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In this paper, the third paragraph of the “Introduction” says that the Standard Model by itself is a natural theory. As I understand, they say there is no quadratic divergence in the Standard Model unless it is extended. But if I understand the gauge hierarchy problem correctly, then it is a problem with the minimal Standard Model where the Higgs mass receives a radiative correction quadratic in the cut-off $\Lambda$. I don't know why they say that the hierarchy problem arises only when the SM must be extended with extra particles?

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Those who write than the Standard Model (SM) is natural interpret it as a fundamental theory, rather than an effective theory or effective field theory, with an unphysical cut-off taken to infinity, $\Lambda\to\infty$.

The bare parameters and loop corrections diverge, but are in any case considered unphysical. Only renormalized Lagrangian parameters (presumably in an MS scheme) and finite loop corrections are considered physical - renormalization is a merely a mathematical trick to remove infinities, which are seen as unphysical artefacts in a calculation (cf. Wilson's ideas about renormalization).

There are no quadratic corrections in the relationship between physical masses and renormalized Lagrangian parameters. Thus, if the finite loop corrections are small, the theory is said to have "physical naturalness" or "finite naturalness". Of course, there must be physics beyond the SM and e.g. Strumia et al attempted to incorporate gravity, inflation and unification in a fundamental theory, without breaking finite naturalness.

This isn't without critics (see e.g the discussion in arXiv:1506.03786). Interpreting a quantum field theory (QFT) as a fundamental theory makes it impossible to explain the values of nature's physical parameters in a more fundamental theory (e.g. string theory). They just are what they are, and there is no physical mechanism that made them that way.

Furthermore, with the above comment in mind, we require extreme fine-tuning such that physical masses, which could have independently been from zero to infinity (there is no mechanism that links or determines them), result in e.g. a QCD scale that is somewhat close to the weak scale. It seems more plausible that a physical mechanism determines the proximity of those scales.

This can be stated more philosophically: if the SM is interpreted as a fundamental, complete theory, there is no physics of which we are ignorant. Yet we are ignorant of the physical parameters and masses, so how are they determined? By God? You might suppose they're random, in some sense, but bear in mind that writing a proper normalizable probability distribution on zero to infinity for a mass would require the introduction of a new mass scale, which is forbidden in this paradigm. This is extremely unsatisfactory.

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