Phenomenological implications of an $SU(5)\times S_4\times U(1)$ SUSY GUT of flavor

We discuss the characteristic low energy phenomenological implications of an $SU(5)$ supersymmetric (SUSY) grand unified theory whose flavor structure is controlled by the family symmetry $S_4\times U(1)$, which provides a good description of all quark and lepton masses, mixings as well as charge parity violation. Although the model closely mimics minimal flavor violation (MFV) as shown in M. Dimou, S. F. King, and C. Luhn, J. High Energy Phys. 02 (2016) 118., here we focus on the differences. We first present numerical estimates of the low energy mass insertion parameters, including canonical normalization and renormalization group running, for well-defined ranges of SUSY parameters and compare the naive model expectations to the numerical scans and the experimental bounds. Our results are then used to estimate the model-specific predictions for electric dipole moments (EDMs), lepton flavor violation (LFV), $B$ and $K$ meson mixing as well as rare $B$ decays. The largest observable deviations from MFV come from the LFV process $\mu \to e\gamma$ and the electron EDM.

Figure 1

The low energy up-type mass insertion parameters plotted against their GUT scale coefficients, defined in Eqs. (a1)–(a2) [except for $({\delta }_{LR}^u{)}_{13,23}$ which are plotted against a coefficient multiplying the RG running contribution, cf. Eqs. (a26)–(a27)]. The blue dashed lines represent our naive numerical expectations according to the second column of Table 2, while the red dotted lines (when available) represent their experimental limits, shown in the third column of Table 2. Since $({\delta }_{RR}^u{)}_{12}\approx ({\delta }_{LL}^u{)}_{12}$, only the $LL$ parameter is plotted. The plots have been produced by scanning over the input parameters listed in Table 1.

Figure 2

The low energy down-type mass insertion parameters $({\delta }_{AB}^d{)}_{ij}$, $A,B=L$, $R$, $i=1$, 2, 3 plotted against their GUT scale coefficients, defined in Eqs. (a1)–(a2). The blue dashed lines represent our naive numerical expectation according to the second columns of Tables 3, 4, 5. The red shaded areas cover the parameter space bounded by the limits shown in the third column of the corresponding tables, with the red dotted lines denoting the weakest limit in each case. The absolute values of ${\delta }_{RR}^d$ are equal in the (12), (23) and (13) sectors and also $|({\delta }_{LR}^d{)}_{12}|=|({\delta }_{RL}^d{)}_{12}|=|({\delta }_{LR}^d{)}_{13}|$. We therefore only show the bounds stemming from the (12) sector as they are the strongest ones. All plots have been produced by scanning over the input parameters shown in Table 1.

Figure 3

The low energy lepton mass insertion parameters $({\delta }_{AB}^e{)}_{ij}$, $A,B=L,R$, plotted against the down-type $\delta \mathrm{s}$ to which they are related via the $SU(5)$ framework. The dashed lines represent their GUT scale relations, while the red shaded areas denote experimental limits on the parameter space according to the third column of Tables 3, 4, 5, 6. Scanning over the input parameters within the ranges shown in Table 1, we observe that in particular $|({\delta }_{LL}^e{)}_{12}|$ exceeds its limit for much of our parameter space. Note that $|({\delta }_{LL}^e{)}_{12}|=|({\delta }_{LL}^e{)}_{23}|=|({\delta }_{LL}^e{)}_{13}|$ and $|({\delta }_{RL}^e{)}_{12}|=|({\delta }_{LR}^e{)}_{12}|=|({\delta }_{RL}^e{)}_{13}|$.

Figure 4

Left panel: the prediction for the SUSY contribution to the electron EDM versus $m_{\tilde{e}}=\sqrt{m_{{\tilde{e}}_{LL}}{m}_{{\tilde{e}}_{RR}}}$. The red dotted line represents the current experimental limit of Eq. (4.3), while the black dotted line corresponds to the expected future limit of $|d_e/e|\lesssim 3\times {10}^{-31}\mathrm{cm}\approx 1.52\times {10}^{-17}{\mathrm{GeV}}^{-1}$ [42]. Right panel: the behavior of the functions $h_1$, $k_1$ and (in anticipation of the discussion in Sec. 4b) $h_2$.

Figure 5

Left panel: the contour lines for $\overline{R}$, the approximate ratio of the $SU(2)$ over the $U(1)$ contributions to the $({\delta }_{LL}^e{)}_{12}$ term in Eq. (4.8), as defined in Eq. (4.10). For the average slepton mass $m_{\tilde{e}}=\sqrt{m_{{\tilde{e}}_{LL}}{m}_{{\tilde{e}}_{RR}}}$, $\overline{x}=(M_1/m_{\tilde{e}}{)}^2\approx {0.43}^{2}x/(1+0.3x)$, with $x=(M_{1/2}/m_0{)}^2$. Right panel: the ratio $R$ (without approximation), as defined in Eq. (4.9) and produced in our scan. The dependence of $(M_2/\mu {)}^2$ and $\overline{x}$ on $x$ is such that the $SU(2)$ contributions dominate for most of the parameter space.

Figure 6

The supersymmetric contribution to the branching ratio of $\mu \to e\gamma$ versus the average slepton mass $m_{\tilde{e}}=\sqrt{m_{{\tilde{e}}_{LL}}{m}_{{\tilde{e}}_{RR}}}$ (left panel) as well as $|d_e/e|$ (right panel). The red dotted lines represent the current experimental limits given in Eqs. (4.3) and (4.7) while the black dotted lines show the expected future limits, that is $\mathrm{BR}(\mu \to e\gamma )\lesssim 6\times {10}^{-14}$ [44] and $|d_e/e|\lesssim 1.52\times {10}^{-17}{\mathrm{GeV}}^{-1}$ [42].

Figure 7

The range of the (12) lepton mass insertion parameters as produced in our scan, together with the resulting prediction for the branching ratio of $\mu \to e\gamma$. The grey points do not satisfy the current experimental limit given in Eq. (4.3).

Figure 8

The dependence of the individual contributions in Eq. (4.29) on $y=(m_{\tilde{g}}/m_{\tilde{q}}{)}^2$. The average squark mass $m_{\tilde{q}}$ is defined in Eq. (4.19) while the functions $A_i^{B_{s,d},(\tilde{g})}$ can be found in Eq. (4.18).

Figure 9

The dependence of the loop functions as well as $|A_{2,3}^{B_q,(\mathrm{DP})}|$ appearing in Eq. (4.30) on $y=(m_{\tilde{g}}/m_{\tilde{q}}{)}^2$, $y_{\mu }=(\mu /m_{\tilde{q}}{)}^2$ and $y_2=(M_2/m_{\tilde{q}}{)}^2\approx 0.11y$. The blue lines correspond to $y_{\mu }/y=30$ and the magenta ones to $y_{\mu }/y=0.3$. In the plots for $|A_2^{B_q,(\mathrm{DP})}|$, we have assumed that $A_t\approx m_{\tilde{q}}$.

Figure 10

The absolute value of the gluino and double penguin contributions to $\mathrm{\Delta }{M}_{B_{s(d)}}$ versus the average squark mass as defined in Eq. (4.19). The color coding corresponds to different values of $x=(M_{1/2}/m_0{)}^2$. The red dotted lines denote the experimental central values of Eqs. (4.25)–(4.26), while the blue dotted lines indicate the maximum allowed NP contributions according to Eq. (4.28).

Figure 11

The dependence of the individual contributions in Eq. (4.42) on $y=(m_{\tilde{g}}/m_{\tilde{q}}{)}^2$. The average squark mass $m_{\tilde{q}}$ is defined in Eq. (4.19) while the functions $A_i^{K,(\tilde{g})}$ can be found in Eq. (4.18).

Figure 12

Upper panels: the absolute value of SUSY contributions to $\mathrm{\Delta }{M}_K$ (left) and ${\epsilon }_K$ (right) plotted against the average squark mass defined in Eq. (4.19), with the different colors corresponding to different values of $x=(M_{1/2}/m_0{)}^2$. Lower panels: the most important mass insertion parameters, relevant for $K$ mixing (left) with different colors representing the produced value of $|{\epsilon }_K^{\mathrm{SUSY}}|$; $|\mathrm{\Delta }{M}_K^{\mathrm{SUSY}}|$ versus $|{\epsilon }_K^{\mathrm{SUSY}}|$ (right), with the grey shaded points being excluded by $\mathrm{BR}(\mu \to e\gamma )$. The red dotted lines indicate the experimentally observed values, while the blue dotted lines show the limits on NP contributions.

Figure 13

The SUSY contributions to the branching ratios of $B_q\to {\mu }^{+}{\mu }^{-}$ versus the average squark mass $m_{\tilde{q}}$, defined in Eq. (4.19). The red dotted lines denote the experimental measurements, while the blue dotted lines indicate the maximum NP contributions.

Figure 14

The neutron EDM versus the average squark mass $m_{\tilde{q}}=\sqrt{m_{\tilde{u}}{m}_{\tilde{d}}}$, with $m_{\tilde{u}}$ and $m_{\tilde{d}}$ as defined in Eq. (4.59) (left panel) and versus the electron EDM (right panel). The red dotted lines denote the current experimental limits as given in Eqs. (4.63) and (4.3) and the black dotted lines the future limits $|d_n/e|\lesssim 10^{-28}\mathrm{cm}\approx 5\times {10}^{-15}{\mathrm{GeV}}^{-1}$ and $|d_e/e|\lesssim 3\times {10}^{-31}\mathrm{cm}\approx 1.52\times {10}^{-17}{\mathrm{GeV}}^{-1}$ [42].