48.4.5 Why the Yukawas?
The fundamental fermion masses and mixing angles in the Standard Model are
determined by Yukawa terms in the Hamiltonian coupling the Higgs field to the
fermions. These terms involve matrices with a significant number of parameters
and the origin of these parameters is unknown. This is related to the mystery
of the Higgs field itself, with our understanding of the nature of the Higgs field
not able to constrain these parameters.
48.4.6 What is the dynamics of the gravitational field?
Our understanding of gravitational forces in classical physics is based on Ein-
stein’s theory of general relativity, which has fundamental degrees of freedom
that describe the geometry of space-time (which is no longer just Minkowski
space-time). These degrees of freedom can be chosen so as to include a connec-
tion (called the “spin connection”) and its curvature, much like the connection
variables of gauge theory. The fields of the Standard Model can be consistently
coupled to the space-time geometry by a minimal coupling prescription using
the spin connection. The Hamiltonian with Einstein’s equations as equations of
motion is however not of the Yang-Mills form. Applying standard perturbation
theory and renormalization methods to this Hamiltonian leads to problems with
defining the theory (it is not asymptotically free). There are a number of pro-
posals for how to deal with this problem and consistently handle quantization of
the space-time degrees of freedom, but none so far have any compelling evidence
in their favor.
48.5 For further reading
The details of the Standard Model are described in just about every textbook
on high energy physics, and most modern textbooks (such as [67]) of relativistic
quantum field theory.