Prospects for mass unification at low energy scales

A simple Pati-Salam SU(4) model with a low symmetry breaking scale of about 1000 TeV is presented. The analysis concentrates on calculating radiative corrections to tree-level mass relations for third generation fermions. The tree-level relation $\frac{m_b}{m_{\tau }}=1\mathrm{}$ predicted by such models can receive large radiative corrections up to about 50% due to threshold effects at the mass unification scale. These corrections are thus of about the same importance as those that give rise to renormalization group running. The high figure of 50% can be achieved because one-loop graphs involving the physical charged Higgs boson give corrections to $m_{\tau }-m_b$ that are proportional to the large top quark mass. These corrections can either increase or decrease $\frac{m_b}{m_{\tau }}$ depending on the value of an unknown parameter. They can also be made to vanish through a fine-tuning. A related model of tree-level $t-b-\tau$ unification which uses the identification of $\mathrm{SU}{\mathrm{}\left(2\right)\mathrm{}}_R$ with custodial SU(2) is then discussed. A curious relation $m_b\simeq \sqrt{2}{m}_{\tau }$ is found to be satisfied at the tree level in this model. The overall conclusion of this work is that the tree-level relation $m_b=m_{\tau }$ at low scales such as 1000 TeV or somewhat higher can produce a successful value for $\frac{m_b}{m_{\tau }}$ after corrections, but one must be mindful that radiative corrections beyond those incorporated through the renormalization group can be very important. This motivates that an ongoing search for the rare decays $K_L^0\to {\mu }^{\pm }{e}^{\mp }$ be maintained.