Searching for signatures of ${\mathrm{E}}_6$

The grand unified group ${\mathrm{E}}_6$ is a predictive scheme for physics beyond the standard model (SM). It offers the possibility of extra $Z$ bosons, new vectorlike fermions, sterile neutrinos, and neutral scalars in addition to the SM Higgs boson. Some previous discussions of these features are updated and extended. Their relevance to present searches at the CERN Large Hadron Collider and in patterns of neutrino masses is noted. Addition of a small set of scalar bosons at the TeV scale permits gauge unification near a scale of $10^{16}\mathrm{GeV}$ and leads to bounds on masses of particles beyond those in the standard model.

Figure 1

${\tilde{X}}_{15}$ is a diquark belonging to the 15 of SU(5) in the 54 of SO(10) contained in the scalar $351^{\prime }$ of ${\mathrm{E}}_6$, which couples to two SM quarks at one-loop level. The loop involves particles with GUT-scale masses and therefore is suppressed. Another multi-TeV-scale diquark, ${\tilde{X}}_{10}$ (not shown) belongs to the $\overline{10}$ of SU(5) in the $\overline{16}$ of SO(10) contained in the scalar $351^{\prime }$ of ${\mathrm{E}}_6$. It does not couple to two SM quarks at the one-loop level. For quantum numbers of ${\tilde{X}}_{15}$ and ${\tilde{X}}_{10}$ see Table 11.

Figure 2

Dark [red] and lighter [dark orange] shaded regions show the exclusion regions for exotics by ATLAS [39, 40] and CMS [41], respectively, in the diphoton channel. Black (dashed) curve: Expected 95% exclusion limit for ATLAS (upper curve, $15.6{\mathrm{fb}}^{-1}$ at 13 TeV) and CMS (lower curve, $16.2{\mathrm{fb}}^{-1}$ at $13\mathrm{TeV}+19.7{\mathrm{fb}}^{-1}$ at 8 TeV). Red (dotted) curve: Contours showing ATLAS reach for nondiscovery for luminosities of 30, 300 and $3000{\mathrm{fb}}^{-1}$ at 13 TeV. Blue (dashed) curve: Contours of constant cross section ${\sigma }_0\equiv \sigma (pp\to \tilde{n}X\to \gamma \gamma X)$ in the plane of $⟨\tilde{n}⟩$ versus $m_{\tilde{n}}$ for values of ${\sigma }_0$, top to bottom, ranging from 0.01 to 10 fb.

Figure 3

Mechanism for hadronic production and decay of scalar boson $\tilde{n}$.

Figure 4

Left: Functions $F_{\pm }(\beta )$ governing decays of scalar ($F_{+}$) or pseudoscalar ($F_{-}$) to two photons. Solid curve, scalar $\tilde{n}$; dashed curve, pseudoscalar $\tilde{n}$. Middle: Effective gluon-gluon luminosity function for proton-proton production of a state with mass $M$ at center-of-mass energy $\sqrt{s}=13\mathrm{TeV}$. Right: Cross section for $pp\to D{D}^c+X$ via two-gluon intermediate state at $\sqrt{s}=13\mathrm{TeV}$.

Figure 5

Main diagrams contributing to ${\mathbb{Z}}_2$-preserving decays of exotic quark $D$.

Figure 6

Drell-Yan processes contributing to exotic lepton pair production via (a) virtual $\gamma ,Z^0$; (b) virtual $W^{+}$; and (c) virtual $W^{-}$.

Figure 7

Predicted cross sections in femtobarns for production of $L\overline{L}$ pairs by proton-proton collisions at $\sqrt{s}=13\mathrm{TeV}$. Left: Leading-order; right: calculation for Higgsinos using Madgraph [168].

Figure 8

Forward-backward asymmetries for several different $Z$’s of mass 3 (left) and 4 (right) TeV.

Figure 9

Left: Dashed [red] contours correspond to minimum required value of $b_{17}$ for stability. Shaded [colored] backgrounds with boundaries labeled by boxed numbers indicate the total range of $b_{17}$ values, i.e., the difference between maximum and minimum values of $b_{17}$ allowed at that point. The inner [orange] region corresponds to the range of at least 0.8 above the minimum value given by the [red] dashed contour. The middle [yellow] region implies that the allowed $b_{17}$ values are 0.5 to 0.8 more than the minimum values required by the stability condition as indicated by the [red] dashed contour. The outer [green] region denotes the corresponding range of 0.1 to 0.5. Right: Shaded [light blue] region excluded by stability constraints. Solid [red] contours indicate maximum exotic quark mass allowed in TeV. Dashed [green] contours indicate maximum exotic lepton mass allowed in TeV.

Figure 10

RGE for the gauge couplings. Dashed lines: SM; solid lines: ${\mathrm{E}}_6$ model with three 27-plet fermions $+$ one 27-plet scalar $+$ one 78-plet scalar $+$ one $351^{\prime }$-plet scalar. Black (uppermost solid) curve, ${\alpha }_N^{-1}$ of $\mathrm{U}(1{)}_N$; red (second from the top for solid, topmost for dashed at low energies) curve, ${\alpha }_1^{-1}$ of $\mathrm{U}(1{)}_Y$; green (third from the top for solid, second from the top for dashed at low energies) curve, ${\alpha }_2^{-1}$ of ${\mathrm{U}(2)}_L$; blue (third from the top for solid, third from the top for dashed at low energies) curve, ${\alpha }_3^{-1}$ of ${\mathrm{U}(3)}_c$. The vertical line corresponds to the experimental lower bound on the $Z_N$ mass.

Figure 11

RGE for the Yukawa couplings when $g_N$ at MTeV scale is set to be the value required by unification. Dashed curves (upper to lower) correspond to $g_3$ (blue), $g_2$ (green), $g_1$ (red) and $g_N$ (black). Solid curves (upper to lower) correspond to $b_{17}$ (yellow), $y_L$ (orange) and $y_D$ (teal). Electroweak-scale Yukawa couplings are $y_D=0.95$, $y_L=0.95$ and quartic $b_{17}=1.4$.

Figure 12

RGE for the Yukawa couplings when $g_N$ at MTeV scale is set to be the value required by unification. Dashed curves (upper to lower) correspond to $g_3$ (blue), $g_2$ (green), $g_1$ (red) and $g_N$ (black). Solid curves (upper to lower) correspond to $b_{17}$ (yellow), $y_D$ (teal) and $y_L$ (orange). Electroweak scale Yukawa couplings are $y_D=1.3$, $y_L=0.2$, $b_{17}=2.1$.