Gravitational corrections to fermion masses in grand unified theories

There are several hints that the strong and electroweak forces unify at some very large energy typically assumed to be at around 10 GeV. The quantum fields of the standard model fit nicely into simple representations of a grand unified theory [1] such as e.g. SUð5Þ or SOð10Þ. The idea of unification is extremely attractive for several reasons. For example, a grand unification drastically reduces the number of independent coupling constants. Furthermore, when extrapolated using renormalization group equations, the value of the strong and electroweak interactions measured at low energy seem to converge amazingly to some common value at around 10 GeV [2–4] if the standard model is replaced by the minimal supersymmetric standard model at around a TeV. An important feature of grand unified theories is that they predict the existence of many, potentially heavy, new particles. This is due to the very nature of grand unified theories that need to be based on groups large enough to incorporate the standard model SUð3Þ SUð2Þ Uð1Þ groups. Besides having to be large as such, unified theories often incorporate multiplets with a large number of fields to obtain viable phenomenology. When the unified theory is supersymmetric, the number of fundamental fields is even larger. It has been argued that the LHC data could be used to reconstruct, using renormalization group techniques, the fundamental grand unified theory, see e.g. [5], or differentiate between different supersymmetry breaking patterns [5]. In [6,7] it was shown that there are potentially sizable quantum gravitational corrections to the unification conditions for the gauge couplings of the standard model. The thresholds have been known for a while [8–11], but it had not been realized that they could potentially be larger than the two-loop corrections [6]. The aim of this work is to show that this quantum gravity blur has a similar effect on the unification conditions for the masses of the fermions in a grand unified framework. An important consequence of the large number of fundamental fields mentioned above, which can easily reach 1000, is that the scale at which quantum gravitational effects are expected to become large is not necessarily as expected some 10 GeV but is given by the renormalized Planck mass: