$SU(5)\times U(1{)}_X$ axion model with observable proton decay

We propose a $SU(5)\times U(1{)}_X\times U(1{)}_{\mathrm{PQ}}$ model, where $U(1{)}_X$ is the generalization of the $B-L$ (baryon minus lepton number) gauge symmetry and $U(1{)}_{\mathrm{PQ}}$ is the global Peccei-Quinn (PQ) symmetry. There are four fermions families in $\overline{\mathbf{5}}+\mathbf{10}$ representations of $SU(5)$, a mirror family in $\mathbf{5}+\overline{\mathbf{10}}$ representations, and three $SU(5)$ singlet Majorana fermions. The $U(1{)}_X$ related anomalies all cancel in the presence of the Majorana neutrinos. The $SU(5)$ symmetry is broken at $M_{\mathrm{GUT}}\simeq (6–9)\times {10}^{15}\mathrm{GeV}$ and the proton lifetime ${\tau }_p$ is estimated to be well within the expected sensitivity of the future hyper-Kamiokande experiment, ${\tau }_p\lesssim 1.3\times {10}^{35}\mathrm{years}$. The $SU(5)$ breaking also triggers the breaking of the PQ symmetry, resulting in axion dark matter (DM), with the axion decay constant $f_a$ of order $M_{\mathrm{GUT}}$ or somewhat larger. The CASPEr experiment can search for such an axion DM candidate. The Hubble parameter during inflation must be low, $H_{\inf }\lesssim {10}^9\mathrm{GeV}$, in order to successfully resolve the axion domain wall, axion DM isocurvature and $SU(5)$ monopole problems. With the identification of the $U(1{)}_X$ breaking Higgs field with the inflaton field, we implement inflection-point inflation, which is capable of realizing the desired value for $H_{\inf }$. The vectorlike fermions in the model are essential for achieving successful unification of the SM gauge couplings as well as the phenomenological viability of both axion DM and inflation scenario.

Figure 1

For fixed masses of the new lepton doublet $L$ and quarks ($D^c$ and $Q$) with masses $4.5\times 10^{12}$ and 5000 GeV, respectively, the plot shows the RG running of SM couplings. In the left panel the diagonal solid (dotted) lines labeled ${\alpha }_i=g_i^2/4\pi$ ($i=1$, 2, 3) depict the SM $U(1{)}_Y$, $SU(2{)}_L$, and $SU(3{)}_c$ gauge couplings with (without) new fermions, respectively, which are unified at $M_{\mathrm{GUT}}\simeq 7.54\times {10}^{15}\mathrm{GeV}$. In the right panel, the solid (dotted) curve depicts the RG running of the SM Higgs quartic coupling with (without) vectorlike fermions along with the horizontal dashed line depicting ${\lambda }_h=0$.

Figure 2

The blue (cyan) shaded region in the top left panel denote the range for the new quarks mass, $M_Q$, and the new lepton mass, $M_L$, to achieve the SM gauge couplings unification with an accuracy of 5% (1%) or less (see text for details). For reference, the yellow diagonal lines depict the contours for fixed $M_L=5.8\times {10}^8\mathrm{GeV}$ for achieving the unification within 5% accuracy. For the mass values to the left of the solid black line, the SM Higgs potential is stabilized. For the new quarks and lepton masses identified in the top left panels, top right, and bottom left panels, respectively, show the unification scale $M_{\mathrm{GUT}}$ and the value of the unified coupling ${\alpha }_{\mathrm{GUT}}$ as a function of $M_Q$. The bottom right panel shows the proton lifetime where the gray shaded region (${\tau }_p\le 1.6\times {10}^{35}$) in the bottom right panel denote the exclusion from super-Kamiokande experiment. The horizontal dashed line (${\tau }_p=1.3\times {10}^{35}$) is the expected reach of the future hyper-Kamiokande experiment.

Figure 3

For fixed value of $M=0.05M_P$, the left panel shows the RG running of the inflaton quartic coupling ${\lambda }_{\varphi }$ as a function of $\varphi /M$. The dashed horizontal line corresponds to ${\lambda }_{\varphi }=0$. The right panel shows the RG improved effective inflaton potential with an approximate inflection point at $\varphi \simeq M$ (vertical dashed-dotted line).