Safe trinification

We provide a UV safe trinification theory in which the Standard Model is embedded. Using recently developed large number-of-flavor techniques, safety is achieved by adding to the theory gauged vectorlike fermions. We find that all gauge, scalar quartic, and Yukawa couplings achieve an interacting ultraviolet fixed point below the Planck scale. We find renormalization group flow solutions matching the Standard Model in the IR, indicating a truly UV completion of the Standard Model. Imposing constraints that realistic top, Higgs, bottom, and tau masses are recovered, we find the set of allowed solutions to be quite restrictive. Furthermore, we find there exists a lower bound of about 10 TeV for the symmetry breaking scale, making this model vulnerable to experiment.

Figure 1

This figure shows the one-loop radiative correction to the neutrino mass.

Figure 2

Feynman diagrams for gauge field renormalization at order $1/N_F$. Diagrams (a) and (b) are present in both the Abelian and non-Abelian two-point functions, while (c) and (d) only exist in the non-Abelian theory.

Figure 3

RG running of the gauge couplings by using the UV to IR approach. We have chosen $N_{FC}=95$, $N_{FL}=165$, $N_{FR}=62$. We set the initial conditions for $g_L$, $g_R$, $g_c$ at the IR using the matching conditions there [see Eq. (2.2)]. For simplification, we have assumed that the vectorlike fermions under different gauge symmetry groups are exactly introduced at the trinification breaking scale, $t_{SB}=10\mathrm{TeV}$, marked by a vertical dashed line.

Figure 4

RG running of the Yukawa couplings by using the UV to IR approach. Sub-Figs. (a), (b), (c), (d) correspond respectively to the Yukawa couplings of $y_{{\psi }_{Q1}},y_{{\psi }_{E1}},y_{{\psi }_{Q2}},y_{{\psi }_{E2}}$. We have chosen $N_{FC}=95$, $N_{FL}=165$, $N_{FR}=62$. For simplification, we have assumed that the vectorlike fermions under different gauge symmetry groups are exactly introduced at the trinification breaking scale, $t_{SB}=10\mathrm{TeV}$, marked by a vertical dashed line.

Figure 5

RG running of the scalar quartic couplings using the UV to IR approach for $N_{FC}=95$, $N_{FL}=165$, $N_{FR}=62$. Sub-Figs. (a), (b), (c), (d) correspond respectively to the scalar couplings of ${\lambda }_{1a}$, ${\lambda }_{1b}$, ${\lambda }_{2a}$, ${\lambda }_{2b}$. All the vectorlike fermions appear (dashed line) at the symmetry breaking scale of the trinification group, which is around 10 TeV.

Figure 6

RG running of the scalar quartic couplings using the UV to IR approach for $N_{FC}=95$, $N_{FL}=165$, $N_{FR}=62$. Sub-Figs. (a), (b), (c), (d), (e), (f) correspond respectively to the scalar couplings of ${\lambda }_a$, ${\lambda }_b$, ${\lambda }_c$, ${\lambda }_d$, ${\lambda }_e$, ${\lambda }_f$. All the vectorlike fermions appear at the symmetry breaking scale of the trinification group, which is around 10 TeV (dashed line).

Figure 7

In this figure, we present two mass eigenvalues of the $CP$-even neutral Higgs mass matrix as a function of $m_{12}$. $M_{H1}$ denotes the lighter Higgs, while $M_{H2}$ denotes the heavier one.

Figure 8

In this figure, we show the 3D parameter space constructed by the values of $N_{FC}\in (70,105);N_{FL}\in (160,170);N_{FR}\in (50,68)$ in the asymmetric case. The orange dot corresponds to a UV fixed point solution which does not match the SM at IR while the black dots correspond to the viable UV fixed point solutions.