$CP$ violation in pseudo-Dirac fermion oscillations

Supersymmetric theories with a $U(1{)}_R$ symmetry have Dirac gauginos, solve the supersymmetric flavor and $CP$ problems, and have distinctive collider signatures. However, when supergravity is included, the $U(1{)}_R$ must be broken, adding small Majorana mass terms which split the mass of the two components of the Dirac gaugino and lead to oscillations between $U(1{)}_R$ charge eigenstates. We present a general study of fermion-antifermion oscillations in this system, including the effects of decays and $CP$ violation. We consider the effects of such oscillations in the case where the two $U(1{)}_R$ charge eigenstates can decay into the same final state and show that $O(1)$ $CP$ violation is allowed. In the case of decays into final states containing leptons, such $CP$ violation can be observed as a same-sign dilepton asymmetry.

Figure 1

Definitions of the self-energy functions for $i$, $j=\lambda$, $\mathcal{O}$. The shaded circles represent the sum of all one-particle irreducible, connected Feynman diagrams. External legs are amputated. $\alpha$ and $\beta$ are spinor indices. Arrows (and dots) denote left- or right-handed chiralities.

Figure 2

The one-loop contribution to $\mathbf{\Xi }$ for $i$, $j=\lambda$, $\mathcal{O}$ arising from the Yukawa interaction in Eq. (10). Contributions to the chirality-flipping two-point function $\mathbf{\Omega }$ are proportional to $m_{\overline{d}}$, which we neglect.

Figure 3

Two-loop corrections to the two-point functions, ${\mathrm{\Xi }}_{ij}$, for $i$, $j=\lambda$, $\mathcal{O}$, that arise due to the couplings in Eq. (54).

Figure 4

The same-sign dimuon asymmetry, $A$, of pseudo Dirac gluino decays as defined in Eq. (64) (upper/black solid curve) and the approximate expression for $A$ in Eq. (71) (upper/black dashed curve) along with the product of $A$ and the ratio of same-sign muon decays, $R$, defined in Eq. (65) (lower/red solid curve) and the product of the approximate expressions for $A$ [Eq. (71)] and $R$ [Eq. (72)] (lower/red dashed curve) as functions of $2M_M/\mathrm{\Gamma }\simeq x=\mathrm{\Delta }m/\mathrm{\Gamma }$. We have taken $\mathrm{\Gamma }=300\mathrm{eV}$, $r=0.14$, and ${\varphi }_{\mathrm{\Gamma }}=-\pi /3$.