What defines the mass of elementary particle?

The electron is particle. The mass of electron is $9.10938215(45)\times 10^{−31}\, {\rm kg}$.

But why is the mass exactly what it is?

What in physics defines the mass of elementary particle?

Answer

Dear GJ, the mass of every electron is exactly the same - as long as our experiments and theories may say - and the same value of mass manifests itself

1. 
   

   in the energy/mass conservation law;

   
   


2. 
   

   in the source of the gravitational field that the object containing massive particles such as the electron produces;

   
   


3. 
   

   in the resistance towards acceleration that the massive objects have.

   
   

All those versions of the "mass" are the same quantity.

In our modern theories of the world, the mass of the electron is the coefficient of a "mass term" in the Lagrangian or Hamiltonian, $-m\bar\psi\psi$, that is almost fundamental and can't have any deeper explanation.

Well, almost. In fact, in the "electroweak theory" - unifying all electromagnetic phenomena (including light) with the weak nuclear interactions (notorious as the origin of the beta-decay) - the mass of electrons (and other particles) arises from the interactions with the Higgs field. So the actual term is actually $$-y h \bar\psi\psi$$ where $\psi$ is the Dirac field for the electron. Here, $h$ is the Higgs field which takes the nonzero value of $h=v$ in the vacuum - the spontaneous symmetry breaking (Higgs mechanism). This value of $v$, about $247$ GeV, is universal for all particles. However, the difficult information about the electron mass is moved to the parameter $y$, the so-called Yukawa coupling, which is dimensionless (no units) and whose value is much smaller than one.

When the Universe is described by string theory, the value of the Yukawa coupling $y$ and the Higgs value in the vacuum, $v$, can be in principle calculated from first principles (in the natural units, e.g. the Planck units, not in kilograms - the unit of "kilogram" was a randomly chosen social convention that the farmers chose to sell vegetables, without any justification why it's not e.g. $1.3$ times different).

However, there are many (but discretely) possible Universes according to string theory and we can't say which one is right, so in practice, we can't calculate the value of the electron mass from purely theoretical considerations. So far, we need to measure it to know the value. Chances are that the situation won't change for quite some time.