Higgs vacuum stability with vectorlike fermions

We present the effects of vectorlike fermions (VLFs) on the stability of the Higgs electroweak vacuum, using the renormalization group improved Higgs effective potential. We review the calculation of the one-loop beta-functions of the standard model couplings, paying particular attention to the fermion contributions. From this, we derive the VLF contributions to the beta-functions. Using these beta-functions, we determine the scale at which the effective Higgs quartic coupling becomes zero and goes negative, signaling vacuum instability. We find that for certain VLF masses and Yukawa couplings, the Higgs quartic stays positive for field values all the way up to the Planck scale, implying that the metastable vacuum of the standard model can be rendered absolutely stable if VLFs are present with certain parameters. For other values of VLF parameters, the Higgs vacuum is metastable as in the standard model. For cases where the vacuum is metastable, we compute the probability of quantum tunneling from the false electroweak vacuum into a deeper true vacuum in our Hubble volume by numerically solving for the bounce configuration in Euclidean space-time and computing the bounce action for it. We compare our numerical solution with the analytical approximation for the bounce action commonly used in the literature and comment on when the latter may be used.

Figure 1

The evolution of $\lambda$, $y_t$, and $g_3$ with Higgs field value $\mu$ for ($n$) number of degenerate $SU(2)$ singlet VLQs of mass 3 TeV (first three plots) and $\lambda$ with three degenerate singlet VLQs of mass $3\times {10}^3\mathrm{GeV}$, $10^5\mathrm{GeV}$, and $10^7\mathrm{GeV}$ shown, respectively, as $3E3$, $1E5$, and $1E7$ (last plot).

Figure 2

The evolution of $\lambda$, $y_t$, and $g_3$ with the Higgs field value $\mu$ for ($n$) number of degenerate $SU(2)$ doublet VLQs of mass 3 TeV (first three plots) and $\lambda$ with a doublet VLQ of mass $3\times {10}^3\mathrm{GeV}$, $10^5\mathrm{GeV}$, and $10^7\mathrm{GeV}$ shown, respectively, as $3E3$, $1E5$, and $1E7$ (last plot).

Figure 3

The evolution of $\lambda$, $y_t$, and $\tilde{y}$ with Higgs field value $\mu$ for a degenerate family of one $SU(2)$ doublet VLL and one singlet VLL, for $M_{VL}=1\mathrm{TeV}$ and various $\tilde{y}$ (first two plots) and for $\tilde{y}(M_{VL})=1$ and various $M_{VL}$ (in GeV) (last two plots).

Figure 4

The evolution of $\lambda$, $y_t$, $g_3$, and $\tilde{y}$ with Higgs field value $\mu$ for a family of one $SU(2)$ doublet VLQ and one singlet VLQ for various $M_{VL}$ and $\tilde{y}$. The top row is for $M_{\mathrm{VLQ}}=3\mathrm{TeV}$, while the bottom row is for different $M_{VL}$ (in GeV) for $\tilde{y}(M_{VL})=\{0.1,0.5\}$.

Figure 5

For the SM, the effective potential as a function of the field value $h=\mu$ and the bounce configuration $h_B(\widehat{\rho })$. The blue (red) dot shows the starting (ending) value of the bounce.

Figure 6

For a VLL family with $M_{VL}={10}^3\mathrm{GeV}$ and $\tilde{y}=0.6$ (top row), $M_{VL}={10}^5\mathrm{GeV}$ and $\tilde{y}=0.57$ (middle row), and $M_{VL}={10}^7\mathrm{GeV}$ and $\tilde{y}=0.6$ (bottom row), the effective potential as a function of the field value $h=\mu$ and the bounce configuration $h_B(\widehat{\rho })$. The blue (red) dot shows the starting (ending) value of the bounce.

Figure 7

With the addition of a VLQ family, the regions of stability, metastability, and instability as a function of $M_{VL}$ (in GeV) and $\tilde{y}$.

Figure 8

For a VLQ family with $M_{VL}=3\times {10}^3\mathrm{GeV}$ and $\tilde{y}=0.75$, with a second minimum in the effective potential, the effective potential as a function of the field value $h=\mu$, and the bounce configuration $h_B(\widehat{\rho })$. The blue (red) dot shows the starting (ending) value of the bounce.

Figure 9

The ${\beta }_{\lambda }(\mu )$ as a function of the field value $h(\rho )=\mu$ (left column) and the integrand of the bounce action integral Eq. (29) vs $\lambda (h(\rho ))$ (right column), for the SM (cf. Fig. 5) (top row) and the remaining for the different cases with a VLF family as follows: a VLL family with $M_{VL}={10}^3\mathrm{GeV}$ and $\tilde{y}=0.6$ (cf. Fig. 6) (top row), a VLQ family with $M_{VL}=3\times {10}^3\mathrm{GeV}$ and $\tilde{y}=0.57$, and a VLQ family with $M_{VL}=3\times {10}^3\mathrm{GeV}$ and $\tilde{y}=0.75$ with a second minimum (cf. Fig. 8). The blue dot shows the starting value $h_0$ for the bounce configuration.