Framework for an asymptotically safe standard model via dynamical breaking

We present a consistent embedding of the matter and gauge content of the Standard Model into an underlying asymptotically safe theory that has a well-determined interacting UV fixed point in the large color/flavor limit. The scales of symmetry breaking are determined by two mass-squared parameters with the breaking of electroweak symmetry being driven radiatively. There are no other free parameters in the theory apart from gauge couplings.

Figure 1

The renormalization group flow of the couplings from the UV fixed point and around the critical curve, toward the Gaussian IR fixed point. The black line is fixed by matching the desired electroweak couplings at the low scale.

Figure 2

One-loop diagram and the leading resummed pole contributions for the $SU(2{)}_L\times SU(2{)}_R$ fixed points, where $\tilde{\alpha }=N_f{g}_{SU(2)}^2/16{\pi }^2$. The plain lines represent both quarks and scalars.

Figure 3

Subleading bubbles in the renormalization of the $SU(2{)}_{L/R}$ gauge couplings (where plain lines represent quarks and/or scalars) are suppressed with respect to the terms in the resummation in Fig. 2. The first diagrams exist also in the pure $SU(2{)}_{L/R}$ theory and are suppressed by a factor of $1/N_f$: this admits the procedure of [10] that establishes a fixed point by balancing the resummed pole of Fig. 2 against the one-loop diagram. On the second row, the insertion of an $SU(N)$ “gluon” line on the quark loops, or a Higgs scalar via Yukawa couplings, gives a factor $\frac{g^2}{(4\pi {)}^2}\sum _A(T^A{T}^A)\sim \frac{g^{2}N}{(4\pi {)}^2}\sim \epsilon$ or a factor $\frac{y^2{N}_F}{(4\pi {)}^2}\sim \epsilon$, respectively, compared to diagrams with the same power of $\tilde{\alpha }$ in Fig. 2, where the $\epsilon$ scalings apply when one is near the fixed points of ${\alpha }_g$, ${\alpha }_y$, ${\alpha }_{u_1}$, ${\alpha }_{u_2}$ couplings. Note that due to the large number of $SU(2)$ doublets, $\epsilon$ is of order $5/N_f$. On the third row, the introduction of a pair of scalars with quartic interactions introduced terms suppressed even more, by factors of order ${\alpha }_{u_2}^2\sim {\epsilon }^2$.

Figure 4

Factors contributing to the beta functions of the $SU(N)$ gauge coupling to two-loops (which get multiplied by an overall ${\alpha }_g$ factor in ${\beta }_g$). The leading term is of course canceled to order ${\epsilon }^2$ against the gauge loops in the Banks-Zaks limit by the choice of colors and flavors. Noting that $\epsilon \gtrsim 1/N\gtrsim 5/N_f$, the $SU(2{)}_{L/R}$ gauging can be neglected in the $N_f\to \infty$ limit. The plain lines represent both quarks and scalars.