In Chapter 1 of his book String Theory in a Nutshell, Kiritsis states the following.
The [Standard M]odel is unstable as we increase the energy (hierarchy problem of mass scales) and the theory loses predictivity as one starts moving far from current accelerator energies and closer to the Planck scale. Gauge bosons are protected from destabilizing corrections because of gauge invariance. The fermions are equally protected due to chiral symmetries. The real culprit is the Higgs boson.
Let $\phi$ be a quantum field and define $G(p,q):=\langle \widehat{\phi}(p)\widehat{\phi}(q)\rangle$ (this is of course a vacuum expectation value). Then, it turns out that, $G(p,q)=\delta (p+q)H(p^2)$. Under the assumption $H$ has a unique pole which is a non-negative real number (anybody have a proof of why this would be true in general?), define the mass of $\phi$ to be the square-root of this pole.
My understanding was that this mass is given to us by experiment; instead, in calculations, we use this condition to calculate the counter-terms, not the mass. Is my understanding of this in-correct?
For some reason I was interpreting his words to mean something along the lines of "In quantum field theory, you can calculate particle masses via perturbation theory, and sometimes certain symmetries tells us that anomalies cannot arise, but in the case of the Higgs boson, there is no such symmetry, and so for the result to come out correct, some crazy cancellation has to occur." (now that I think about it, my assumption regarding his meaning here must have been heavily influenced by what I have been told about the Hierarchy Problem from other sources). This of course, doesn't jive with my aforementioned understanding of masses in quantum field theory (i.e. that you don't calculate them).
Can somebody explain to me the Hierarchy Problem given my understanding of mass in quantum field theory? What precisely are the theorems that "protect" gauge bosons and fermions? I've seen time and again people talk about some how finely-tuned cancellation needs to happen regarding the Higgs. Could somebody tell me very explicitly just what this cancellation is, and where and how it comes up? How does this imply the instability of the Standard Model?
Thanks much in advance.