In QCD, the gauge-invariant Lagrangian under the transformation

$$ \psi \to \psi' = e^{ig T^a \theta^a(x)} \psi$$

is written as:

$$\mathcal{L} = \bar{\psi}(i\gamma^\mu D_\mu - m)\psi - \frac{1}{4}G^a_{\mu\nu}G_a^{\mu\nu}$$

where the covariant derivative is:

$$D_\mu = \partial_\mu - ig T^a G^a_\mu$$

and the field strength tensor is defined as:

$$ G^a_{\mu\nu} = \partial_\mu G^a_\nu - \partial_\nu G^a_\mu + g f_{abc} G^b_\mu G^c_\nu $$

If I impose the gauge-invariance, I find that the gauge field transforms as:

$$ G^a_\mu \to G'^a_\mu = G^a_\mu + \partial_\mu \theta^a $$

Am I correct? I think I am, but if I look at how the field strength transforms, I expect it to remain invariant, but instead I find an extra term:

$$ G^a_{\mu\nu} \to G^a_{\mu\nu} + g_s f_{abc}(\partial_\mu \theta^b \partial_\nu \theta^c + \partial_\mu \theta^b G^c_\nu + \partial_\nu \theta^c G^b_\mu) $$

Does this term vanish? If does then Why? Or am I totally wrong on the transformation of the gauge field?

In QCD, the gauge-invariant Lagrangian under the transformation

$$ \psi \to \psi' = e^{ig T^a \theta^a(x)} \psi$$

is written as:

$$\mathcal{L} = \bar{\psi}(i\gamma^\mu D_\mu - m)\psi - \frac{1}{4}G^a_{\mu\nu}G_a^{\mu\nu}$$

where the covariant derivative is:

$$D_\mu = \partial_\mu - ig T^a G^a_\mu$$

and the field strength tensor is defined as:

$$ G^a_{\mu\nu} = \partial_\mu G^a_\nu - \partial_\nu G^a_\mu + g f_{abc} G^b_\mu G^c_\nu $$

If I impose the gauge-invariance, I find that the gauge field transforms as:

$$ G^a_\mu \to G'^a_\mu = G^a_\mu + \partial_\mu \theta^a $$

Am I correct? I think I am, but if I look at how the field strength transforms, I expect it to remain invariant, but instead I find an extra term:

$$ G^a_{\mu\nu} \to G^a_{\mu\nu} + g_s f_{abc}(\partial_\mu \theta^b \partial_\nu \theta^c + \partial_\mu \theta^b G^c_\nu + \partial_\nu \theta^c G^b_\mu) $$

Does this term vanish? If does then Why? Or am I totally wrong on the transformation of the gauge field?

In QCD, the gauge-invariant lagrangian under the trasformation

$$ \psi \to \psi' = e^{ig T^a \theta^a(x)} \psi$$

is written as:

$$\mathcal{L} = \bar{\psi}(i\gamma^\mu D_\mu - m)\psi - \frac{1}{4}G^a_{\mu\nu}G_a^{\mu\nu}$$

where the covariant derivative is:

$$D_\mu = \partial_\mu - ig T^a G^a_\mu$$

and the field strength tensor is defined as:

$$ G^a_{\mu\nu} = \partial_\mu G^a_\nu - \partial_\nu G^a_\mu + g f_{abc} G^b_\mu G^c_\nu $$

If I impose the gauge-invariance, I find that the gauge field transforms as:

$$ G^a_\mu \to G'^a_\mu = G^a_\mu + \partial_\mu \theta^a $$

Am I correct? I think I am, but if I look at how the field strength transforms, I expect it to remain invariant, but instead I find an extra term:

$$ G^a_{\mu\nu} \to G^a_{\mu\nu} + g_s f_{abc}(\partial_\mu \theta^b \partial_\nu \theta^c + \partial_\mu \theta^b G^c_\nu + \partial_\nu \theta^c G^b_\mu) $$

Does this term vanish? If does then Why? Or am I totally wrong on the transformation of the gauge field?

In QCD, the gauge-invariant Lagrangian under the transformation

$$ \psi \to \psi' = e^{ig T^a \theta^a(x)} \psi$$

is written as:

$$\mathcal{L} = \bar{\psi}(i\gamma^\mu D_\mu - m)\psi - \frac{1}{4}G^a_{\mu\nu}G_a^{\mu\nu}$$

where the covariant derivative is:

$$D_\mu = \partial_\mu - ig T^a G^a_\mu$$

and the field strength tensor is defined as:

$$ G^a_{\mu\nu} = \partial_\mu G^a_\nu - \partial_\nu G^a_\mu + g f_{abc} G^b_\mu G^c_\nu $$

If I impose the gauge-invariance, I find that the gauge field transforms as:

$$ G^a_\mu \to G'^a_\mu = G^a_\mu + \partial_\mu \theta^a $$

Am I correct? I think I am, but if I look at how the field strength transforms, I expect it to remain invariant, but instead I find an extra term:

$$ G^a_{\mu\nu} \to G^a_{\mu\nu} + g_s f_{abc}(\partial_\mu \theta^b \partial_\nu \theta^c + \partial_\mu \theta^b G^c_\nu + \partial_\nu \theta^c G^b_\mu) $$

Does this term vanish? If does then Why? Or am I totally wrong on the transformation of the gauge field?

In QCD, the gauge-invariant lagrangian under the trasformation

$ \psi \to \psi' = e^{ig T^a \theta^a(x)} \psi$

is written as:

$\mathcal{L} = \bar{\psi}(i\gamma^\mu D_\mu - m)\psi - \frac{1}{4}G^a_{\mu\nu}G_a^{\mu\nu}$

where the covariant derivative is:

$D_\mu = \partial_\mu - ig T^a G^a_\mu$

and the field strength tensor is defined as:

$ G^a_{\mu\nu} = \partial_\mu G^a_\nu - \partial_\nu G^a_\mu + g f_{abc} G^b_\mu G^c_\nu $

If I impose the gauge-invariance, I find that the gauge field transforms as:

$ G^a_\mu \to G'^a_\mu = G^a_\mu + \partial_\mu \theta^a $

Am I correct? I think I am, but if I look at how the field strength transforms, I expect it to remain invariant, but instead I find an extra term:

$ G^a_{\mu\nu} \to G^a_{\mu\nu} + g_s f_{abc}(\partial_\mu \theta^b \partial_\nu \theta^c + \partial_\mu \theta^b G^c_\nu + \partial_\nu \theta^c G^b_\mu) $

Does this term vanish? Why? Or am I totally wrong on the transformation of the gauge field...?

In QCD, the gauge-invariant lagrangian under the trasformation

$$ \psi \to \psi' = e^{ig T^a \theta^a(x)} \psi$$

is written as:

$$\mathcal{L} = \bar{\psi}(i\gamma^\mu D_\mu - m)\psi - \frac{1}{4}G^a_{\mu\nu}G_a^{\mu\nu}$$

where the covariant derivative is:

$$D_\mu = \partial_\mu - ig T^a G^a_\mu$$

and the field strength tensor is defined as:

$$ G^a_{\mu\nu} = \partial_\mu G^a_\nu - \partial_\nu G^a_\mu + g f_{abc} G^b_\mu G^c_\nu $$

If I impose the gauge-invariance, I find that the gauge field transforms as:

$$ G^a_\mu \to G'^a_\mu = G^a_\mu + \partial_\mu \theta^a $$

Am I correct? I think I am, but if I look at how the field strength transforms, I expect it to remain invariant, but instead I find an extra term:

$$ G^a_{\mu\nu} \to G^a_{\mu\nu} + g_s f_{abc}(\partial_\mu \theta^b \partial_\nu \theta^c + \partial_\mu \theta^b G^c_\nu + \partial_\nu \theta^c G^b_\mu) $$

Does this term vanish? If does then Why? Or am I totally wrong on the transformation of the gauge field?