 2016
PX430: Gauge Theories for Particle Physics
Tim Gershon
([email protected])
Handout 5: Non-Abelian Gauge Symmetries
Groups and Symmetries
So far, we have only discussed one gauge symmetry – that of the QED Lagrangian under local
U(1) transformations. This means that
L̂QED = L̂D + L̂INT + L̂em ,
(1)
L̂D = ψ̄ˆ(iγ µ ∂µ − m)ψ̂ ,
L̂
= −q ψ̄ˆγ µ ψ̂ Â ,
(2)
where
µ
INT
1
L̂em = − F̂µν F̂ µν ,
4
(3)
(4)
is invariant under transformations of the form
ψ̂ 7→ ψ̂ 0 = e−ieχ̂(x) ψ̂ .
(5)
Refer to earlier handouts for more details and definitions of the symbols. The symmetry group U(1) is
the group of all 1 × 1 unitary matrices. [We will not go into the mathematical details of group theory,
since although it is relevant, it is not essential for this module. Interested students are encouraged
to read into this topic – a very useful summary of the relevant parts of the topic can be found in
Appendix M of Aitchison and Hey, volume 2.]
Q1 Show that the members of the group U(1) are all complex numbers with unit magnitude.
Q2 Show that successive U(1) gauge transformations commute.
Since U(1) gauge transformations commute, this is known as an Abelian gauge symmetry. The
U(1) group is an Abelian group (also known as a commutative group).
Let us consider a system consisting of two fermion fields ψ1 and ψ2 (we will not bother with hats,
for convenience), and demand that it possesses symmetry under transformations that mix them
together:
ψ1 7→ ψ10 = αψ1 + βψ2 ,
ψ2 7→
ψ20
= γψ1 + δψ2 ,
(6)
(7)
where α, β, γ and δ are complex numbers. Mixing together two fields might seem a bit strange, but
in fact we have already done this, since in Handout 3 we showed that a complex field can be written
in terms of two real fields (which we called φR and φI ). A U(1) transformation on a complex field is
nothing more than the mixing together of the φR and φI components. Now we are extending this to
mix complex fields instead of real – or scalar – fields, but the concept is the same.
To preserve normalization, we require hψ10 |ψ10 i = hψ20 |ψ20 i = 1 which gives |α|2 +|β|2 = |γ|2 +|δ|2 =
1. To preserve orthogonality, we require hψ10 |ψ20 i = 0 which gives α∗ γ + β ∗ δ = 0. We can express this
in an alternate mathematical form, as
0 ψ1
ψ1
α β
0
Ψ=
7→ Ψ =
=
Ψ = UΨ ,
(8)
ψ2
ψ20
γ δ
where U is a unitary matrix, since
†
UU =
α β
γ δ
α∗ γ ∗
β ∗ δ∗
=
1 0
0 1
.
(9)
The field Ψ is referred to as a doublet, since it contains two fields. More generally, if we mix together
an arbitrary number of fields, we describe Ψ as a multiplet.
A 2 × 2 unitary matrix has four free parameters, so consequently we can, quite generally, write
U = eiω0 +iω1 τ1 +iω2 τ2 +iω3 τ3
where ω0,1,2,3 are real parameters, and τ1,2,3 are the Pauli matrices:
1 0
0 −i
0 1
.
,
τ3 =
,
τ2 =
τ1 =
0 −1
i 0
1 0
(10)
(11)
Two comments:
• We previously used σ1,2,3 to denote the Pauli matrices; both notations are in frequent use in the
literature. It is convenient to use τ here, since then the discussion can be made more generic,
and in particular can be extended to the case of mixing together three fields. We will see below
that this is of interest.
• To remind those who are not accustomed to finding matrices in exponents, recall that using
a Taylor expansion it can be shown that eA = 1 + A + 12 A2 + 3!1 A3 + . . . Interpreting “1” as
a general unit element, we can use the above to define eA for any (dimensionless) A that can
operate on itself (i.e. for which A2 , A3 , . . . are defined). Square matrices, such as τi , clearly fall
into this category.
The group of 2 × 2 unitary matrices is denoted U(2), and since multiplication of such matrices is
not, in general, commutative, this is called a non-Abelian group.
Q3 Show that members of the symmetry group U(2) do not commute.
Q4 Paul Hewson, Dave Evans and friends were so impressed with this symmetry group that they
named their band after it. True or false?
From Eq. (10), we can see that we can separate off a global phase transformation
U = eiω0 V ,
V = eiω1 τ1 +iω2 τ2 +iω3 τ3 = eiω.τ .
(12)
The global, i.e. U(1), phase rotation eiω0 represents the same physics as discussed for QED, so we can
ignore it here. The interesting new physics is containing in V , which is a member of the symmetry
group SU(2) – the special unitary group of degree 2, i.e. the group of 2 × 2 unitary matrices with
unit determinant. SU(2) is also a non-Abelian group.
Q5 Show that the general matrix V defined above has unit determinant.
Q6 What are the members of the group SU(1)?
Global SU(2) Symmetry
Consider the Dirac Lagrangian density for our two fermion system.
LD = ψ̄1 (iγ µ ∂µ − m)ψ1 + ψ̄2 (iγ µ ∂µ − m)ψ2 ,
µ
= Ψ̄(iγ ∂µ − m)Ψ ,
(13)
(14)
with Ψ defined as in Eq. (8), and Ψ̄ its Hermitian conjugate (Ψ̄ = ψ̄1 ψ̄2 ).
This Lagrangian density is invariant under global SU(2) transformations
g
Ψ 7→ Ψ0 = ei 2 ωj τj Ψ ,
0
Ψ̄ 7→ Ψ̄
−i g2 ωj τj
= Ψ̄e
(15)
,
(16)
with g and ωj real constants.
Q7 Derive the symmetry current associated with the SU(2) gauge symmetry. [hint: refer to the
discussion of complex scalar fields in Handout 3.]
The presence of the Pauli matrices τj , previously encountered in quantum mechanics, suggests
that this symmetry has something to do with spin. Indeed, global SU(2) invariance represents
a symmetry in particle physics called “hadronic isospin” (it also has some uses in other areas of
physics). Hadronic isospin was first introduced by Heisenberg in 1932, in an attempt to explain
properties of nuclear physics, such as the similarity of the masses of the proton and the neutron. In
this scenario, the Lagrangian possesses a symmetry under mixing of the proton and neutron fermion
fields. Based on our current understanding (that these particles consist of quarks), the success of
hadronic isospin is just a coincidence, due to the near equality of the up and down quark masses
(i.e. hadronic isospin arises due to a near symmetry of the Lagrangian – technically, the strong
interaction Lagrangian – under mixing of the up and down quark fermion fields). However, the
SU(2) gauge symmetry turned out to be extremely important in particle physics to understand the
weak interaction (discussed later).
It is worth briefly noting that hadronic SU(2) was later extended to hadronic SU(3) by Gell-Mann
(in 1961, for which he won the 1969 Nobel Prize) and independently by Ne’eman (in 1962, for which
he didn’t), to explain the structure of subatomic particles that had been discovered at that time. This
structure was denoted “the eight-fold way” by Gell-Mann, and its understanding was an essential
step towards uncovering the quark substructure of hadronic particles. The phrase “eight-fold way” is
used since symmetry under SU(3) rotations results in particles (mesons and spin- 12 baryons) forming
an octet structure. Since one particle was missing from the baryon octet, Gell-Mann was able to
predict the existence and properties of the Ω− baryon, which was subsequently discovered in 1964.
Again, the success of this symmetry – sometimes called “flavour SU(3)” – is now understood as a
coincidence due to the similarity of the up, down and strange quark masses (i.e. it arises due to a
near symmetry of the strong interaction Lagrangian under mixing of the up, down and strange quark
fermion fields). However, SU(3) gauge symmetry turned out to be extremely important in particle
physics to understand the strong interaction. It seems ironic that symmetries introduced in an
attempt to comprehend the flavour structure of fundamental particles (i.e. to understand the curious
nature of the quark and lepton generations) have led to profound successes in the descriptions of both
weak and strong interactions of particles, while leaving the flavour structure a complete mystery.
Local SU(2) Symmetry
In 1954, Yang and Mills tried to promote SU(2) from a global symmetry to a local symmetry (they
were attempting to explain the binding of hadrons, though it turns out they explained something
else instead). We can write local SU(2) transformations as
g
Ψ 7→ Ψ0 = ei 2 ωj (x)τj Ψ ,
(17)
where the only difference compared to Eq. (15) is that ωj (x) is now an arbitrary function of spacetime (as χ(x) was in the U(1) case). We will write the space-time dependence explicitly in some
places to remind ourselves of it. Although it is not trivial that such finite transformations can be
built up from infinitesimally small transformations, it turns out that they can, and so we can consider
ωj (x) to be very small and hence
g
Ψ 7→ Ψ0 = (1 + i ωj (x)τj )Ψ ,
2
g
0
Ψ̄ 7→ Ψ̄ = Ψ̄(1 − i ωj (x)τj ) .
2
(18)
(19)
The Lagrangian density transforms to
g
g
LD 7→ L0D = Ψ̄(1 − i ωj (x)τj )(iγ µ ∂µ − m)(1 + i ωj (x)τj )Ψ ,
2
2
g µ
µ
= Ψ̄(iγ ∂µ − m)Ψ − Ψ̄γ (∂µ ωj (x))τj Ψ .
2
(20)
(21)
(In the above we have neglected terms of ω 2 as well as many subtle details of the derivation –
readers are encouraged to follow carefully the more detailed explanations that can be found in many
textbooks, for example Aitchison and Hey.)
The Dirac density alone is therefore not invariant, due to the presence of the new term at the
end of Eq. (21). As was the case for QED, this is accommodated by replacing ∂ µ with the covariant
derivative Dµ , i.e. the minimal substitution, which for SU(2) is given by
g
Dµ ≡ ∂ µ + i Wjµ (x)τj ,
2
(22)
which contains three vector fields Wjµ (x) (equivalent to the Aµ (x) of QED), excitations of which
correspond to gauge bosons. Note that, using the standard definition of the Pauli matrices, we can
write
√
g
g
Wµ3
Wµ1 − iWµ2
Wµ3
2Wµ+
µ
µ
µ
√
D =∂ +i
=∂ +i
,
(23)
−Wµ3
2Wµ− −Wµ3
2 Wµ1 + iWµ2
2
where we have defined the complex gauge fields Wµ± = √12 Wµ1 ∓ iWµ2 . This will become useful
later on.
The meaning of the covariant derivative is that Dµ Ψ should transform in the same way as Ψ (this
guarantees that the Lagrangian will be invariant under the transformation), i.e.
Dµ0 Ψ0 = U Dµ Ψ .
(24)
Since
g
Dµ0 = ∂ µ + i Wjµ0 (x)τj ,
2
this constrains the transformation of Wj to satisfy
g
g
(∂ µ + i Wjµ0 (x)τj )U Ψ = U (∂ µ + i Wjµ (x)τj )Ψ ,
2
2
(25)
(26)
for U = 1 + i g2 ωj (x)τj . The solution is not trivial, but it is worthwhile to work through. [hint: refer
to a good textbook, and put Wjµ0 (x) = Wjµ (x) + δWjµ (x).] The result is
Wlµ (x) 7→ Wlµ0 (x) = Wlµ (x) − ∂ µ ωl (x) − gjkl ωj (x)Wkµ (x) ,
(27)
where jkl is the Levi-Civita symbol. Note that the last term is equal to g(ω(x) × W µ (x))l .
Q8 Derive Eq. (27) from Eqs. (22) and (24).
It is instructive to compare the above to the corresponding result in QED: Aµ 7→ Aµ0 (x) =
− ∂ µ χ(x). The additional term is related to the SU(2) symmetry of the Wkµ triplet, i.e.
that the W field carries the quantum number that it interacts with, and therefore we expect selfinteractions of the associated gauge bosons. Contrast this with the photon (the QED gauge boson),
which being electrically neutral, does not interact with itself. The quantum number in question is
known as “weak isospin” in particle physics. The three bosons of the Wkµ triplet have weak isospin
magnitude one, with z-component +1 for the W + , −1 for the W − and 0 for the neutral current W 3
(see Eq. (23) for definitions of W ± ). As the conserved quantity is related to a gauge symmetry, weak
isospin is absolutely conserved.
Where do the self-interactions appear in the Lagrangian? Since the analogy with QED is serving
us well so far, we might expect that we need a term like − 41 Fjµν Fjµν with Fjµν = ∂ µ Wjν − ∂ ν Wjµ .
However, this is not SU(2) invariant, as can be seen by inserting the expression for Wlµ0 from Eq. (27),
and does not contain any self-interaction terms (due to the fact that the U(1) symmetry is Abelian).
In fact, we require
1
L = − Gjµν Gµν
(28)
j ,
4
with
µ
µ
ν
µ
ν
ν
(29)
Gµν
j = ∂ Wj − ∂ Wj − gjkl Wk Wl .
Aµ (x)
It can be checked that this is invariant under local SU(2) transformations, as required: the transµν0
formation Gµν
proceeds by transforming all W 7→ W 0 . Note that the last term is equal to
j 7→ Gj
µ
ν
g(W × W )j .
Q9 Expand the Lagrangian of Eq. (28) using the expression given in Eq. (29), and demonstrate
that it is invariant under local SU(2) transformations.
We have now obtained the full SU(2) Lagrangian
1
µ
LSU(2) = − Gjµν Gµν
j + Ψ̄(iγ Dµ − m)Ψ ,
4
= L0 + LINT ,
(30)
(31)
where
1
L0 = − (∂µ Wjν − ∂ν Wjµ ) ∂ µ Wjν − ∂ ν Wjµ + iΨ̄γ µ ∂µ Ψ − mΨ̄Ψ ,
(32)
4
g
g2
g
LINT =
jkl ∂ µ Wjν − ∂ ν Wjµ Wkµ Wlν − jkl jmn Wkµ Wlν Wmµ Wnν − Ψ̄γ µ Wjµ τj Ψ . (33)
2
4
2
Examining the interaction Lagrangian, we see the first term describes interactions between three W
bosons (so-called triple gauge or trilinear couplings), the second describes interactions between four
W bosons (quadrilinear couplings) and the third describes the interactions between the W fields and
our doublet of fermion fields Ψ. The Feynman diagrams for the self-interactions are given below.
Note that there is no mass term for the W fields (such a term would look like 21 m2 Wjµ Wjµ ), just
as there is no mass term for the A field in the QED Lagrangian. In fact, such terms are forbidden
by gauge symmetry. Hence the Yang-Mills theory (in its attempt to extend hadronic isospin to a
local gauge symmetry) predicts massless, and hence longlived, charged particles. Since these were
not (and are not) observed, the theory was put aside until new ideas on symmetry breaking in the
1960s, notably by Goldstone, Nambu (Nobel Prize 2008) and Jona-Lasinio, showed the solution to
this problem. This will be discussed later on. (It is worth noting that there is another potential
problem, that all components of the multiplet field Ψ are assumed to have the same mass. This will
also be addressed in due couse.)
W3
W1
W1
W2
W1
W2
W2
g
2 jkl
∂ µ Wjν − ∂ ν Wjµ Wkµ Wlν
2
− g4 jkl jmn Wkµ Wlν Wmµ Wnν
Q10 Considering the (roman lettered) subscripts that appear in the self-interaction diagrams given
above, show that there is no self-interaction between three (or four) gauge bosons carrying the
same index. Show also that weak isospin is conserved in these self-interactions.
Q11 Show that gauge boson masses are forbidden by gauge symmetry.
While the discussion above has been vague on some issues (as will be the discussion below), it
is important to realise that all the arguments can be put on a firm mathematical framework in the
context of quantum field theory. We could work through the equations much as we did for QED: we
can calculate conserved currents, make Fourier expansions of the fields introducing annihilation and
creation operators, obtain scattering amplitudes and derive Feynman rules and diagrams, etc. Due
to lack of time however, we will not. Interested readers are encouraged to refer to the recommended
texts for more details.
Q12 Obtain the Feynman rules for the weak interaction based on SU(2) gauge symmetry.
It should be obvious by now that local SU(2) symmetry has something to do with the weak
interaction of particle physics. However, there are clearly a few issues that need to be worked out,
most notably that a boson mass term in forbidden by gauge symmetry, but that it is known from
experiment that the W and Z bosons that mediate the weak interaction are massive (indeed, this is
the reason why the weak interaction appears weak). We will return to these issues later.
Local SU(3) Symmetry – Quantum Chromodynamics (The Strong Interaction)
It is fairly easy to extend the discussion of SU(2) gauge invariance – related to the symmetry of the
Lagrangian under mixing of two fermion fields – to other symmetries. Only slightly more complicated
would be a Lagrangian that possesses symmetry under the mixing of three fermion fields – i.e. under
the SU(3) gauge group. Let us be up front about what the fermion fields are in this case (alert
readers will have noticed that we have avoided this question for the SU(2) gauge symmetry so far
– it will be addressed later): three independent copies of the same quark, distinguished by a new
quantum number that is called colour. (This colour of course has nothing to do with the everyday
usage of the word “colour”, related to the wavelength of visible light. It is simply a label. The three
states are usually called “red”, “green” and “blue”.) Extending the colour metaphor, the theory of
the strong interaction is referred to as quantum chromodynamics, or QCD.
These three fields can mix into each other via a unitary matrix, as for U(2) in Eq. (8). However,
in place of the Pauli matrices, we find a set of eight 3 × 3 matrices, usually denoted λ. The algebra
of SU(3) is defined by the commutation relations of the λ matrices:
1
1
1
λj , λk = ifjkl λl .
(34)
2
2
2
[The commutation relations of the Pauli matrices define the algebra of SU(2), in a similar way as
seen in Eq. (34), although we did not say so explicitly before. This explains the factors of 12 that have
appeared in various equations.] In Eq. (34) above there is, of course, an implied summation over the
repeated index; this was also present in the SU(2) case but it is worth explicitly noting it for the less
familiar SU(3). The precise definitions of the λ matrices, sometimes referred to as the Gell-Mann
matrices, and the values of the structure constants fjkl , are not important for the discussion here,
but can be found in a good textbook (they are anyway conventional). Note that since there are eight
λ matrices, there are in principle 83 = 512 different structure constants, but most of them turn out
to be zero.
A general SU(3) gauge transformation can then be written
g
Ψ 7→ Ψ0 = ei 2 ω.λ Ψ ,
(35)
in terms of a single gauge coupling g (which is different to that of the weak interaction, for which
we used the same symbol), and eight free parameters ωi . In order to make this a local gauge
transformation, we simply require ω = ω(x).
To maintain gauge invariance of the Lagrangian, we need to introduce the covariant derivative,
which in this case is (cf. Eq. (22))
g
Dµ ≡ ∂ µ + i W µ .λ ,
(36)
2
with transformations of the eight gauge fields given by (cf. Eq. (27))
Wlµ 7→ Wlµ0 = Wlµ − ∂ µ ωl − gfjkl ωj Wkµ .
(37)
We have reused the notation W µ to represent the gauge fields, but the eight fields of SU(3) are of
course completely different to the three of SU(2). The field strength tensor is given by (cf. Eq. (29))
µ
µ
µ
ν
ν
ν
Gµν
j = ∂ Wj − ∂ Wj − gfjkl Wk Wl .
(38)
Many of the principle features of the theory of the strong interaction are therefore very similar
to those of the weak interaction. The theory predicts eight massless gauge bosons (gluons), which
self-interact (trilinear and quadrilinear couplings are similar to those shown in the diagrams for the
weak interaction above). A single coupling constant describes the strength of all strong interactions.
The quantum number associated with the gauge symmetry (colour) is absolutely conserved. The
markedly different phenomenology of the strong and weak interactions governed by SU(3) and SU(2)
respectively arises principally from a very simple empirical fact: the strong interaction is strong,
while the weak interaction is weak. The strength of the former has several profound consequences.
Perturbation theory cannot in general be used to calculate strong interaction processes. Notably, only
colour singlet states (i.e. states with no net colour quantum number) can be observed, explaining why
we never observe free quarks, only those bound into mesons (qi q̄j ) or baryons (qi qj qk ). However, on
short distance scales (or, equivalently, at high energies) quarks behave as free particles – perturbative
QCD calculations can be used to predict various phenomena at the Large Hadron Collider, for
example, and these predictions are being tested with impressive accuracy.
The self-interaction amongst gluons makes the force appear stronger at longer distances (lower
energies), causing quark confinement in hadrons. This result, known as asymptotic freedom, and its
experimental verification, led to the 2004 Nobel Prize being awarded to Gross, Polizer and Wilczek.
Q13 Explain how the properties of the strong interaction based on local SU(3) gauge symmetry
constrain quarks to be bound into mesons or baryons.
Chiral Structure of the Weak Interaction
Returning now to the weak interaction (i.e. SU(2)), it is time to face up to one extremely important question that we have until now skirted around: what are the fermion fields that possess
symmetry under mixing through local SU(2) rotations? Since we know that the weak interaction
mediates transitions between νe ↔ e leptons and u ↔ d quarks (considering only the lightest generation), it is tempting to suggest that the weak isospin doublets should be (νe e− ) and (u d) – or
more precisely the doublets should be composed of the corresponding fermion fields. This is not quite
right, as discussed in more detail below, but is close enough for now. We know that if the members
of the doublet have different masses, then the mass term in the Lagrangian will break its gauge
invariance, but since we already have a problem with gauge boson mass, perhaps we can overlook
that and hope that everything will work out alright in the end (spoiler: it will, but not yet). There
are some further subtleties which have also not yet been mentioned, but require at least a (far too)
brief explanation.
First, it is experimentally observed that the charged current weak interaction (i.e. the interactions
of the W ± bosons) is chiral, and acts on the left-handed components of particles only. The chirality
of a particle is left-handed if its spin is opposite to its direction of motion, and right-handed if its
spin and motion are aligned. Left- and right-handed components of a fermion field can be obtained
using the chirality operators:
ψL = 12 (1 − γ 5 )ψ ,
(39)
ψR = 21 (1 + γ 5 )ψ ,
where γ 5 is defined as γ 5 = iγ 0 γ 1 γ 2 γ 3 .
Q14 Using the standard definitions of the gamma matrices (given, for example, in handout 1), find
an expression for γ 5 , in block form or otherwise.
Q15 What are the commutators γ 5 , γ i (i = 0, 1, 2, 3)?
Q16 Consider the four plane-wave solutions for ψ (the solutions to the Dirac equation). For each,
what is γ 5 ψ? What is 12 (1 − γ 5 )ψ? What is 12 (1 + γ 5 )ψ?
Q17 Write the (free-field) Dirac Lagrangian of Eq. (2) in terms of the left- and right-handed fields.
The fact that the charged current weak interactions act only on the left-handed components of
fermion fields is accommodated in the theory by having ψL and ψR transform differently under SU(2).
Left-handed fermion fields have weak isospin IW = ± 21 , while right-handed fermion fields have weak
isospin IW = 0, so that the transformations are given by
g
ΨL 7→ Ψ0L = ei 2 ω.τ ΨL ,
ΨR 7→ Ψ0R =
ΨR .
(40)
More generally, we could allow ΨR to have similar transformations to ΨL , but with a different
coupling strength. The case we are looking at, which is the one realised in nature, is a special case
where the coupling for ΨR is equal to zero.
The possibility for a fundamental interaction to distinguish between left- and right-handedness
(known as parity violation) was first proposed by Yang and Lee in 1956 (two years after the Yang-Mills
theory was developed, but well before anything about the SU(2) gauge symmetry of the weak interaction was known). Soon thereafter, the experimental observation of parity violation was achieved
by Wu, studying the β decay of 60 Co, and Yang and Lee were (promptly) awarded the Nobel prize
in 1957. In fact, parity violation is found to be maximal in the weak interaction, which is related
to the fact that the “vector”-like interaction (the “1” part of Eq. (39)) and the “axial-vector”-like
interaction (the “γ 5 ” part of Eq. (39)) are equal in magnitude. This is sometimes referred to as the
“V − A structure of the weak interaction.” (Clearly V − A refers to γ µ (1 − γ 5 ), not just (1 − γ 5 ).)
Due to the maximal parity violation of the V − A theory, the right-handed components of the
fermion fields do not transform under the SU(2) symmetry and do not carry weak-isospin quantum
numbers. Therefore, they do not interact with the weak interaction. This is particularly notable for
neutrinos: since they also do not interact with either of the strong or electromagnetic interactions,
right-handed neutrinos do not interact with anything at all (except gravity – and since their masses
are so small, they hardly interact with that, either)! This aspect of neutrino physics is one of the
least understood, and most interesting, parts of current research into fundamental particles.
The distinction between left- and right-handed components of the fermion fields has many interesting consequences, one of which is worth commenting on here. The fermion mass terms of the
Lagrangian (i.e. terms like mΨ̄Ψ) couple together left- and right-handed fields. Since these fields
transform differently under the SU(2) gauge symmetry, such terms are no longer gauge invariant –
i.e. they are forbidden under gauge symmetry. The problem with fermion mass has got even worse
than it was before, and we still have the problem with gauge boson mass. Clearly, something needs
to be done! We will get to this momentarily.
Let us conclude this section by commenting that one might reasonably wonder why it should be
that the weak interaction should distinguish between left and right (i.e. violate parity) in this way,
while the other interactions do not. In fact, there is no real reason known to explain this – although
Sakharov (1967) showed that C and CP violation are necessary conditions for the evolution of a
matter dominated Universe, and therefore anthropic arguments can be applied, the behaviour of the
weak interaction in this respect remains one of the mysteries of the Standard Model. We can only
hope that future experiments will shed some light upon it.
The Weak Interaction Doublets – the CKM and PMNS Matrices
Coming back to the question of what exactly are the fields being mixed by the SU(2) symmetry,
we might now expect that the SU(2) doublets for the quark fields are (u d), (c s) and (t b) – i.e. one
doublet per quark “family”. However, it is experimentally observed that the charged current weak
interactions allow flavour changes between families, such as a b quark decaying to a c quark. The
solution to this problem was first proposed by Cabibbo in 1964, when only the u, d and s quarks
were known. He suggested that the “d” in the SU(2) doublet should be given by
d0 = d cos θC + s sin θC ,
(41)
where the Cabibbo angle is determined experimentally to be given by sin θC ≈ 0.22.
Parenthetically, it is worth noting that Cabibbo’s theory, while solving one problem, introduced
another – although it could successfully explain the observed flavour changing charged currents, it
predicted also the existence of flavour changing neutral currents, which were not observed. The
solution to this problem was found by Glashow, Iliopoulos and Maiani (1970): if a fourth quark field
(the charm quark) was introduced, so that all fermion fields appear in doublets, the flavour changing
neutral currents would be suppressed.
Although we are not following the chronology exactly, the next chapter in the story was the 1964
observation of violation of the combined symmetry CP – a product of C and P operations, where
C represents charge-conjugation, under which particles are replaced with anti-particles, and vice
versa, and P is parity. Although parity violation was understood to occur in the weak interaction,
it was widely believed that the CP symmetry should be conserved. This surprising observation
resulted in the Nobel prize of 1980 being awarded to Cronin and Fitch. Several theoretical schemes
were proposed to accommodate CP violation, but it took several years until the correct (as we now
understand it) solution was found. In 1973, Kobayashi and Maskawa extended the quark mixing
scheme to a 3 × 3 mixing matrix, known as the CKM matrix.
 0 
 
d
d
 s0  = VCKM  s  .
(42)
b0
b
The CKM matrix has four free parameters, that must be determined experimentally. The theoretical
work of Kobayashi and Maskawa thus predicted the existence of six quark fields before the discovery
of the charm (1974), bottom (1977) and top (1995), and also predicts a pattern of CP violation
phenomena in the B meson system that, over the past 15 years or so, has been confirmed by experiment. Thus, the Kobayashi-Maskawa mechanism to explain CP violation is highly predictive and
extremely successful, and well-deserving of the 2008 Nobel Prize in Physics (awarded jointly with
Nambu).
For completeness, note that the observation of neutrino oscillations shows that leptons also mix.
The lepton mixing matrix is usually called the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix.
While the underlying physics of lepton mixing is similar to that for the quark sector, its phenomenology is dramatically different, due to the fact that quarks are bound in hadrons while leptons are
not. There is a whole separate module on Neutrino Physics (PX435) to discuss the many interesting
aspects of neutrino phenomenology.