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>
A particle is just an excitation of its own quantum field. For example, an electron is just an excitation of the electron field. Same for other fundamental particles.
The properties of these particles are intrinsically depends upon the behaviour and interation of respective quantum field with the other quantum fields. For example, an electron's mass is intrinsically affected by the coupling of electron field with the Higgs field and Higgs VEV. If the VEV and/or coupling of the Higgs field does depends on space-time coordinates (i.e., if it's not 246GeV everywhere and everytime?) and vary throughout the space-time than the mass of the fundamental particles like electron would be different place to place and time to time. See the point is the properties of a particle which depends on interactions of two or more fields also depends on the properties of participating fields. Higgs field has been hypothesized to have a constant 246GeV VEV which in turns says that every excitation of a field will have the identical mass, leaving all electrons with identical mass.