$\mathrm{\Delta }L=3$ processes: Proton decay and the LHC

We discuss lepton number violation in three units. From an effective field theory point of view, $\mathrm{\Delta }L=3$ processes can only arise from dimension 9 or higher operators. These operators also violate baryon number, hence many of them will induce proton decay. Given the high dimensionality of these operators, in order to have a proton half-life in the observable range, the new physics associated to $\mathrm{\Delta }L=3$ processes should be at a scale as low as 1 TeV. This opens up the possibility of searching for such processes not only in proton decay experiments but also at the LHC. In this work we analyze the relevant $d=9$, 11, 13 operators which violate lepton number in three units. We then construct one simple concrete model with interesting low- and high-energy phenomenology.

Figure 1

Proton decay $p\to {\pi }^{-}{\pi }^{-}{e}^{+}{e}^{+}{e}^{+}$ induced by a $d=13$ operator in our model. Diagrams permuting the 4 external up-quarks are equally allowed. Note the presence of the trilinear coupling $\mu S_u{S}_d{S}_d^{\prime }$ in the diagram.

Figure 2

Proton decay into four bodies, $p\to {\pi }^{-}{e}^{+}{e}^{+}\overline{\nu }$ and $p\to {\pi }^0{e}^{+}\overline{\nu }\overline{\nu }$, induced by the same $d=13$ operator as in Fig. 1.

Figure 3

Proton decay lifetime as function of $m_N$, for $\mu =10\mathrm{TeV}$, $m_{S_u}=m_{S_d}=m_{S_d^{\prime }}=1\mathrm{TeV}$, $Y_1=Y_2=1$, $Y_3=0.26$ and $Y_4=0.27$. The figure also shows existing constraints from proton decay searches, for the chosen benchmark point. Inclusive searches exclude $m_N$ smaller than $\sim 150\mathrm{GeV}$. Also, for reference, are shown the current Super-Kamiokande limits on three body decay mode $p\to e^{+}\overline{\nu }\overline{\nu }$, which is currently the most stringent limit for a three body decay mode. Note, however, that this limit does not apply to our case. We have also shown the expected reach of the Hyper-Kamiokande and DUNE experiments for two body decay modes.

Figure 4

Feynman diagrams for $S_u$ decay at the LHC.

Figure 5

Loop generation of the dimension 11 operator $\partial Q\overline{u^c}\overline{u^c}LL\overline{e^c}H$ in our model. This leads to $p\to e^{+}\overline{\nu }\overline{\nu }$, for example.