$CP$ violation in bilinear $R$-parity violation and its consequences for the early universe

Supersymmetric models with bilinear $R$-parity violation (BRpV) provide a framework for neutrino masses and mixing angles to explain neutrino oscillation data. We consider $CP$ violation within the new physical phases in BRpV and discuss their effect on the generation of neutrino masses and the decays of the lightest supersymmetric particle (LSP), being a light neutralino with mass $\sim 100\mathrm{GeV}$, at next-to-leading order. The decays affect the lepton and via sphaleron transitions the baryon asymmetry in the early universe. For a rather light LSP, asymmetries generated before the electroweak phase transition via e.g. the Affleck-Dine mechanism are reduced up to 2 orders of magnitude, but are still present. On the other hand, the decays of a light LSP themselves can account for the generation of a lepton and baryon asymmetry, the latter in accordance with the observation in our universe, since the smallness of the BRpV parameters allows for an out-of-equilibrium decay and sufficiently large $CP$ violation is possible consistent with experimental bounds from the nonobservation of electric dipole moments.

Figure 1

Dominant NLO virtual contribution generating a $CP$ asymmetry between the different final states for (a) ${\tilde{\chi }}_1^0\to l^{\pm }{W}^{\mp }$; (b) ${\tilde{\chi }}_1^0\to \nu Z$; (c) ${\tilde{\chi }}_1^0\to \nu h$.

Figure 2

Values of $R$-parity violating parameters as a function of $\varphi ≔{\varphi }_1={\varphi }_2={\varphi }_3$ in degrees for scenario $P_2$, where in (a) we give $|{\epsilon }_i|$ in GeV with $i=1$ (black, circle), 2 (blue, square) and 3 (red, diamond) and in (b) $v_i$ in $10^{-3}\mathrm{GeV}$ with the same coding.

Figure 3

$J_{CP}$ as a function $\varphi ≔{\varphi }_1={\varphi }_2={\varphi }_3$ in degrees for (a) $P_1$ and (b) $P_2$.

Figure 4

Decay widths ${\mathrm{\Gamma }}^{+}$ for the final states $e^{+}{W}^{-}$ (black, circle), ${\mu }^{+}{W}^{-}$ (blue, square), ${\tau }^{+}{W}^{-}$ (red, diamond), ${\overline{\nu }}_{3}Z$ (green, triangle) as a function of $\varphi ≔{\varphi }_1={\varphi }_2={\varphi }_3$ in degrees for (a) $P_1$ and (b) $P_2$.

Figure 5

$CP$ asymmetry $|{\delta }_{\mathrm{\Gamma }}|$ for the final states $e^{\pm }{W}^{\mp }$ (black, circle), ${\mu }^{\pm }{W}^{\mp }$ (blue, square), ${\tau }^{\pm }{W}^{\mp }$ (red, diamond), ${\nu }_3({\overline{\nu }}_3)Z$ (green, triangle) defined in Eq. (43) as a function of $\varphi ≔{\varphi }_1={\varphi }_2={\varphi }_3$ in degrees for (a) $P_1$ and (b) $P_2$.

Figure 6

$CP$ asymmetry $|{\delta }_{\mathrm{\Gamma }}|$ for the final states $e^{\pm }{W}^{\mp }$ (black, empty circle), ${\mu }^{\pm }{W}^{\mp }$ (blue, empty square), ${\tau }^{\pm }{W}^{\mp }$ (red, filled diamond), ${\nu }_3({\overline{\nu }}_3)Z$ (green, filled triangle), ${\nu }_1({\overline{\nu }}_1)h$ (yellow, filled circle), ${\nu }_2({\overline{\nu }}_2)h$ (purple, filled square) and ${\nu }_3({\overline{\nu }}_3)h$ (brown, empty triangle) defined in Eq. (43) as a function of $\varphi ≔{\varphi }_1={\varphi }_2={\varphi }_3$ in degrees for $P_1^{\prime }$.

Figure 7

(a) Lepton asymmetry $|{\delta }_N|$ as defined in Eq. (30) as a function of $x$ for $CP$ conserving BRpV for all widths set to $5\times 10^{-14}\mathrm{GeV}$ for different asymmetries ${\delta }_N$ at $x=10^{-2}$; (b) LSP density as a function of $x$ for the same cases.

Figure 8

Lepton asymmetry $|{\delta }_N|$ as defined in Eq. (30) as a function of $x$ for $CP$ conserving BRpV for $P_1$ (black) and $P_2$ (red, dashed) for different asymmetries ${\delta }_N$ at $x=10^{-2}$.

Figure 9

Lepton asymmetry $|{\delta }_N|$ as defined in Eq. (30) for (a) $P_1$ with phase $\varphi ≔{\varphi }_1={\varphi }_2={\varphi }_3=5$ degrees; (b) $P_2$ with phase $\varphi ≔{\varphi }_1={\varphi }_2={\varphi }_3=15$ degrees in both cases for different initial asymmetries ${\delta }_N$ at $x=10^{-2}$.

Figure 10

Resulting lepton asymmetry $|{\delta }_N|$ at $x=10^2$ as a function of $\varphi ≔{\varphi }_1={\varphi }_2={\varphi }_3$ for $P_2$ for zero initial asymmetry ${\delta }_N=0$ at $x=10^{-2}$.

Figure 11

(a) Lepton asymmetry $|{\delta }_N|$ (solid) and baryon asymmetry $|{\delta }_B|$ (dashed) for scenario $P_2$ with phase $\varphi ≔{\varphi }_1={\varphi }_2={\varphi }_3=15$ degrees for two different initial asymmetries ${\delta }_N,{\delta }_B$ at $x=10^{-2}$; (b) Resulting baryon asymmetry $|{\delta }_B(x=0.66)|$ as a function of $\varphi ≔{\varphi }_1={\varphi }_2={\varphi }_3$ for $P_2$ for zero initial asymmetry ${\delta }_N={\delta }_B=0$ at $x=10^{-2}$.

Figure 12

(a) Cross section $⟨{\sigma }_{ij}v⟩$ in $1/{\mathrm{GeV}}^2$ as a function of $x$ for $P_2$ for different final states: $ij=d\overline{d}/s\overline{s}/b\overline{b}$ (black), $u\overline{u}/c\overline{c}/t\overline{t}$ (black, dashed), ${\tau }^{+}{\tau }^{-}$ (red, dot-dashed), $e^{+}{e}^{-}/{\mu }^{+}{\mu }^{-}$ (red, dotted); (b) $K_1/K_2{N}^{\mathrm{eq}}\mathrm{\Gamma }$ in GeV with $\mathrm{\Gamma }=10^{-14}\mathrm{GeV}$ (red, dashed) and $\widehat{\sigma }({\tilde{\chi }}_1^0{\tilde{\chi }}_1^0\to b\overline{b})N^{\mathrm{eq},2}$ in GeV (black) as a function of $x$.

Figure 13

(a) LSP number density $|N_{{\tilde{\chi }}_1^0}-N_{{\tilde{\chi }}_1^0}^{\mathrm{eq}}|$ (solid) and $N_{{\tilde{\chi }}_1^0}$ (dashed) as a function of $x$ for only annihilation (green), only decays (red), annihilation and decays (black) for $P_1$ with phase ${\varphi }_i=5$ degrees. (b) Evolution of ${\delta }_N$ as defined in Eq. (30) as a function of $x$ for only decays (red, dashed), and annihilation and decays (black) for the same case.