421
22.8. The Higgs sector
22.8.1 Introduction
The Lagrangian for an unbroken SU(2) L × U(1) gauge theory of vector bosons
and fermions is rather simple and elegant, all the interactions being deter′
mined by just two Lagrangian parameters g and g in a ‘universal’ way. All
the particles in this hypothetical world are, however, massless. In the real
world, while the electroweak interactions are undoubtedly well described by
the SU(2) L × U(1) theory, neither the mediating gauge quanta (apart from the
photon) nor the fermions are massless. They must acquire mass in some way
that does not break the gauge symmetry of the Lagrangian, or else the renormalizability of the theory is destroyed, and its remarkable empirical success (at
a level which includes loop corrections) would be some kind of freak accident.
In chapter 19 we discussed how such a breaking of a gauge symmetry does
happen, dynamically, in a superconductor. In that case ‘electron pairing’ was
a crucial ingredient. In particle physics, while a lot of effort has gone into examining various analogous ‘dynamical symmetry breaking’ theories, none has
yet emerged as both theoretically compelling and phenomenologically viable.
However, a simple count of the number of degrees of freedom in a massive vector field, as opposed to a massless one, indicates that some additional fields
must be present in order to give mass to the originally massless gauge bosons.
And so, in the Standard Model, it is simply assumed, following the original
ideas of Higgs and others (Higgs 1984, Englert and Brout 1964, Guralnik et
al. 1964; Higgs 1966) that a suitable scalar (‘Higgs’) field exists, with a potential which causes the ground state (the vacuum) to break the symmetry
spontaneously. Furthermore, rather than (as in BCS theory) obtaining the
fermion mass gaps dynamically, they too are put in ‘by hand’ via Yukawa-like
couplings to the Higgs field.
It has to be admitted that this part of the Standard Model appears to be
the least satisfactory. Consider the Higgs couplings, which are listed in appendix Q, section Q.2.3. While the couplings of the Higgs field to the gauge
fields are all determined by the gauge symmetry, the Higgs self-couplings
(trilinear and quadrilinear) are not gauge interactions and are unrelated to
anything else in the theory. Likewise, the Yukawa-like fermion couplings are
not gauge interactions either, and they are both unconstrained and uncomfortably different in orders of magnitude. True, all these are renormalizable
couplings – but this basically means that their values are not calculable and
have all to be taken from experiment.
Such considerations may indicate that the ‘Higgs Sector’ of the Standard
Model is on a somewhat different footing from the rest of it – a commonly held
view, indeed. Perhaps it should be regarded as more a ‘phenomenology’ than
a ‘theory’, much as the current–current model was. In this connection, we may
mention a point which has long worried many theorists. In section 22.6 we
noted that figure 22.26 gives a quadratically divergent (O(Λ 2 )) and positive
contribution to the ˆ φ term in the Lagrangian, at one loop order. This term
φ
† ˆ
would ordinarily, of course, be just the mass term of the scalar field. But in
22.8. The Higgs sector
22.8.1 Introduction
The Lagrangian for an unbroken SU(2) L × U(1) gauge theory of vector bosons
and fermions is rather simple and elegant, all the interactions being deter′
mined by just two Lagrangian parameters g and g in a ‘universal’ way. All
the particles in this hypothetical world are, however, massless. In the real
world, while the electroweak interactions are undoubtedly well described by
the SU(2) L × U(1) theory, neither the mediating gauge quanta (apart from the
photon) nor the fermions are massless. They must acquire mass in some way
that does not break the gauge symmetry of the Lagrangian, or else the renormalizability of the theory is destroyed, and its remarkable empirical success (at
a level which includes loop corrections) would be some kind of freak accident.
In chapter 19 we discussed how such a breaking of a gauge symmetry does
happen, dynamically, in a superconductor. In that case ‘electron pairing’ was
a crucial ingredient. In particle physics, while a lot of effort has gone into examining various analogous ‘dynamical symmetry breaking’ theories, none has
yet emerged as both theoretically compelling and phenomenologically viable.
However, a simple count of the number of degrees of freedom in a massive vector field, as opposed to a massless one, indicates that some additional fields
must be present in order to give mass to the originally massless gauge bosons.
And so, in the Standard Model, it is simply assumed, following the original
ideas of Higgs and others (Higgs 1984, Englert and Brout 1964, Guralnik et
al. 1964; Higgs 1966) that a suitable scalar (‘Higgs’) field exists, with a potential which causes the ground state (the vacuum) to break the symmetry
spontaneously. Furthermore, rather than (as in BCS theory) obtaining the
fermion mass gaps dynamically, they too are put in ‘by hand’ via Yukawa-like
couplings to the Higgs field.
It has to be admitted that this part of the Standard Model appears to be
the least satisfactory. Consider the Higgs couplings, which are listed in appendix Q, section Q.2.3. While the couplings of the Higgs field to the gauge
fields are all determined by the gauge symmetry, the Higgs self-couplings
(trilinear and quadrilinear) are not gauge interactions and are unrelated to
anything else in the theory. Likewise, the Yukawa-like fermion couplings are
not gauge interactions either, and they are both unconstrained and uncomfortably different in orders of magnitude. True, all these are renormalizable
couplings – but this basically means that their values are not calculable and
have all to be taken from experiment.
Such considerations may indicate that the ‘Higgs Sector’ of the Standard
Model is on a somewhat different footing from the rest of it – a commonly held
view, indeed. Perhaps it should be regarded as more a ‘phenomenology’ than
a ‘theory’, much as the current–current model was. In this connection, we may
mention a point which has long worried many theorists. In section 22.6 we
noted that figure 22.26 gives a quadratically divergent (O(Λ 2 )) and positive
contribution to the ˆ φ term in the Lagrangian, at one loop order. This term
φ
† ˆ
would ordinarily, of course, be just the mass term of the scalar field. But in