Why the quark mass is not the Planck mass

We investigate whether quantum gravity fluctuations can break chiral symmetry for fermions that are charged under a U(1) and an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> <mrow> <mi>SU</mi> <mo stretchy="false">(</mo> <msub> <mrow> <mi>N</mi> </mrow> <mi mathvariant="normal">c</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math> gauge symmetry and thus closely resemble Standard-Model fermions. Unbroken chiral symmetry in the quantum-gravity regime is a necessary prerequisite to recover the Standard Model from a joint gravity-matter theory; if chiral symmetry is broken by quantum gravity, fermions cannot generically be much lighter than the Planck mass, and the theory is ruled out. To answer this, we work in a Fierz-complete basis of four-fermion interactions and explore whether they are driven to criticality. We discover that the interplay of quantum gravity with the non-Abelian gauge theory results in chiral symmetry breaking, because gravitational and gauge field fluctuations act together to produce bound states. Chiral symmetry breaking is triggered by four-fermion channels that first appear when non-Abelian charges are introduced and that become critical if the non-Abelian symmetry is gauged. Extrapolating our result to the Standard Model fermions, we conjecture that the non-Abelian gauge coupling, Abelian gauge coupling and Newton coupling are all bounded from above at trans-Planckian scales, if Standard Model fermions are to remain much lighter than the Planck mass. In contrast, fermions that are charged under a global non-Abelian group can remain light for arbitrarily large values of the Newton coupling. We find that different chiral symmetries are emergent at low energies, depending on the strength of the gravitational coupling. This is an example of fixed points with different degrees of enhanced global symmetry trading stability in fixed-point collisions driven by gravitational fluctuations.