18
Chiral Symmetry Breaking
In section 12.4.2 we arrived at a puzzle: there seemed good reason to think
that a world consisting of u and d quarks and their antiparticles, interacting
via the colour gauge fields of QCD, should exhibit signs of the non-Abelian
chiral symmetry SU(2) f 5 , which was exact in the massless limit m u , m d → 0.
But, as we showed, one of the simplest consequences of such a symmetry
should be the existence of nucleon parity doublets, which are not observed.
We can now resolve this puzzle by making the hypothesis (section 18.1) first
articulated by Nambu (1960) and Nambu and Jona-Lasinio (1961a), that this
chiral symmetry is spontaneously broken as a dynamical effect – presumably,
from today’s perspective, as a property of the QCD interactions, as discussed
in section 18.1.1. If this is so, an immediate physical consequence should be
the appearance of massless (Goldstone) bosons, one for every symmetry not
respected by the vacuum. Indeed, returning to (12.168) which we repeat here
for convenience,
1
2
+5
( )
ˆ
T |d› = |u ˜›,
(18.1)
we now interpret the state |u ˜› (which is degenerate with |d›) as |d + ‘π
+ ’›
where ‘π
+ ’ is a massless particle of positive charge, but a pseudoscalar (0
− )
rather than a scalar (0
+ ) since, as we saw, |˜
‘π
− ’ and ‘π
0 ’ will be associated with ˆ (
1
2
1
2
u› has opposite parity to |u›. In
)
( )
and T ˆ
3 5
the same way,
T
. Of course,
− 5
no such massless pseudoscalar particles are observed: but it is natural to hope
that when the small up and down quark masses are included, the real pions
(π
+ , π
− , π
0 ) will emerge as ‘anomalously light’, rather than strictly massless.
This is indeed how they do appear, particularly with respect to the octet of
mesons, which differ only in q¯ q spin alignment from the 0
− octet. As Nambu
and Jona-Lasinio (1961a) said, ‘it is perhaps not a coincidence that there
exists such an entity [i.e. the Goldstone state(s)] in the form of the pion’.
If this was the only observable consequence of spontaneously breaking chiral symmetry, it would perhaps hardly be sufficient grounds for accepting
the hypothesis. But there are two circumstances which greatly increase the
phenomenological implications of the idea. First, the vector and axial vec1
2
1
2
tor symmetry currents T ˆ
( )μ
( )μ
and T ˆ
of the u-d strong interaction SU(2)
5
symmetries (see (12.109) and (12.165)) happen to be the very same currents
which enter into strangeness-conserving semileptonic weak interactions (such
as n → pe
− ν ¯ e and π
−
→ μ
− ν ¯ μ ), as we shall see in chapter 20. Thus some remarkable connections between weak- and strong-interaction parameters can be
227
Chiral Symmetry Breaking
In section 12.4.2 we arrived at a puzzle: there seemed good reason to think
that a world consisting of u and d quarks and their antiparticles, interacting
via the colour gauge fields of QCD, should exhibit signs of the non-Abelian
chiral symmetry SU(2) f 5 , which was exact in the massless limit m u , m d → 0.
But, as we showed, one of the simplest consequences of such a symmetry
should be the existence of nucleon parity doublets, which are not observed.
We can now resolve this puzzle by making the hypothesis (section 18.1) first
articulated by Nambu (1960) and Nambu and Jona-Lasinio (1961a), that this
chiral symmetry is spontaneously broken as a dynamical effect – presumably,
from today’s perspective, as a property of the QCD interactions, as discussed
in section 18.1.1. If this is so, an immediate physical consequence should be
the appearance of massless (Goldstone) bosons, one for every symmetry not
respected by the vacuum. Indeed, returning to (12.168) which we repeat here
for convenience,
1
2
+5
( )
ˆ
T |d› = |u ˜›,
(18.1)
we now interpret the state |u ˜› (which is degenerate with |d›) as |d + ‘π
+ ’›
where ‘π
+ ’ is a massless particle of positive charge, but a pseudoscalar (0
− )
rather than a scalar (0
+ ) since, as we saw, |˜
‘π
− ’ and ‘π
0 ’ will be associated with ˆ (
1
2
1
2
u› has opposite parity to |u›. In
)
( )
and T ˆ
3 5
the same way,
T
. Of course,
− 5
no such massless pseudoscalar particles are observed: but it is natural to hope
that when the small up and down quark masses are included, the real pions
(π
+ , π
− , π
0 ) will emerge as ‘anomalously light’, rather than strictly massless.
This is indeed how they do appear, particularly with respect to the octet of
mesons, which differ only in q¯ q spin alignment from the 0
− octet. As Nambu
and Jona-Lasinio (1961a) said, ‘it is perhaps not a coincidence that there
exists such an entity [i.e. the Goldstone state(s)] in the form of the pion’.
If this was the only observable consequence of spontaneously breaking chiral symmetry, it would perhaps hardly be sufficient grounds for accepting
the hypothesis. But there are two circumstances which greatly increase the
phenomenological implications of the idea. First, the vector and axial vec1
2
1
2
tor symmetry currents T ˆ
( )μ
( )μ
and T ˆ
of the u-d strong interaction SU(2)
5
symmetries (see (12.109) and (12.165)) happen to be the very same currents
which enter into strangeness-conserving semileptonic weak interactions (such
as n → pe
− ν ¯ e and π
−
→ μ
− ν ¯ μ ), as we shall see in chapter 20. Thus some remarkable connections between weak- and strong-interaction parameters can be
227