17.7. The BCS superconducting ground state
219
Oscillations about (17.103) are conveniently parametrized by
(
)
0
ˆ
φ = exp(− ˆ
iθ(x) · τ /2v)
1
,
(17.107)
ˆ
√ (v + H(x))
2
which is to be compared with (17.84). Inserting (17.107) into (17.97) (see
problem 17.6) we easily find that no mass term is generated for the θ fields,
while the H field piece is
1
L ˆ
ˆ
= ∂ H∂
μ ˆ
H − μ
2 ˆ
H
μ
H
2 + interactions
(17.108)
2
just as in (17.85), showing that m H =
√
2μ.
Let us now note carefully that whereas in the ‘normal symmetry’ case
with the opposite sign for the μ
2 term in (17.97), the free-particle spectrum
consisted of a degenerate doublet of four degrees of freedom all with the same
mass μ, in the ‘spontaneously broken’ case no such doublet structure is seen:
instead, there is one massive scalar field, and three massless scalar fields.
The number of degrees of freedom is the same in each case, but the physical
spectrum is completely different.
In the application of this to the electroweak sector of the Standard Model,
the SU(2) × U(1) symmetry will be ‘gauged’ (i.e. made local), which is easily
done by replacing the ordinary derivatives in (17.97) by suitable covariant
ones. We shall see in chapter 19 that the result, with the choice (17.107),
will be to end up with three massive gauge fields (those mediating the weak
interactions) and one massless gauge field (the photon). We may summarize
this (anticipated) result by saying, then, that when a spontaneously broken
non-Abelian symmetry is gauged, those gauge fields corresponding to symmeˆ
tries that are broken by the choice of ‹0|φ|0› acquire a mass, while those that
ˆ
correspond to symmetries that are respected by ‹0|φ|0› do not. Exactly how
this happens will be the subject of chapter 19.
We end this chapter by considering a second important example of spontaneous symmetry breaking in condensed matter physics, as a preliminary to
our discussion of chiral symmetry breaking in the following chapter.
17.7 The BCS superconducting ground state
We shall not attempt to provide a self-contained treatment of the Bardeen–
Cooper–Schrieffer (1957) – or BCS – theory; rather, we wish simply to focus
on one aspect of the theory, namely the occurrence of an energy gap separating
the ground state from the lowest excited levels of the fermionic energy spectrum. The existence of such a gap is a fundamental ingredient of the theory
of superconductivity; in the following chapter we shall see how Nambu (1960)
219
Oscillations about (17.103) are conveniently parametrized by
(
)
0
ˆ
φ = exp(− ˆ
iθ(x) · τ /2v)
1
,
(17.107)
ˆ
√ (v + H(x))
2
which is to be compared with (17.84). Inserting (17.107) into (17.97) (see
problem 17.6) we easily find that no mass term is generated for the θ fields,
while the H field piece is
1
L ˆ
ˆ
= ∂ H∂
μ ˆ
H − μ
2 ˆ
H
μ
H
2 + interactions
(17.108)
2
just as in (17.85), showing that m H =
√
2μ.
Let us now note carefully that whereas in the ‘normal symmetry’ case
with the opposite sign for the μ
2 term in (17.97), the free-particle spectrum
consisted of a degenerate doublet of four degrees of freedom all with the same
mass μ, in the ‘spontaneously broken’ case no such doublet structure is seen:
instead, there is one massive scalar field, and three massless scalar fields.
The number of degrees of freedom is the same in each case, but the physical
spectrum is completely different.
In the application of this to the electroweak sector of the Standard Model,
the SU(2) × U(1) symmetry will be ‘gauged’ (i.e. made local), which is easily
done by replacing the ordinary derivatives in (17.97) by suitable covariant
ones. We shall see in chapter 19 that the result, with the choice (17.107),
will be to end up with three massive gauge fields (those mediating the weak
interactions) and one massless gauge field (the photon). We may summarize
this (anticipated) result by saying, then, that when a spontaneously broken
non-Abelian symmetry is gauged, those gauge fields corresponding to symmeˆ
tries that are broken by the choice of ‹0|φ|0› acquire a mass, while those that
ˆ
correspond to symmetries that are respected by ‹0|φ|0› do not. Exactly how
this happens will be the subject of chapter 19.
We end this chapter by considering a second important example of spontaneous symmetry breaking in condensed matter physics, as a preliminary to
our discussion of chiral symmetry breaking in the following chapter.
17.7 The BCS superconducting ground state
We shall not attempt to provide a self-contained treatment of the Bardeen–
Cooper–Schrieffer (1957) – or BCS – theory; rather, we wish simply to focus
on one aspect of the theory, namely the occurrence of an energy gap separating
the ground state from the lowest excited levels of the fermionic energy spectrum. The existence of such a gap is a fundamental ingredient of the theory
of superconductivity; in the following chapter we shall see how Nambu (1960)