32
1. The Particles and Forces of the Standard Model
this symmetry is gauge invariance as we shall explain in section 7.3.1. It
turns out that this symmetry is vital in rendering QED renormalizable. It is
natural then to ask whether in the case of QED, a situation ever arises where
the photon acquires mass, while retaining fully gauge-invariant interactions –
and hence renormalizability (we would hope). If so, we would then have an
analogue of what is needed for a renormalizable theory of weak interactions.
The answer is that this can indeed happen, but it requires some extra dynamics
to do it. Nature has actually provided us with a working model of what we
want, in the phenomenon of superconductivity. There, the Meissner effect can
be interpreted as implying that the photons propagating in a thin surface layer
of the material have non-zero mass (see section 19.2). The dynamics behind
this is subtle, and required many years of theoretical efforts before it was
finally understood by Bardeen, Cooper and Schrieffer (1957). In simple terms,
the mechanism is a two-step process. First, lattice interactions cause electrons
to bind into pairs; then these pairs undergo Bose-Einstein condensation. This
‘condensate’ is the BCS superconducting ground state. The essential point is
that although the electromagnetic interactions are fully gauge invariant, the
ground state is not. When a symmetry is broken by the ground state, it is
said to be ‘spontaneously’ broken. We shall provide an introduction to the
BCS ground state in chapter 17 of volume 2.
The BCS theory is an example of spontaneous symmetry breaking occurring dynamically (through the particular lattice interactions). Many of
the physically important phenomena can, however, be very satisfactorily described in terms of an effective theory, which treats only the electrodynamics
of the condensate. Such a description was proposed by Ginzburg and Landau
(1950), well before the BCS paper, in fact.
How can this be applied in particle physics? Recall the idea, mentioned
in section 1.3.1, that the analogue of the many-body ground state is the qft
vacuum (Nambu 1961). In the Standard Model, the weak interactions are
indeed described by a gauge-invariant theory, and the assumption is made
that the vacuum breaks the gauge symmetry. The simplest way this idea
can be implemented is along the lines of the Ginzburg-Landau theory, as
suggested by Weinberg (1967) and by Salam (1968), and their proposal is embodied in the Glashow-Salam-Weinberg electroweak theory, which is part of
the SM. It requires the introduction of four new spin-0 fields, which are called
Higgs fields (Higgs 1964, Englert and Brout 1964, Guralnik et al. 1964),
and which we may think of as playing the role of the BCS condensate (but
not for electromagnetism, of course). The combined theory of quarks, leptons, electroweak gauge fields, and Higgs fields is gauge invariant, but one of
the Higgs fields is supposed to have a non-zero average value in the physical
vacuum, which breaks the gauge symmetry. The other three Higgs fields effectively become the longitudinal parts of the massive spin-1 W ± and Z
0 fields,
while the quantized excitations of the fourth Higgs field away from its vacuum value appear physically as neutral spin-0 particles, called Higgs bosons
(Higgs 1964).
1. The Particles and Forces of the Standard Model
this symmetry is gauge invariance as we shall explain in section 7.3.1. It
turns out that this symmetry is vital in rendering QED renormalizable. It is
natural then to ask whether in the case of QED, a situation ever arises where
the photon acquires mass, while retaining fully gauge-invariant interactions –
and hence renormalizability (we would hope). If so, we would then have an
analogue of what is needed for a renormalizable theory of weak interactions.
The answer is that this can indeed happen, but it requires some extra dynamics
to do it. Nature has actually provided us with a working model of what we
want, in the phenomenon of superconductivity. There, the Meissner effect can
be interpreted as implying that the photons propagating in a thin surface layer
of the material have non-zero mass (see section 19.2). The dynamics behind
this is subtle, and required many years of theoretical efforts before it was
finally understood by Bardeen, Cooper and Schrieffer (1957). In simple terms,
the mechanism is a two-step process. First, lattice interactions cause electrons
to bind into pairs; then these pairs undergo Bose-Einstein condensation. This
‘condensate’ is the BCS superconducting ground state. The essential point is
that although the electromagnetic interactions are fully gauge invariant, the
ground state is not. When a symmetry is broken by the ground state, it is
said to be ‘spontaneously’ broken. We shall provide an introduction to the
BCS ground state in chapter 17 of volume 2.
The BCS theory is an example of spontaneous symmetry breaking occurring dynamically (through the particular lattice interactions). Many of
the physically important phenomena can, however, be very satisfactorily described in terms of an effective theory, which treats only the electrodynamics
of the condensate. Such a description was proposed by Ginzburg and Landau
(1950), well before the BCS paper, in fact.
How can this be applied in particle physics? Recall the idea, mentioned
in section 1.3.1, that the analogue of the many-body ground state is the qft
vacuum (Nambu 1961). In the Standard Model, the weak interactions are
indeed described by a gauge-invariant theory, and the assumption is made
that the vacuum breaks the gauge symmetry. The simplest way this idea
can be implemented is along the lines of the Ginzburg-Landau theory, as
suggested by Weinberg (1967) and by Salam (1968), and their proposal is embodied in the Glashow-Salam-Weinberg electroweak theory, which is part of
the SM. It requires the introduction of four new spin-0 fields, which are called
Higgs fields (Higgs 1964, Englert and Brout 1964, Guralnik et al. 1964),
and which we may think of as playing the role of the BCS condensate (but
not for electromagnetism, of course). The combined theory of quarks, leptons, electroweak gauge fields, and Higgs fields is gauge invariant, but one of
the Higgs fields is supposed to have a non-zero average value in the physical
vacuum, which breaks the gauge symmetry. The other three Higgs fields effectively become the longitudinal parts of the massive spin-1 W ± and Z
0 fields,
while the quantized excitations of the fourth Higgs field away from its vacuum value appear physically as neutral spin-0 particles, called Higgs bosons
(Higgs 1964).