Ghosts in QCD Lagrangian

Question

The QCD Lagrangian is $$ \mathcal{L}_{\text{QCD}} = -\dfrac{1}{2} \text{Tr}\, G_{\mu\nu}G^{\mu\nu} + \sum_i^{N_f} \bar{q_i} \left(i \gamma^\mu \mathcal{D}_\mu - m_i\right)\,q_i, \tag{1} $$ where $\mathcal{D}_\mu = \partial_\mu + ig G_\mu$, the trace has an implicit factor of $1/2$, $G_\mu$ is the gluonic field and $G_{\mu\nu}$ is the gluonic field strength.

Now, in my QFT course, we've seen that for quantizing a $SU(3)$ QFT, we must use the Faddeev-Popov method, which introduces ghosts through the change of the measure of the path integral. Now, I do not see any trail of these (very spooky!) ghosts in the all-present QCD Lagrangian (1).

Could someone please explain me why? Is (1) supposed to be a "classical" Lagrangian that we still have to "properly quantize"? Or are they hidden somewhere else? Are they re-introduced after doing any gauge-fixing? We shouldn't be able to ignore them, as they contribute to amplitudes!

Answer 1

It's conventional to specify the classical Lagrangian, which does not include ghost terms. (Ghosts only contribute at loop level).

One reason not to write ghost terms, when one is speaking generically about the classical Lagrangian of QCD, is that the ghost terms depend on your gauge fixing procedure, so it depends on how you choose to do the calculation, while the classical gauge invariant QCD Lagrangian does not.

Another reason is simplicity; if you are writing the classical QCD Lagrangian down, it is probably just to make some very general, high-level points about the degrees of freedom or couplings, or to fix conventions. In other words, equations are tools to express ideas, and you don't want to write an equation down that has superfluous technical content you don't need for your discussion. As an extreme example, the Standard Model can fit on a coffee cup, but only if you're willing to use a very abstract notation that leaves very many technical details implicit. However, the abstract notation is fine, for "coffee cup level physics", in the sense that it expresses there is a single equation that can be used to derive all particle physics results, and the various conceptual pieces of the Standard Model are represented.

On the other hand, if you are writing a paper about loop level calculations in QCD, you should specify how you are doing the gauge fixing and show the ghost Lagrangian explicitly, but then the classical QCD Lagrangian will only be one of many equations in your paper.

Answer 2

• Briefly speaking, OP's Lagrangian density (1) is the un-gauge-fixed original Lagrangian density ${\cal L}_0$ for QCD. ${\cal L}_0$ defines the classical theory.

• Now let us quantize the theory. In the exponent of the Faddeev-Popov path integral, the full Lagrangian density $$ {\cal L}~=~{\cal L}_0+{\cal L}_{FP}+{\cal L}_{gf}$$ also contains a Faddeev-Popov ghost term ${\cal L}_{FP}$ and a gauge-fixing term ${\cal L}_{gf}$.

• For the analougous situation in QED, see e.g. this and this Phys.SE posts.