Asymptotically safe Pati-Salam theory

We provide an asymptotically safe Pati-Salam embedding of the standard model. Safety is achieved by adding to the theory gauged vectorlike fermions and by employing recently developed large number-of-flavor techniques and results. We show that the gauge, scalar quartic and Yukawa couplings achieve an interacting ultraviolet fixed point below the Planck scale. The minimal model is a relevant example of a standard model extension in which unification of all type of couplings occurs because of a dynamical principle, i.e., the presence of an ultraviolet fixed point. This extension differs from the usual grand unified theories scenario in which only gauge couplings unify and become free with the remaining couplings left unsafe. We find renormalization group flow solutions that match the standard model couplings values at low energies allowing for realistic safe extensions of the standard model.

Figure 1

Feynman diagrams for gauge field renormalization at order $1/N_F$. Diagrams (a) and (b) are present in both the Abelian and non-Abelian 2-point functions, while (c) and (d) only exist in the non-Abelian theory.

Figure 2

We show a sample case of gauge unification, where we have chosen $N_{F_{2L}}=35$, $N_{F_{2R}}=N_{F_4}=140$. The dashed line represents the Pati-Salam symmetry breaking scale at 2000 TeV where all the vectorlike fermions are introduced. The three couplings $g_Y$, $g_2$, $g_3$ at the left-hand side of the dashed line are determined by the running of the SM gauge couplings.

Figure 3

RG running of the gauge and Yukawa couplings by using the UV to IR approach. We have chosen $N_{F2}=40$, $N_{F3}=130$, $N_{F4}=130$. We have used the matching conditions at IR [see Eqs. (2) and (9)] to set the initial conditions of $g_L$, $g_R$, $g_4$ at IR. For simplification, we have assumed that the vectorlike fermions under gauge different symmetry groups are exactly introduced at the Pati-Salam breaking scale, $v_R=2000\mathrm{TeV}$, marked by a vertical dashed line.

Figure 4

RG running of the scalar quartic couplings using the UV to IR approach for $N_{F2}=40$, $N_{F3}=130$, $N_{F4}=130$. All the vectorlike fermions appear (dashed line) at the symmetry breaking scale of the Pati-Salam group, which is around 2000 TeV.

Figure 5

In this figure, we show 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.