Quantum Mechanics for Mathematicians

defining representations. The Hamiltonian is

hHiggs=

∫

R^3

(|Π|^2 +|(∇−iA)Φ|^2 −m^2 |Φ|^2 +λ|Φ|^4 )d^3 x

whereAare vector potential fields forU(1)⊗SU(2) acting in the defining
representation, scaled by the coefficientsg 1 ,g 2.
Note the unusual sign of the mass term and the existence of a quartic term.
This means that the dynamics of such a theory cannot be analyzed by the
methods we have used so far. Thinking of the mass term and quartic term as
a potential energy for the field, for constant fields this will have a minimum at
some non-zero values of the field. To analyze the physics, one shifts the field Φ
by such a value, and approximates the theory by a quadratic expansion of the
potential energy about that point. Such a theory will have states corresponding
to a new scalar particle (the Higgs particle), but will also require a new analysis
of the gauge symmetry, since gauge transformations act nontrivially on the space
of minima of the potential energy. For how this “Anderson-Higgs mechanism”
affects the physics, one should consult a standard textbook.
Finally, the Higgs field and the spinor fields are coupled by cubic terms called
Yukawa terms, of a general form such as

hY ukawa=

∫

R^3

MΦΨ†Ψd^3 x

where Ψ,Ψ†are the spinor fields, Φ the Higgs field andMa complicated matrix.
When one expresses this in terms of the shifted Higgs field, the constant term
in the shift gives terms quadratic in the fermion fields. These determine the
masses of the spin^12 particles as well as the so-called mixing angles that appear
in the coupling of these particles to the gauge fields.

48.4 Unanswered questions and speculative ex-

tensions

While the Standard Model has been hugely successful, with no conflicting ex-
perimental evidence yet found, it is not a fully satisfactory theory, leaving unan-
swered a short list of questions that one would expect a fundamental theory to
address. These are:

48.4.1 Why these gauge groups and couplings?

We have seen that the theory has aU(1)×SU(2)×SU(3) gauge group acting
on it, and this motivates the introduction of gauge fields that take values in
the Lie algebra of this group. An obvious question is that of why this precise
pattern of groups appears. When one introduces the Hamiltonian 48.1 for these
gauge fields, one gets three different coupling constantsg 1 ,g 2 ,g 3. Why do these
have their measured values? Such coupling constants should be thought of as