What is the relationship between the different aspects of mass?

Question

There seem to be several different ways in which mass manifests itself, in particular with reference to fundamental particles:

1. Gravitational mass.
2. Inertial mass.
3. The coupling to the Higgs field.
4. The energy and lifetime of virtual particles.

Values for the mass of the fundamental particles are shown on the table in the Standard Model.

As I understand it, gravitational mass relates to the energy of particles through $E=MC^2$ and inertia is a consequence of interaction with the Higgs field and therefore (I presume) a consequence of some constant property of the particle that must be determined by experiment.

But how does this relate to the mass/energy required to create a virtual particle? Is it again the coupling with the Higgs field (and therefore related to inertia) that affects energy required for a particle, or is that relationship indirect? Why are these values the same, if they are the same?

Answer 1

First there was the empirical mass, the weight, that humanity used for thousands of years, various civilizations using a standard of weight so as to count their produce.

Then Newton came and weight became a subset of one of his laws, as the result of the force of gravity on an intrinsic mass, given by the famous $F=ma$ , where m is the inertial mass of classical mechanics. This mass is conserved.

Then special relativity came for very high velocities and energies in general, where the much discussed $E=mc^2$ has a variable mass dependent on velocity, the relativistic mass, which is out of fashion for particle physics since it is not invariant under Lorentz transformations.

What is called the invariant mass is invariant under Lorentz transformations and is the length of the four vector describing an elementary particle or a system of particles given by $(E,p_x,p_y,p_z)$ .

In our present physics knowledge the mass of the elementary particles, is the mass assigned to them when the electroweak symmetry was broken back in the current cosmological model , a fixed invariant mass for each particle in the table.