Question: In Peskin & Schroeder, p.523, they give the diagram contributing to the gluon self-energy that arises from the 3-gluon vertex, and they claim that the $1/2$ factor is a symmetry factor:

[![Gluon loop contributing to 1-loop gluon self-energy](https://i.stack.imgur.com/SNhke.png)](https://i.stack.imgur.com/SNhke.png)

How can this symmetry factor be calculated? I get all sorts of things except the correct result. The relevant part of the Lagrangian is:

$$\mathcal{L} = -\frac{1}{4} F^a\_{\mu\nu} F\_a^{\mu\nu}\tag{1}$$

with

$$F^a\_{\mu\nu} = \partial\_\mu A^a\_\nu - \partial\_\nu A\_\mu^a + gf^{abc} A\_\mu^b A\_\nu^c \tag{2}$$

Answer: 
The identity of fundamental particles is a basic hypothesis of all theories, but we don't really know this, neither can we disprove it. Just like the homogeneity and isotropy of spacetime. These assumptions are good starting points of physics theories and can make things harmonic, simple and elegant.

Related link: <https://www.quora.com/Are-fundamental-particles-of-the-same-type-all-identical-to-each-other-If-so-to-what-precision-and-why-should-that-be-the-case>
We have arrived at the [standard model of particle physics](https://en.wikipedia.org/wiki/Standard_Model) which encapsulates all particle data up to now. In the standard model the basic hypothesis is that the particles in the table are point particles, with specific quantum numbers.

[![enter image description here](https://i.stack.imgur.com/i6eJGm.png)](https://i.stack.imgur.com/i6eJGm.png)

Thus by construction of the theory there is no way to distinguish one elementary particle of the same type of particle, as they have no extent in spacetime to be able to do so.

When one starts with composites of elementary particles, as the proton and neutron, because they are in a specific quantum mechanical bound state, they have to be indistinguishable under the same boundary conditions. 

It is only at the level of complexity of solid state that matter can be distinguished into identifiable characteristics, the ones that lead to DNA identification.

So no, there could not be different mass electrons or other elementary particles as long as the standard model holds. The same holds true for composites of these elementary particles. 

Of course history of physics shows that when new frontiers are reached, unexpected effects may appear, but at the moment the standard model has not been falsified.