228
18. Chiral Symmetry Breaking
established, such as the Goldberger–Treiman (1958) relation (see section 18.2)
and the Adler–Weisberger (Adler 1965, Weisberger 1965) relation. Second, it
turns out that the dynamics of the Goldstone modes, and their interactions
with other hadrons such as nucleons, are strongly constrained by the underlying chiral symmetry of QCD; indeed, surprisingly detailed effective theories
(see section 18.3) have been developed, which provide a very successful description of the low energy dynamics of the Goldstone degrees of freedom.
Finally we shall introduce the subject of chiral anomalies in section 18.4.
It would take us too far from our main focus on gauge theories to pursue
these interesting avenues in any detail. But we hope to convince the reader, in
this chapter, that chiral symmetry breaking is an integral part of the Standard
Model, being a fundamental property of QCD.
18.1 The Nambu analogy
We recall from section 12.4.2 that for ‘almost massless’ fermions it is natural
to use the representation (3.40) for the Dirac matrices, in terms of which the
Dirac equation reads
Eφ = σ · pφ + mχ
(18.2)
Eχ = −σ · pχ + mφ.
(18.3)
Nambu (1960) and Nambu and Jona-Lasinio (1961a) pointed out a remarkable
analogy between (18.2) and (18.3) and equations (17.122) and (17.123) which
describe the elementary excitations in a superconductor (in the case Δ is real),
and which we repeat here for convenience:
ω l cos θ l = ∈ l cos θ l + Δ sin θ l
(18.4)
ω l sin θ l = −∈ l sin θ l + Δ cos θ l .
(18.5)
In (18.4) and (18.5), cos θ l and sin θ l are respectively the components of the
†
electron destruction operator ˆ
c l and the electron creation operator ˆ
c in the
−l
quasiparticle operator β ˆ
l (see (17.111)):
†
β ˆ
l = cos θ l c ˆ l − sin θ l c ˆ
(18.6)
−l
.
The superposition in β ˆ
l combines operators which transform differently under
the U(1) (number) symmetry. The result of this spontaneous breaking of the
U(1) symmetry is the creation of the gap Δ (or 2Δ for a number-conserving
excitation), and the appearance of a massless mode. If Δ vanishes, (17.126)
implies that θ l = 0, and we revert to the symmetry-respecting operators
†
c ˆ l , c ˆ . Consider now (18.2) and (18.3). Here φ and χ are the components of
−l
18. Chiral Symmetry Breaking
established, such as the Goldberger–Treiman (1958) relation (see section 18.2)
and the Adler–Weisberger (Adler 1965, Weisberger 1965) relation. Second, it
turns out that the dynamics of the Goldstone modes, and their interactions
with other hadrons such as nucleons, are strongly constrained by the underlying chiral symmetry of QCD; indeed, surprisingly detailed effective theories
(see section 18.3) have been developed, which provide a very successful description of the low energy dynamics of the Goldstone degrees of freedom.
Finally we shall introduce the subject of chiral anomalies in section 18.4.
It would take us too far from our main focus on gauge theories to pursue
these interesting avenues in any detail. But we hope to convince the reader, in
this chapter, that chiral symmetry breaking is an integral part of the Standard
Model, being a fundamental property of QCD.
18.1 The Nambu analogy
We recall from section 12.4.2 that for ‘almost massless’ fermions it is natural
to use the representation (3.40) for the Dirac matrices, in terms of which the
Dirac equation reads
Eφ = σ · pφ + mχ
(18.2)
Eχ = −σ · pχ + mφ.
(18.3)
Nambu (1960) and Nambu and Jona-Lasinio (1961a) pointed out a remarkable
analogy between (18.2) and (18.3) and equations (17.122) and (17.123) which
describe the elementary excitations in a superconductor (in the case Δ is real),
and which we repeat here for convenience:
ω l cos θ l = ∈ l cos θ l + Δ sin θ l
(18.4)
ω l sin θ l = −∈ l sin θ l + Δ cos θ l .
(18.5)
In (18.4) and (18.5), cos θ l and sin θ l are respectively the components of the
†
electron destruction operator ˆ
c l and the electron creation operator ˆ
c in the
−l
quasiparticle operator β ˆ
l (see (17.111)):
†
β ˆ
l = cos θ l c ˆ l − sin θ l c ˆ
(18.6)
−l
.
The superposition in β ˆ
l combines operators which transform differently under
the U(1) (number) symmetry. The result of this spontaneous breaking of the
U(1) symmetry is the creation of the gap Δ (or 2Δ for a number-conserving
excitation), and the appearance of a massless mode. If Δ vanishes, (17.126)
implies that θ l = 0, and we revert to the symmetry-respecting operators
†
c ˆ l , c ˆ . Consider now (18.2) and (18.3). Here φ and χ are the components of
−l