13.3. Local non-Abelian symmetries in Lagrangian quantum field theory 53
in which our local SU(2) Lagrangian is not suitable (yet) for describing weak
interactions. First, weak interactions violate parity, in fact ‘maximally’, by
ˆ
which is meant that only the ‘left-handed’ part ψ L of ( the ) fermion field enters
ˆ
ˆ
the interactions with the W
μ fields, where ψ L ≡
1−γ5 ψ; for this reason
2
the weak isospin group is called SU(2) L . Secondly, the physical W
± are of
course not massless, and therefore cannot be described by propagators of the
form (13.69). And thirdly, the fermion mass term violates the ‘left-handed’
SU(2) gauge symmetry, as the discussion in section 12.3.2 shows. In this
case, however, the chiral symmetry which is broken by fermion masses in the
Lagrangian is a local, or gauge, symmetry (in section 12.3.2 the chiral flavour
symmetry was a global symmetry). If we want to preserve the chiral gauge
symmetry SU(2) L – and it is necessary for renormalizability – then we shall
have to replace the simple fermion mass term in (13.66) by something else, as
will be explained in chapter 22.
The locally SU(3) c -invariant Lagrangian for one quark triplet (cf (12.137))
(
)
ˆ
f r
q ˆ = ( ˆ
f
f )
b
,
(13.70)
ˆ
f g
where ‘f’ stands for ‘flavour’, and ‘r, b, and g’ for ‘red, blue, and green’, is
1
1
¯ ˆ
q ˆ f (iD −
ˆ
ˆ
ˆ
ˆ
m f )ˆ q − F aμν F
μν
μ
ν
f
4
a − (∂ μ A
2ξ
a )(∂ ν A a )
(13.71)
ˆ
where D
μ
ˆ
μ
is given by (13.53) with A
μ replaced by A , and the footnote
before equation (13.68) also applies here. This leads to the interaction term
(cf (13.59))
− ¯
ˆ
g s q ˆ f γ
μ
λ/2ˆ q f · A μ
(13.72)
and the Feynman rule (13.60) for figure 13.2. Once again, the gluon quanta
must be massless, and their propagator is the same as (13.69), with δ ij →
δ ab (a, b = 1, 2, . . . 8). The different quark flavours are included by simply
repeating the first term of (13.71) for all flavours:
∑ ¯ ˆ
q ˆ f (iD
f
− m f )ˆ q f ,
(13.73)
which incorporates the hypothesis that the SU(3) c -gauge interaction is ‘flavourblind’, i.e. exactly the same for each flavour. Note that although the flavour
masses are different, the masses of different ‘coloured’ quarks of the same
flavour are the same (m u = m d , m u,r = m u,b = m u,g ).
The Lagrangians (13.66)–(13.68), and (13.71), though easily written down
after all this preparation, are unfortunately not adequate for anything but
tree graphs. We shall indicate why this is so in section 13.3.3. Before that, we
want to discuss in more detail the nature of the gauge-field self-interactions
contained in the Yang-Mills pieces.
/
/
/
in which our local SU(2) Lagrangian is not suitable (yet) for describing weak
interactions. First, weak interactions violate parity, in fact ‘maximally’, by
ˆ
which is meant that only the ‘left-handed’ part ψ L of ( the ) fermion field enters
ˆ
ˆ
the interactions with the W
μ fields, where ψ L ≡
1−γ5 ψ; for this reason
2
the weak isospin group is called SU(2) L . Secondly, the physical W
± are of
course not massless, and therefore cannot be described by propagators of the
form (13.69). And thirdly, the fermion mass term violates the ‘left-handed’
SU(2) gauge symmetry, as the discussion in section 12.3.2 shows. In this
case, however, the chiral symmetry which is broken by fermion masses in the
Lagrangian is a local, or gauge, symmetry (in section 12.3.2 the chiral flavour
symmetry was a global symmetry). If we want to preserve the chiral gauge
symmetry SU(2) L – and it is necessary for renormalizability – then we shall
have to replace the simple fermion mass term in (13.66) by something else, as
will be explained in chapter 22.
The locally SU(3) c -invariant Lagrangian for one quark triplet (cf (12.137))
(
)
ˆ
f r
q ˆ = ( ˆ
f
f )
b
,
(13.70)
ˆ
f g
where ‘f’ stands for ‘flavour’, and ‘r, b, and g’ for ‘red, blue, and green’, is
1
1
¯ ˆ
q ˆ f (iD −
ˆ
ˆ
ˆ
ˆ
m f )ˆ q − F aμν F
μν
μ
ν
f
4
a − (∂ μ A
2ξ
a )(∂ ν A a )
(13.71)
ˆ
where D
μ
ˆ
μ
is given by (13.53) with A
μ replaced by A , and the footnote
before equation (13.68) also applies here. This leads to the interaction term
(cf (13.59))
− ¯
ˆ
g s q ˆ f γ
μ
λ/2ˆ q f · A μ
(13.72)
and the Feynman rule (13.60) for figure 13.2. Once again, the gluon quanta
must be massless, and their propagator is the same as (13.69), with δ ij →
δ ab (a, b = 1, 2, . . . 8). The different quark flavours are included by simply
repeating the first term of (13.71) for all flavours:
∑ ¯ ˆ
q ˆ f (iD
f
− m f )ˆ q f ,
(13.73)
which incorporates the hypothesis that the SU(3) c -gauge interaction is ‘flavourblind’, i.e. exactly the same for each flavour. Note that although the flavour
masses are different, the masses of different ‘coloured’ quarks of the same
flavour are the same (m u = m d , m u,r = m u,b = m u,g ).
The Lagrangians (13.66)–(13.68), and (13.71), though easily written down
after all this preparation, are unfortunately not adequate for anything but
tree graphs. We shall indicate why this is so in section 13.3.3. Before that, we
want to discuss in more detail the nature of the gauge-field self-interactions
contained in the Yang-Mills pieces.
/
/
/