"Dimensional analysis" arguments in quantum field theory

A good way to do dimension analysis in computing amplitudes relies on a good power-counting of the action. Let me explain how it works before answering your question. For simplicity, in the following, I will consider a QFT in $d=4$.

Usually, it is very useful to distinguish between couplings, decay constant and mass scales. In order to do so, we restore the dimensions of $\hbar$. What I mean is that the action of the system has to have dimension of $\hbar$, namely

$$ [\mathcal{S}] = [\hbar].\qquad\qquad\qquad (1) $$

Let us consider the very simple example of a One Coupling One Scales scalar theory (1C1S scalar theory), characterized by one coupling $g_*$ and one scale $\Lambda$ (that you interpret as a cut-off which may drive the derivative expansion of the EFT in powers of $\partial/\Lambda$).

Having restored the powers of $\hbar$, the dimensions of the fields are not just those of powers of energy. Indeed, from the kinetic term

$$ [\hbar]=[\mathcal{S}] = [\int d^4x \, (\partial\phi)^2] = M^{-4} M^2 [\phi]^2 $$

where $M$ just means "dimension of a mass". From this dimensional analysis we see that $[\phi] = \hbar^{1/2} M $, and analogously for spinorial fields $[\Psi] = \hbar^{1/2} M^{3/2}$.

We define then a coupling $g_*$ whose dimension is $[g_*] = [\hbar]^{-1/2}$. For the scalar theory, you can see this as an equivalent definition of $g_*$ as the square of the coefficient of the marginal $\phi^4$ interaction. Indeed, the $\phi^4$ interaction has the correct dimension as in Eq.(1)

$$ \int d^4 x\, [g_*^2 \phi]^4 = M^{-4} [\hbar]^{-1} M^4 [\hbar]^{2} = [\hbar]. $$

You see then the Lagrangian can be written as

$$ \mathcal{L}(\phi,\partial\phi) = \frac{\Lambda^4}{g_*^2}\hat{\mathcal{L}}\left(g_* \frac{\phi}{\Lambda},\frac{\partial}{\Lambda}\right) $$ where $\hat{\mathcal{L}}$ is a dimensionless function of its dimensionless arguments. In general, any insertion of the field can carry a $O(1)$-coefficient that is not fixed by the power-counting (these coefficients will be called $a_1,a_2,a_3....$; see below). So, for example, you can write the most general terms compatible with the symmetries of the theory. In this case, we do not have any symmetry, and we can write

\begin{align} \mathcal{L} &\supset \frac{\Lambda^4}{g_*^2}\left( a_1 g_*^2 \frac{(\partial\phi)^2}{\Lambda^4} + a_2 \,g_*^2 \Lambda^{-2} \phi^2 + a_3\, g_*^3 \Lambda^{-3}\phi^3+a_4\, g_*^4 \Lambda^{-4} \phi^4 + a_5\, g_*^4 \Lambda^{-6}(\partial\phi)^2\phi^2\right)\\ &= a_1(\partial\phi)^2 + a_2 \, \Lambda^{2} \phi^2 + a_3\, g_* \Lambda\phi^3+a_4\, g_*^2 \phi^4 + a_5\, g_*^2 \frac{(\partial\phi)^2\phi^2}{\Lambda^2}\,.\qquad \qquad \qquad(2) \end{align}

To have a canonically normalized scalar field we can set $a_1=1/2$.

Let's come to your question. First of all, you have to identify the correct dimension of the scatting amplitude (and this depends on the definition of the S-matrix and the normalization of the states of the Hilbert space). I use the same relativistic normalization of states and definition of $S-$matrix as in the Schwartz's book of QFT. For a $2\rightarrow 2$ scattering in $d=4$, the amplitude has dimension

$$ [\mathcal{A}_{2\rightarrow 2}] = [\hbar]^{-1} = [g_*]^2\,\qquad \qquad \qquad (3) $$ Indeed, my scattering amplitude contribution from $g_*^2\phi^4$ goes like $\sim g_*^2$

Please, note the dimension of the amplitude depends on the space-time dimensions and the number of scattered particles. In general $d$ dimension the $n-$point amplitude has dimension $d-n\,d/2+n$.

Eq.(3) means that terms like $g_*^2 (E^2/\Lambda^2 + E^4/\Lambda^4)$ are generically allowed if there are irrelevant operators which contribute at these orders in the energy; here $E$ is the energy of the center of mass $E= \sqrt{s}$ with $s$ the Mandelstam variable.

In the example of 4-Fermi theory, you can identify (in my normalization)

$$ G\simeq \frac{g_*^2}{\Lambda^2} $$

If you have the operator $ \frac{g_*^2}{\Lambda^2} (\bar{\psi}\gamma^\mu \psi)^2$, the only contribution at tree-level is $g_*^2\, s/\Lambda^2 $. The contribution proportional to $G^2$ comes from a loop. This is because loops carry a power of $\hbar$ which cancels the dimension of the extra $G$ insertion. When dealing with loops and renormalization, you have to fix a renormalization scheme.

$$ \text{loop contribution}\simeq \frac{g_*^4}{\Lambda^4} (\#) $$

What do we put in $(\#)$ ?

If you use cut-off regularization (i.e. you cut-off the momenta integral of the loop) you can have a combination of powers of $s$ and $\Lambda$; However, these contributions are local and can be eliminated by choosing counter-terms; they are not physical. The only physical contribution from the loop-integral is a logarithm, namely $s^2 log(s/\Lambda^2)$.

I believe that Zee is discussing two different scenarios in a very general way, without entering in details and without discussing renormalization of fields

• Very high cut-off (but you know the theory has a physical cut-off), small probing energies. The loop integral $I = \int d^4 k \frac{1}{k}\frac{1}{k+p}$ where $p$ is an external momenta, has dimension $2$. Then, the result can be $s$ or $\Lambda^2$. In the assumption $s\ll \Lambda$ you take $\Lambda^2$ possibly multiplying a log.

• You might believe your theory is UV-completed and that it can be probed at any energy; namely, we do not have a cut-off and the only dimensionful quantity is the center of mass energy (and maybe the renormalization scale or particle masses which are however assumed to be small); then, the result of the loop can be $s$ possibly multiplying a log.

Note that in a massive (UV-completed) QFT the amplitude is constrained by the Froissart-Bound to be bounded from above, namely $|\mathcal{A}(s)| < s\cdot log^2(s)$.

I agree that Zee's presentation is unsatisfactory, and like Srednicki's presentation (Chapter 18) better. Rather than just making hand-waving arguments based on dimensional analysis, Srednicki presents the dimensional analysis as just a useful shortcut for figuring out whether a given Feynman diagram contains a divergent loop integal, using some straightforward but non-obvious graph-theory arguments. You might also like Lecture 1 of these notes, which gives a rougher argument in the spirit of Zee's, but which I think is much clearer than Zee's.