278
19. Spontaneously Broken Local Symmetry
ˆ
ˆ
and this is why A μ does not acquire a mass when ‹0|φ|0› = 0 (gauge fields
ˆ
coupled to unbroken symmetries of ‹0|φ|0› do not become massive). Although
ˆ
certainly not unique, this choice of φ and ‹0| ˆ
φ|0› is undoubtedly very economical and natural. We are interpreting the zero eigenvalue of (1 + τ 3 ) as the
electromagnetic charge of the vacuum, which we do not wish to be non-zero.
We then make the identification
e = g sin θ W
(19.97)
in order to get the right ‘electromagnetic D μ ’ in (19.96).
We emphasize once more that the particular form of (19.87) corresponds
to a choice of gauge, namely the unitary one (cf the discussions in sections
19.3 and 19.5). There is always the possibility of using other gauges, as in
the Abelian case, and this will in general be advantageous when doing loop
calculations involving renormalization. We would then return to a general
parametrization such as (cf (19.65) and (17.95))
(
)
(
)
0
1
ˆ
φ =
√
φ
+ √
2 − ˆ
iφ
ˆ
1
,
(19.98)
v/ 2
ˆ
2
σ ˆ − iφ 3
and add ’t Hooft gauge-fixing terms
(
)
1
∑
−
ˆ μ
ˆ
(∂ μ W i + ξM W φ
2
ˆ
ˆ
ˆ
i ) + (∂
μ
μ Z + ξM Z φ 3 )
2 + (∂ μ A
μ )
2 . (19.99)
2ξ
i=1,2
In this case the gauge boson propagators are all of the form (19.74), and ξdependent. In such gauges, the Feynman rules will have to involve graphs
ˆ
corresponding to exchange of quanta of the ‘unphysical’ fields φ i , as well as
those of the physical Higgs scalar σ ˆ. These will also have to be suitable ghost
interactions in the non-Abelian sector as discussed in section 13.3.3. The
complete Feynman rules of the electroweak theory are given in Appendix B
of Cheng and Li (1984), for example.
The model introduced here is actually the ‘Higgs sector’ of the Standard
Model, but without any couplings to fermions. We have seen how, by supposing that the potential in (19.78) has the symmetry-breaking sign of the
parameter μ
2 , the W
± and Z
0 gauge bosons can be given masses. This seems
to be an ingenious and even elegant ‘mechanism’ for arriving at a renormalizable theory of massive vector bosons. One may of course wonder whether
this ‘mechanism’ is after all purely phenomenological, somewhat akin to the
GL theory of a superconductor. In the latter case, we know that it can be
derived from ‘microscopic’ BCS theory, and this naturally leads to the question whether there could be a similar underlying ‘dynamical’ theory, behind
the Higgs sector. It is, in fact, quite simple to construct a theory in which the
ˆ
Higgs fields φ appear as bound, or composite, states of heavy fermions.
But generating masses for the gauge bosons is not the only job that the
Higgs sector does, in the Standard Model: it also generates masses for all
/
19. Spontaneously Broken Local Symmetry
ˆ
ˆ
and this is why A μ does not acquire a mass when ‹0|φ|0› = 0 (gauge fields
ˆ
coupled to unbroken symmetries of ‹0|φ|0› do not become massive). Although
ˆ
certainly not unique, this choice of φ and ‹0| ˆ
φ|0› is undoubtedly very economical and natural. We are interpreting the zero eigenvalue of (1 + τ 3 ) as the
electromagnetic charge of the vacuum, which we do not wish to be non-zero.
We then make the identification
e = g sin θ W
(19.97)
in order to get the right ‘electromagnetic D μ ’ in (19.96).
We emphasize once more that the particular form of (19.87) corresponds
to a choice of gauge, namely the unitary one (cf the discussions in sections
19.3 and 19.5). There is always the possibility of using other gauges, as in
the Abelian case, and this will in general be advantageous when doing loop
calculations involving renormalization. We would then return to a general
parametrization such as (cf (19.65) and (17.95))
(
)
(
)
0
1
ˆ
φ =
√
φ
+ √
2 − ˆ
iφ
ˆ
1
,
(19.98)
v/ 2
ˆ
2
σ ˆ − iφ 3
and add ’t Hooft gauge-fixing terms
(
)
1
∑
−
ˆ μ
ˆ
(∂ μ W i + ξM W φ
2
ˆ
ˆ
ˆ
i ) + (∂
μ
μ Z + ξM Z φ 3 )
2 + (∂ μ A
μ )
2 . (19.99)
2ξ
i=1,2
In this case the gauge boson propagators are all of the form (19.74), and ξdependent. In such gauges, the Feynman rules will have to involve graphs
ˆ
corresponding to exchange of quanta of the ‘unphysical’ fields φ i , as well as
those of the physical Higgs scalar σ ˆ. These will also have to be suitable ghost
interactions in the non-Abelian sector as discussed in section 13.3.3. The
complete Feynman rules of the electroweak theory are given in Appendix B
of Cheng and Li (1984), for example.
The model introduced here is actually the ‘Higgs sector’ of the Standard
Model, but without any couplings to fermions. We have seen how, by supposing that the potential in (19.78) has the symmetry-breaking sign of the
parameter μ
2 , the W
± and Z
0 gauge bosons can be given masses. This seems
to be an ingenious and even elegant ‘mechanism’ for arriving at a renormalizable theory of massive vector bosons. One may of course wonder whether
this ‘mechanism’ is after all purely phenomenological, somewhat akin to the
GL theory of a superconductor. In the latter case, we know that it can be
derived from ‘microscopic’ BCS theory, and this naturally leads to the question whether there could be a similar underlying ‘dynamical’ theory, behind
the Higgs sector. It is, in fact, quite simple to construct a theory in which the
ˆ
Higgs fields φ appear as bound, or composite, states of heavy fermions.
But generating masses for the gauge bosons is not the only job that the
Higgs sector does, in the Standard Model: it also generates masses for all
/