Grand unification and the Planck scale: An $\mathit{SO}(10)$ example of radiative symmetry breaking

Grand unification of gauge couplings and fermionic representations remains an appealing proposal to explain the seemingly coincidental structure of the Standard Model. However, to realise the Standard Model at low energies, the unified symmetry group has to be partially broken by a suitable scalar potential in just the right way. The scalar potential contains several couplings, whose values dictate the residual symmetry at a global minimum. Some (and possibly many) of the corresponding symmetry-breaking patterns are incompatible with the Standard Model and therefore non-admissible. Here, we initiate a systematic study of radiative symmetry breaking to thereby constrain viable initial conditions for the scalar couplings, for instance, at the Planck scale. We combine these new constraints on an admissible scalar potential with well-known constraints in the gauge-Yukawa sector into a general blueprint that carves out the viable effective-field-theory parameter space of any underlying theory of quantum gravity. We exemplify the constraining power of our blueprint within a non-supersymmetric $\mathit{SO}(10)$ GUT containing a $\mathbf{16}_H$- and a $\mathbf{45}_H$-dimensional scalar representation. We explicitly demonstrate that the requirement of successful radiative symmetry breaking to the correct subgroups significantly constraints the underlying microscopic dynamics. The presence of non-admissible radiative minima can even entirely exclude specific breaking chains: In the $\mathit{SO}(10)$ example, Pati-Salam breaking chains cannot be realised since the respective minima are never the deepest ones.