Renormalization group invariant of lepton Yukawa couplings

By using quark Yukawa matrices only, we can construct renormalization invariants that are exact at the one-loop level in the standard model. One of them, $I^q$, is accidentally consistent with unity, even though quark masses are strongly hierarchical. We calculate a lepton version of the invariant $I^l$ for Dirac and Majorana neutrino cases and find that $I^l$ can also be close to unity. For the Dirac neutrino and inverted hierarchy case, if the lightest neutrino mass is 3.0 meV to 8.8 meV, an equality $I^q=I^l$ can be satisfied. These invariants are not changed even if new particles couple to the standard model particles, as long as those couplings are generation independent.

Figure 1

Neutrino mass dependence of the renormalization group invariant $I^l$ for the Dirac neutrino case. Upper and lower curves show the IH and NH cases with $1\sigma$ deviation roughly estimated from Eqs. (11) and (12). The straight line shows the lower limit of $I^q$.

Figure 2

Majorana neutrino, NH case. The blue solid curve is the case of real $R$, or $\overrightarrow{a}=0$. The upper and the lower dashed curves are cases in which $I^l(0.1\mathrm{eV})$ is largest and smallest [${\overrightarrow{a}}^T=(5.00,2.89,0.572)$ and ${\overrightarrow{a}}^T=(-1.98,3.80,-5.00)$]. The lower limit of the invariant $I^q$ is also shown by a straight line.

Figure 3

Majorana neutrino, IH case. The upper and the lower dashed curves are taking ${\overrightarrow{a}}^T=(5.00,2.60,-0.235)$ and ${\overrightarrow{a}}^T=(-3.58,5.00,-5.00)$. Straight horizontal lines show the central value and lower limit of $I^q$. Other notations are the same as in Fig. 2.