How is there no hierarchy problem without UV cutoff?

Question

I can understand the quadratic divergent corrections to Higgs bare mass which is referred to as the hierarchy problem.

But I don't understand how there won't be any hierarchy problem if we do not introduce a UV cutoff in the loop integrals, why won't we have such a problem when we integrate up to infinity...how can see it mathematically?

Answer 1

The specific regularization scheme or its concomitant cutoff scale $\Lambda$ ($\frac{1}{\epsilon}$ in DR) has nothing to do with the hierarchy problem.

I don't blame you, actually even some professional physics papers got confused (for instance, some might erroneously argue that there is no hierarchy problem with dimensional regularization).

The hierarchy problem has to be framed in the context of beyond standard model physics. You have to distinguish between 5 mass scales, namely

1. $m$: the mass of the particle in concern, e.g. Higgs mass $m_H$.
2. $\Lambda$: the UV cutoff scale of the regularization scheme (in dimensional regularization (DR), $\frac{1}{\epsilon}$ plays the role of $\Lambda$, where $\epsilon = d -4$). At the end of the renormalization procedure, $\Lambda$ can be safely sent to infinity (or $\epsilon$ sent to zero in DR), thanks to the painstakingly crafted counter terms.
3. $Q$: the energy scale of the incoming/outgoing particles involved in a scattering process.
4. $\mu$: the renormalization scale, which is an arbitrary scale to anchor the scattering amplitude (or coupling 'constant') as a function of $\frac{Q}{\mu}$ (or $ln(\frac{Q}{\mu}$)). The renormalization scale $\mu$ is a fiat scale that is set forth by human convention/convenience. Usually $\mu$ is set to the typical energy scale $Q_0$ of a scattering process.
5. $M$:the mass scale where beyond standard model (BSM) physics effect comes into the picture. $M$ could be either the grand unification scale $M_{GUT}$ or Planck scale $M_P$. In the effective field theory framework, the BSM Langrangian terms are suppressed by a factor of $(\frac{Q}{M})^n$, with $n>0$.

Assuming that there are BSM Langrangian terms, the hierarchy problem concerns the uncanny fine-tuning to arrive at the tiny value of ${m}$ compared with ${M}$, unless there is a spontaneously broken symmetry (technical naturalness) constraining the otherwise large BSM quantum loop corrections (of order $M$) to $m$.

As you can see, the hierarchy problem has to do with $M$ of energy scale #5, but not $\Lambda$ of energy scale #2. If there is no $M$, "the quadratic divergent corrections to Higgs bare mass" mentioned in OP is of the order $O(\Lambda^2)$, which can be canceled out by the $\Lambda$-dependent mass counter term. And the cutoff $\Lambda$ can be safely sent to infinity without any issue. Thus there is no hierarchy problem if there is no $M$.

Answer 2

The hierarchy problem in its basic form is a statement about large differences in physical scales. If a theory has no UV cut-off (effectively sending it to infinity), and just a single physical scale (in this case the electro-weak scale), there are no scales to compare, and there can be no hierarchy. The single physical scale is just a parameter of the theory.

Now we have good reason to believe the Standard Model is only valid up to a certain cut-off scale $\Lambda$, say at most the Planck scale. Now you can talk about a hierarchy, and look for an explanation as why these scales are separated, even though from an effective field theory point of view it would naively seem that the electro-weak scale should be $\mathcal{O}(\Lambda)$.

Answer 3

You simply have to introduce a UV cutoff to define the theory. The Standard Model Lagrangian is not UV-finite - there are divergent diagrams in perturbation theory, so it's impossible to "integrate up to infinity" unless you introduce a regulator.