Electron electric dipole moment as a sensitive probe of PeV scale physics

We give a quantitative analysis of the electric dipole moments as a probe of high scale physics. We focus on the electric dipole moment of the electron since the limit on it is the most stringent. Further, theoretical computations of it are free of QCD uncertainties. The analysis presented here first explores the probe of high scales via electron electric dipole moment (EDM) within minimal supersymmetric standard model where the contributions to the EDM arise from the chargino and the neutralino exchanges in loops. Here it is shown that the electron EDM can probe mass scales from tens of TeV into the PeV range. The analysis is then extended to include a vectorlike generation which can mix with the three ordinary generations. Here new $CP$ phases arise and it is shown that the electron EDM now has not only a supersymmetric (SUSY) contribution from the exchange of charginos and neutralinos but also a nonsupersymmetric contribution from the exchange of $W$ and $Z$ bosons. It is further shown that the interference of the supersymmetric and the nonsupersymmetric contribution leads to the remarkable phenomenon where the electron EDM as a function of the slepton mass first falls and become vanishingly small and then rises again as the slepton mass increases. This phenomenon arises as a consequence of cancellation between the SUSY and the non-SUSY contribution at low scales while at high scales the SUSY contribution dies out and the EDM is controlled by the non-SUSY contribution alone. The high mass scales that can be probed by the EDM are far in excess of what accelerators will be able to probe. The sensitivity of the EDM to $CP$ phases both in the SUSY and the non-SUSY sectors are also discussed.

Figure 1

The neutralino-slepton exchange diagram (left) and the chargino-sneutrino exchange diagram (right) that contribute to the electric dipole moment of the electron in MSSM.

Figure 2

Left panel: A display of the electron EDM as a function of $m_0$ (where $m_0=M_{\tilde{e}L}=M_{\tilde{e}}$) for different ${\alpha }_{\mu }$ (the phase of the Higgs mixing parameter $\mu$) with the mixings of the vectorlike generation with the regular three generations set to zero. The curves are for the cases ${\alpha }_{\mu }=-3$ (small-dashed, red), ${\alpha }_{\mu }=-0.5$ (solid), ${\alpha }_{\mu }=1$ (medium-dashed, orange), and ${\alpha }_{\mu }=2.5$ (long-dashed, green). The horizontal solid line is the current upper limit on the electron EDM set at $|d_e|=8.7\times 10^{-29}$. The other parameters are $|\mu |=4.1\times 10^2$, $|M_1|=2.8\times 10^2$, $|M_2|=3.4\times 10^2$, $|A_e|=3\times 10^6$, $m_0^{\tilde{\nu }}=4\times 10^6$, $|A_0^{\tilde{\nu }}|=5\times 10^6$, $\tan \beta =30$. All masses are in GeV, phases in rad and EDM in $e\mathrm{cm}$.The analysis shows that improvements in the electron EDM constraint can probe scalar masses in the 100 TeV–1 PeV region and beyond. Right panel: The same as the left panel except that the region below the current experiment limit is blown up. The analysis shows that an improvement by a factor of ten can allow one to probe up to and beyond 1 PeV in mass scales.

Figure 3

Upper diagrams: Supersymmetric contributions to the leptonic EDMs arising from the exchange of the charginos, sneutrinos and mirror sneutrinos (upper left) and the exchange of neutralinos, sleptons, and mirror sleptons (upper right) inside the loop. Lower diagrams: Nonsupersymmetric diagrams that contribute to the leptonic EDMs via the exchange of the $W$, the sequential and vectorlike neutrinos (lower left) and the exchange of the $Z$, the sequential and vectorlike charged leptons (lower right).

Figure 4

Left panel: Exhibition of the individual contributions to the EDM of the electron when there is no mixing between the vectorlike generation and the three regular generations. The parameters chosen for this case are the same as for the solid curve in Fig. 2 where ${\alpha }_{\mu }=-0.5$. As expected the contributions from the W-exchange (the long-dashed curve in orange) and the Z -exchange (dot-dashed purple curve) give vanishing contribution in this case, and the entire contribution arises from the chargino-exchange (the small-dashed curve in red) and the neutralino-exchange (the medium-dashed blue curve). Right panel: The parameter point chosen is the same as for the left panel except that mixing of the vectorlike generation with the regular three generations is allowed. The additional parameters chosen are $m_N=250,m_E=380$ and the f couplings set to $|f_3|=7.20\times 10^{-6}$, $|f_3^{\prime }|=1.19\times 10^{-4}$, $|f_3^{″}|=1.55\times 10^{-5}$, $|f_4|=8.13\times 10^{-4}$, $|f_4^{\prime }|=3.50\times 10^{-1}$, $|f_4^{″}|=6.29\times 10^{-1}$, $|f_5|=8.82\times 10^{-5}$, $|f_5^{\prime }|=5.36\times 10^{-5}$, $|f_5^{″}|=1.27\times 10^{-5}$. Their corresponding $CP$ phases set to ${\chi }_3=9.71\times 10^{-1}$, ${\chi }_3^{\prime }=7.86\times 10^{-1}$, ${\chi }_3^{″}=7.89\times 10^{-1}$, ${\chi }_4=7.66\times 10^{-1}$, ${\chi }_4^{\prime }=8.38\times 10^{-1}$, ${\chi }_4^{″}=8.23\times 10^{-1}$, ${\chi }_5=7.70\times 10^{-1}$, ${\chi }_5^{\prime }=1.47$, ${\chi }_5^{″}=7.82\times 10^{-1}$. All masses are in GeV, phases in rad and EDM in $e\mathrm{cm}$.

Figure 5

An exhibition of the dependence of $|d_e|$ on $m_0$ for various vectorlike masses. The curves correspond to $m_N=m_E=150$ (dot dashed), $m_N=m_E=200$ (solid), $m_N=m_E=250$ (dotted), $m_N=m_E=300$ (dashed). The parameters are $|\mu |=4.1\times 10^2$, $|M_1|=2.8\times 10^2$, $|M_2|=3.4\times 10^2$, $|A_0|=3\times 10^6$, $m_0^{\tilde{\nu }}=4\times 10^6$, $|A_0^{\tilde{\nu }}|=5\times 10^6$, $\tan \beta =50$. The $CP$ phases are ${\theta }_{\mu }=1$, ${\alpha }_1=1.26$, ${\alpha }_2=0.94$, ${\alpha }_{A_0}=0.94$, ${\alpha }_{A_0^{\tilde{\nu }}}=1.88$. The f couplings are $|f_3|=3.01\times 10^{-5}$, $|{f_3}^{\prime }|=8.07\times 10^{-6}$, $|{f_3}^{″}|=2.06\times 10^{-5}$, $|f_4|=8.13\times 10^{-4}$, $|{f_4}^{\prime }|=3.50\times 10^{-1}$, $|{f_4}^{″}|=6.29\times 10^{-1}$, $|f_5|=6.38\times 10^{-5}$, $|{f_5}^{\prime }|=1.03\times 10^{-6}$, $|{f_5}^{″}|=2.44\times 10^{-8}$. Their corresponding $CP$ phases are ${\chi }_3=7.91\times 10^{-1}$, ${\chi }_3^{\prime }=7.87\times 10^{-1}$, ${\chi }_3^{″}=7.78\times 10^{-1}$, ${\chi }_4=7.66\times 10^{-1}$, ${\chi }_4^{\prime }=8.38\times 10^{-1}$, ${\chi }_4^{″}=8.23\times 10^{-1}$, ${\chi }_5=7.57\times 10^{-1}$, ${\chi }_5^{\prime }=7.54\times 10^{-1}$, ${\chi }_5^{″}=7.83\times 10^{-1}$. All masses are in GeV, phases in rad, and $d_e$ in $e\mathrm{cm}$.

Figure 6

An exhibition of the dependence of $|d_e|$ on ${\alpha }_{\mu }$ for various $\tan \beta$. The curves correspond to $\tan \beta =20$ (dashed), $\tan \beta =30$ (dotted), $\tan \beta =40$ (solid), and $\tan \beta =50$ (dot dashed). The parameters used are $|\mu |=3.9\times 10^2$, $|M_1|=3.1\times 10^2$, $|M_2|=3.6\times 10^2$, $m_N=340,m_E=250$, $m_0=1.1\times 10^6$, $|A_0|=3.2\times 10^6$, $m_0^{\tilde{\nu }}=4.3\times 10^6$, $|A_0^{\tilde{\nu }}|=5.1\times 10^6$, ${\alpha }_1=1.88$, ${\alpha }_2=1.26$, ${\alpha }_{A_0}=0.94$, ${\alpha }_{A_0^{\tilde{\nu }}}=1.88$. The mixings are $|f_3|=2.88\times 10^{-4}$, $|f_3^{\prime }|=8.19\times 10^{-6}$, $|f_3^{″}|=9.19\times 10^{-5}$, $|f_4|=8.13\times 10^{-4}$, $|f_4^{\prime }|=3.50\times 10^{-1}$, $|f_4^{″}|=1.29\times 10^{-1}$, $|f_5|=5.75\times 10^{-6}$, $|f_5^{\prime }|=1.00\times 10^{-5}$, $|f_5^{″}|=2.49\times 10^{-7}$, ${\chi }_3=7.74\times 10^{-1}$, ${\chi }_3^{\prime }=7.73\times 10^{-1}$, ${\chi }_3^{″}=7.86\times 10^{-1}$, ${\chi }_4=7.6\times 10^{-1}$, ${\chi }_4^{\prime }=8.40\times 10^{-1}$, ${\chi }_4^{″}=8.20\times 10^{-1}$, ${\chi }_5=7.51\times 10^{-1}$, ${\chi }_5^{\prime }=8.19\times 10^{-1}$, ${\chi }_5^{″}=8.03\times 10^{-1}$. All masses are in GeV, phases in rad, and $d_e$ in $e\mathrm{cm}$.

Figure 7

An exhibition of the dependence of $|d_e|$ on ${\chi }_4^{″}$ for various $f_4^{″}$. The curves correspond to $f_4^{″}$ of 0.1 (dashed), 0.2 (dotted), 0.5 (solid), 1 (dot dashed). The other parameters are $|\mu |=1.1\times 10^6$, $|M_1|=2.8\times 10^6$, $|M_2|=3.4\times 10^6$, $m_N=250$, $m_E=380$, $m_0=1.1\times 10^6$, $|A_0|=3.2\times 10^6$, $m_0^{\tilde{\nu }}=1.4\times 10^6$, $|A_0^{\tilde{\nu }}|=5.1\times 10^6$, ${\alpha }_1=1.26$, ${\alpha }_2=0.94$, ${\alpha }_{A_0}=0.94$, ${\alpha }_{A_0^{\tilde{\nu }}}=1.88$, $\tan \beta =30$. The mixings are $|f_3|=2.93\times 10^{-4}$, $|f_3^{\prime }|=8.19\times 10^{-6}$, $|f_3^{″}|=9.15\times 10^{-5}$, $|f_4|=8.13\times 10^{-1}$, $|f_4^{\prime }|=3.50\times 10^{-1}$, $|f_5|=5.08\times 10^{-6}$, $|f_5^{\prime }|=9.98\times 10^{-6}$, $|f_5^{″}|=2.56\times 10^{-7}$, ${\chi }_3=7.86\times 10^{-1}$, ${\chi }_3^{\prime }=7.80\times 10^{-1}$, ${\chi }_3^{″}=8.02\times 10^{-1}$, ${\chi }_4=7.6\times 10^{-1}$, ${\chi }_4^{\prime }=8.4\times 10^{-1}$, ${\chi }_5=7.39\times 10^{-1}$, ${\chi }_5^{\prime }=7.82\times 10^{-1}$, ${\chi }_5^{″}=7.82\times 10^{-1}$. All masses are in GeV, phases in rad and $d_e$ in $e\mathrm{cm}$.