I do not know what I should think about power divergences from loops.
Most QFT textbooks tell us how to deal with logarithmic divergences from loops $\sim\ln(\Lambda^2/\Delta)$: we can set a counterterm $\sim\ln(\Lambda^2/\mu^2)$ to cancel the divergence at some scale $\mu$, then at any other scale this stuff in terms of renormalized quantities are finite $\sim\ln(\mu^2/\Delta)$.
However, sometimes we encounter power divergences like $\sim\Lambda^2s$ in theories where we have momentum dependent interactions (such as a non-renormalizable theory with interaction $\phi^2\partial_\mu\phi\partial^\mu\phi$), where $s$ is one of the Mandelstam variables which depend on the external momenta. If we try to apply the same method to add counterterms, we may have a counterterm $\sim\Lambda^2\mu^2$ such that the power divergence is cancelled at energy scale $\mu$. But if we go to another scale with $s\neq \mu^2$, we will have $\Lambda^2(\mu^2-s)$, which is divergent and invalidates perturbation theory. This is very bad and it means we cannot cancel the infinity if we only have this type of counterterms.
Fortunately, there is another possibility: we may be able to add counterterms that gives $s\Lambda^2$ (for example, by field strength renormalization in the theory with interaction $\phi^2\partial_\mu\phi\partial^\mu\phi$), which is able to cancel the power divergences at all scales. This is nice, and it is even nicer than the case with logarithmic divergences, because we can cancel power divergences identically at all scales but we can cancel logarithmic divergences only at a particular scale that we choose to be our renormalization scale. However, my concern is that because we can only add terms that is compatible with the symmetries of the system, we may not be able to add some counterterms that are necessary for cancelling some power divergences but inconsistent with the symmetries.
My questions are:
1) Is this (the second type of counterterms) the only way that power divergences will be cancelled?
2) If yes, are we always able to set counterterms with respect to the symmetries of the system in this way, so that power divergences can always be cancelled at all scales (then we do not have to worry about power divergences at all!)?
3) If no, how should we cancel power divergences? Or if it cannot really be cancelled, what is its effect?
By the way, one may say if we use dimensional regularization, power divergences appear in the same way as logarithmic divergences (as $1/\epsilon$ where $\epsilon=4-d$), so we can apply the method of dealing with logarithmic divergences to them. But I think this is too tricky and I want to face this problem with other regularization schemes which have a cutoff.
Best Answer
Yes, you need to add counter terms that look like the divergences you find. So in this case, you are exactly right, you need to add a counter term corresponding to $\phi^2 \partial(\phi)^2$.
I'm not sure what symmetry you are worried about here, but often the power law divergences will break the symmetries you started with and will require you to add counterterms that break the symmetry you started with. In fact, this is precisely why people tend to focus on the logs, or use regularization methods like dim reg, where the power laws don't appear and you don't have this annoying problem.
The general principle is that you will need counterterms that violate the symmetry you started with, if your regularization method itself breaks the symmetry. An example is QED. Regulating with a hard cutoff violates gauge invariance, so the power law divergences also violate gauge invariance. (Just compute the one loop correction to the two point function for the photon, gauge invariance tells you the propagator should be proportional to $\eta_{\mu\nu}-p_\mu p_\nu$, but the power law divergences will give different relative contributions to those two terms).
The best solution is to use a regularization scheme that respects all the symmetries. Dim reg is usually a good choice. If you can't find such a regularization scheme, that may be a sign that the symmetry is anamolous and can't be maintained at the quantum level (eg--massless fermions have a chiral symmetry that is anamolous when you couple them to gauge fields).
If you insist on using a different regularization scheme, you can still get the right answer, but it will take a little more work. You will need to add counterterms that violate the symmetry. However the real question is that when you compute something physical (such as the S matrix), after you renormalize, will the physical quantities respect the symmetry? You will end up finding, if you do things correctly, that the failure of the counterterms to respect the symmetry will exactly cancel the failure of the divergences to respect the symmetry, and the final answer will be symmetric. I would consult Weinberg if you want to get a precise prescription on how to proceed.
The lesson is that to renormalize, you simply add the counterterms you need to cancel the divergences you find, without thinking about what they mean. Later on, you may need to do some re-interpreting to figure out exactly what the physics is.
Also, it is true that this interaction is renormalizable, so you will end up generating an infinite number of counterterms. However, depending on what you are trying to do, that's not necessarily as bad as it sounds.