Accordingly to Yang-Mills theories, after the introduction of a covariant derivative such that
Dμ=∂μ−igAμ,
you can built the kinetic term for the gauge potential Aμ as
LA=−12tr{FμνFμν},Fμν=ig[Dμ,Dν].
The action of the covariant derivative transforms under a local group tranformation Ω in the following way:
Dμψ→ΩDμψ.
And the gauge field as
Aμ→ΩAμΩ†+igΩ∂μΩ†.
Introducing Eq. (4) in Eq. (1) you get that under the group tranformations, the covariant derivate transforms as,
Dμ→∂μ−igΩAμΩ†+Ω∂μΩ†=ΩDμΩ†+∂μ.
Eq. (5) into definition of strength tensor in Eq. (2) gives
Fμν→ΩFμνΩ†+ig(∂μ(ΩDνΩ†)−∂ν(ΩDμΩ†)).
Taking Ω≃1−gθiTi, with {Ti} the set of generators of the group, θi∈R and fijk the structure constants:
Fμν→ΩFμνΩ†+ig(∂μAkν−∂νAkμ)Tk+ig2fajkTk[∂μ(θjAaν)−∂ν(θjAaμ)].
In Particle Physics books, it is said that Fμν→ΩFμνΩ†,
Answer
Ok, I think I found the answer. Eq. (5) has to be written as:
Dμ→∂μ+Ω(∂μΩ†)−igΩAμΩ†
As you can deduce from Eq. (4) that is a consequence of imposing Dμψ→ΩDμψ. Nevertheless, you can re-write Eq. (A) as:
Dμ→(Ω∂μ)Ω†−igΩAμΩ†=ΩDμΩ†
After considering,
(Ω∂μ)Ω†=Ω(∂μΩ†)+ΩΩ†∂μ=Ω(∂μΩ†)+∂μ
So with this, everything makes sense and the strength tensor is perfectly gauge-covariant, keeping in mind Eqs. (B)-(C) when interpreting OP's Eq. (8).
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