  The asymptotic-safety paradigm posits that the symmetry of quantum theories
of gravity and matter is enhanced to quantum scale symmetry, i.e., scale
symmetry in the presence of quantum fluctuations, at very high energies. To
achieve such a symmetry enhancement, the effect of quantum fluctuations must
balance out. It is to be expected that such a balance can only be achieved
within a set of theories with limited field content and interaction structure.
In this chapter, we review how much is known about these limits. From the
quantum scale invariant regime, the theory transits to a theory with distinct
physical scales - most importantly masses for various elementary particles - at
low energies. There, quantum scale invariance can leave its imprint in
relations between various interactions and mass scales of the theory. These
relations can be compared to experimental data, which has two possible
implications: first, if the relations do not match the data, the underlying
quantum theory of gravity and matter, formulated at and beyond the Planck
scale, has been ruled out using experimental data from energies much below the
Planck scale. Second, if the relations match the data, the asymptotic-safety
paradigm provides a first-principles derivation of free parameters of the
Standard Model. Most importantly, this may include the ratios of the Higgs mass
to the electroweak scale as well as the value of the finestructure constant.
Similarly, theories beyond the Standard Model may come with fewer free
parameters than in their effective-field-theory incarnation without gravity.
This may lead to an explanation of the smallness of neutrinos masses and
predictions for the nature and interactions of dark matter.
