Trace of Fermion Loops in Effective Field Theories

Question

I'd like to know whether we need to take the trace of fermion loops in effective theory in the same way that we need to do so for renormalizable theories. At first thought, it seems obvious that should need to since we are dealing with fermions, but I recently saw a calculation and the professor did not bother to take the trace. Here the particular problem at hand:

The theory at hand is, \begin{equation} {\cal L} = \bar{\psi} ( i \partial _\mu \gamma ^\mu - m ) \psi + \frac{1}{2} ( \partial _\mu \phi ) ^2 - \frac{1}{2} M ^2 \phi ^2 + g \phi \bar{\psi} \psi \end{equation} At low energies we have, \begin{equation} {\cal L} = \bar{\psi} ( i \partial _\mu \gamma ^\mu - m ) \psi + \frac{ a }{ M } ( \bar{\psi} \psi ) ^2 + ... \end{equation} Doing matching between the theories at tree level we find that,$ a = g ^2 $. We then move on to calculate the diagram,

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which the professor wrote as (omitting the external legs), \begin{equation} \frac{ i g ^2 }{ M ^2 } \int \frac{ \,d^4k }{ (2\pi)^4 } \frac{ k_\mu\gamma^\mu + m }{ k ^2 - m ^2 } \end{equation} Not taking the trace of the diagram changes the final result by a factor of $ 4 - \epsilon $ (using dim-reg). On the one hand, this makes sense since we didn't need to take a trace in the high energy theory where the corresponding diagram is

$\hspace{6cm}$

on the other hand these are fermions after all.

Is the professor just being sloppy here or do you not need to take the trace for some reason I am missing?

For more context please see my notes under Effective Field Theory in this link (Eq. 4.6)

Answer 1

You don't need to trace over the spinor indices because the self-energy has some: $\Sigma_{ab}$.

But first, let's see why there are traces in the `usual' case of QED. In this case, the fermion-photon index contains a matrix $\gamma^\mu_{ab}$. (I will omit all sign, factor $i$ and $e$, etc.) Therefore, the QED diagram equivalent to the second diagram of the OP's question contains (G_{cd} is the fermionic propagator) $$\gamma^\mu_{ac}G_{cd}\gamma^\nu_{db}, $$ where repeated indices are summed over (this is the usual trace over the spinorial indices) and there should also be a photon propagator.

Now, let's see what happens with the $g\phi\bar\psi\psi$ vertex. There is no matrix $\gamma^\mu_{ab}$, but instead the trivial $\delta_{ab}$. Therefore, the trace over spinor indices of the second diagram is trivial: $$g^2\delta_{ac}G_{cd}\delta_{db}=G_{ab}, $$ and we recover the formula of the OP's professor.

Finally, what about the four-fermions interaction? In this case, the vertex is given by $\frac{a}{M}\delta_{ab}\delta_{cd}$ where the left indices correspond to the $\bar\psi$'s and the right indices to the $\psi$'s. The first diagram gives $$(\delta_{ab}\delta_{cd}+\delta_{ac}\delta_{db})G_{cd} ,$$ and the trace is once again trivial.