$\mathit{CP}$ violation from the standard model to strings

A review of $\mathit{CP}$ violation from the standard model to strings is given which includes a broad landscape of particle physics models, encompassing the nonsupersymmetric four-dimensional (4D) extensions of the standard model, and models based on supersymmetry, on extra dimensions, on strings, and on branes. The supersymmetric models discussed include complex minimal supergravity unified model and its extensions, while the models based on extra dimensions include five-dimensional models including models based on warped geometry. $\mathit{CP}$ violation beyond the standard model is central to achieving the desired amount of baryon asymmetry in the Universe via baryogenesis and leptogenesis. They also affect a variety of particle physics phenomena: electric dipole moments, $g-2$, relic density and detection rates for neutralino dark matter in supersymmetric theories, Yukawa unification in grand unified and string based models, and sparticle production cross sections, and their decay patterns and signatures at hadron colliders. Additionally $\mathit{CP}$ violations can generate $\mathit{CP}$ even–$\mathit{CP}$ odd Higgs mixings, affect the neutral Higgs spectrum, and lead to phenomena detectable at colliders. Prominent among these are the $\mathit{CP}$ violation effects in decays of neutral and charged Higgs bosons. Neutrino masses introduce new sources of $\mathit{CP}$ violation which may be explored in neutrino factories in the future. Such phases can also enter in proton stability in unified models of particle interactions. The current experimental status of $\mathit{CP}$ violation is discussed and possibilities for the future outlined.

Figure 1

The penguin diagram that contributes to $B$ decays.

Figure 2

The phenomena of spontaneous vs explicit breaking in a 5D compactification model (315). The effective potential $V_{\mathrm{eff}}$ for four cases of twist angles with $\alpha =0$, $\pi ∕2$, $\pi$, $3\pi ∕2$. The cases $\alpha =0,\pi$ correspond to spontaneous breaking and $\alpha =\pi ∕2,3\pi ∕2$ correspond to explicit breaking.

Figure 3

Contributions to the electric dipole moment of a lepton or of a quark from the exchange of the charginos, the neutralinos, and the gluino. The internal dashed line in the loop is the scalar field ${\varphi }_k$, the solid line is the fermion field ${\psi }_i$, and the external wiggly line is the external photon line.

Figure 4

The quark-squark-gluino exchange contribution to the purely gluonic dimension 6 operator. The dashed line in the upper semicircle in the loop is the squark $\tilde{q}$, the internal horizontal solid line is the gluino $\tilde{g}$, the solid line on the lower semicircle in the loop is the quark $q$, while the external wiggly lines are the gluons.

Figure 5

Two loop Barr-Zee type diagrams that contribute to the EDMs in supersymmetry (151).

Figure 6

Size estimate of the two loop contribution to the EDMs in supersymmetry with phases only in the third generation (151).

Figure 7

Chargino and neutralino exchanges contributing to the muon $g-2$ which generate dependence of $g_{\mu }-2$ on phases.

Figure 8

The dependence of $a_{\mu }$ on a SUSY $\mathit{CP}$ phase. The curves correspond to the four following cases (361): (1) $m_0=70$, $m_{1∕2}=99$, $\tan \beta =3$, $\mid A_0\mid =5.6$, ${\xi }_1=-1$, ${\xi }_3=0.62$; ${\theta }_{\mu }=2.35$, ${\alpha }_{A_0}=0.4$; (2) $m_0=80$, $m_{1∕2}=99$, $\tan \beta =5$, $\mid A_0\mid =5.5$, ${\xi }_1=-0.8$, ${\xi }_3=0.95$; ${\theta }_{\mu }=1.98$, ${\alpha }_{A_0}=0.4$; (3) $m_0=75$, $m_{1∕2}=132$, $\tan \beta =4$, $\mid A_0\mid =6.6$, ${\xi }_1=-1$, ${\xi }_3=2.74$; ${\theta }_{\mu }=1.2$, ${\alpha }_{A_0}=-1.5$; (4) $m_0=70$, $m_{1∕2}=99$, $\tan \beta =6$, $\mid A_0\mid =3.2$, ${\xi }_1=0.63$, ${\xi }_3=0.47$, ${\theta }_{\mu }=2.7$, ${\alpha }_{A_0}=-0.4$, where all masses are in GeV units and all phases are in rad.

Figure 9

The phenomenon of $\mathit{CP}$ even–$\mathit{CP}$ odd Higgs mixing via the SUSY $\mathit{CP}$ phases. The $\mathit{CP}$ even component ${\varphi }_1$ of $H_1$ (upper curves) and the $\mathit{CP}$ odd component ${\psi }_{1D}$ of $H_1$ (lower curves) including the stop, sbottom, stau, chargino, and neutralino sector contributions as a function of ${\theta }_{\mu }$. The common parameters are $m_A=300$, $Q=320$, $m_0=100$, $m_{1∕2}=500$, ${\xi }_2=0.5$, ${\alpha }_0=0.3$, $\mid A_0\mid =1$. For curves with diamonds $\tan \beta =15$, ${\xi }_1=1.5$, for circles $\tan \beta =8$, ${\xi }_1=1.5$, and for triangles $\tan \beta =8$, ${\xi }_1=0.5$ where all masses are in GeV and all angles are in radians.

Figure 10

Loop diagrams with $\mathit{CP}$ dependent vertices that contribute to charged Higgs decays into charginos and neutralinos.

Figure 11

(Color online) The relic density constrains with large phases from the analysis of 293. The curve labeled (i) is for the case $m_0=1040$, $\mid A_0\mid =0$, $\tan \beta =40$, ${\theta }_{mu}=2.9$, ${\alpha }_A=0$, ${\xi }_1=1.0$, ${\xi }_2=0.15$, ${\xi }_3=0.5$, while the curve labeled (ii) corresponds to $m_0=1080$, $\mid A_0\mid =0$, $\tan \beta =40$, ${\theta }_{\mu }=0.6$, ${\alpha }_A=0$, ${\xi }_1=0.5$, ${\xi }_2=-0.6$, ${\xi }_3=1.6$. For case (i) EDM constraints are satisfied when $m_{1∕2}=1250$ and for case (ii) they are satisfied when $m_{1∕2}=1100$. All masses are in units of GeV and all angles in radians.

Figure 12

The counterterm diagram which produces the leading term in amplitude proportional to ${\tan }^3\beta$ in the branching ratio $B_s^0\to l^{+}{l}^{-}$.

Figure 13

(Color online) The strong dependence on ${\alpha }_A$ of the ratio of the branching ratios $B(B_s^0\to {\mu }^{+}{\mu }^{-})∕B{(B_s^0\to {\mu }^{+}{\mu }^{-})}_0$, where $B{(B_s^0\to {\mu }^{+}{\mu }^{-})}_0$ is the branching ratio when all phases are set to zero (366). The curves in ascending order are for values of $\mid A_0\mid$ of 1,2,3,4,5. The other parameters are $m_0=200\mathrm{GeV}$, $m_{1∕2}=200\mathrm{GeV}$, $\tan \beta =50$, ${\xi }_1={\xi }_2=\pi ∕4$, ${\xi }_3=0$, and ${\theta }_{\mu }=2$.

Figure 14

A sample of one loop diagrams with $\mathit{CP}$ phase dependent vertices that contribute to the decay of the stops.

Figure 15

(Color online) The strong flavor dependence via $A$ nonuniversalities in enhancing the muon EDM relative to the electron EDM in the cancellation region from the analysis of 364. Plotted are the electron EDM $d_e$ (solid line with squares), the neutron EDM $d_n$ (dashed line with plus signs), and the muon EDM $d_{\mu }$ as a function of $\mid A_0\mid$ for $\tan \beta =20$, $m_0=200$, $m_{1∕2}=246$, ${\xi }_1=0.28$, ${\xi }_2=-0.51$, ${\xi }_3=-0.11$, ${\theta }_{\mu }=0.4$, and ${\alpha }_{A_e}=1.02$ where all masses are in GeV. The curve with the dashed line with triangles pointed down is a plot of the muon EDM $d_{\mu }$ which have all the same parameters as for $d_e$ and $d_n$ except that ${\alpha }_{A_{\mu }}=0.0$ and the curve with the dashed line with triangles pointed up is a plot of the muon EDM $d_{\mu }$ which have all the same parameters as for $d_e$ and $d_n$ except that $\mid A_{\mu }\mid =6.0$ and ${\alpha }_{A_{\mu }}=-2.0$.

Figure 16

A sample of diagrams with $\mathit{CP}$ dependent vertices that contribute to the NLO corrections to the epsilons in $b\to s+\gamma$ decay. There are a total of 20 such diagrams.

Figure 17

Generation of lepton number asymmetry via decay of the right handed neutrino $\left(N\right)$ by interference between the tree, vertex, and self-energy loop diagrams.