Sensitivity of lepton number violating meson decays in different experiments

We study the discovery prospect of different three body lepton number violating (LNV) meson decays $M_1^{-}\to {\ell }_1^{-}{\ell }_2^{-}{M}_2^{+}$ in the framework of the right-handed (RH) neutrino extended Standard Model. We consider a number of ongoing experiments, such as NA62 and LHCb at CERN, Belle II at SuperKEK, as well as at the proposed future experiments, SHiP, MATHUSLA and FCC-ee. The RH Majorana neutrino $N$ mediating these meson decays provides a resonant enhancement of the rates, if the mass of $N$ lies in the range (100 MeV–6 GeV). We consider the effect of parent mesons velocity, as well as the effect of finite detector size. Using the expected upper limits on the number of events for the LNV decay modes, $M_1^{-}\to {\ell }_1^{-}{\ell }_2^{-}{\pi }^{+}$ ($M_1=B,B_c,D,D_s$ and $K$), we analyze the sensitivity reach of the mixing angles $|V_{eN}{|}^2$, $|V_{\mu N}{|}^2$, $|V_{\tau N}{|}^2$, $|V_{eN}{V}_{\mu N}|$, $|V_{eN}{V}_{\tau N}|$ and $|V_{\mu N}{V}_{\tau N}|$ as a function of heavy neutrino mass $M_N$. We show that inclusion of parent meson velocity can account to a large difference for active-sterile mixing, especially for $D$, $D_s$ meson decay at SHiP and $K$ meson decay at NA62. Taking into account the velocity of the $D_s$ meson, the future beam dump experiment SHiP can probe $|V_{eN}{|}^2\sim {10}^{-9}$. For RH neutrino mass in between 2–5 GeV, MATHUSLA can provide the best sensitivity reach of active-sterile mixings.

Figure 1

The Feynman diagrams for the lepton number violating meson decays. These processes can produce resonance enhancement. See the text for details.

Figure 2

The t-channel diagram for the lepton number violating meson decay. See the text for details.

Figure 3

Left panel: The total decay width of heavy neutrino N with the assumption, $|V_{eN}{|}^2=|V_{\mu N}{|}^2=|V_{\tau N}{|}^2=1$. Right panel: $a_e$, $a_{\mu }$ and $a_{\tau }$ as function of mass $M_N$.

Figure 4

Dependence of the decay length $L_N$ on parent meson velocity as a function of RH neutrino mass $M_N$. The upper panel is for $B$ meson decay at Belle-II (left) and SHiP (right). The left figure of the lower panel is for $D_s$ meson decay at SHiP and the right figure of the lower panel is for $K$ meson decay at NA62.

Figure 5

Dependence of parameter $x=\frac{L_D}{L_N}$ on parent meson velocity as a function of RH neutrino mass $M_N$. The upper panel is for $B$ meson decay at Belle-II (left) and SHiP (right). The left figure of the lower panel is for $D_s$ meson decay at SHiP and the right figure of the lower panel is for $K$ meson decay at NA62.

Figure 6

Future sensitivity reach and present limits on the mixing angles as a function of RH neutrino mass $M_N$. The black and red lines stand for meson decay at rest and meson decay with finite momentum, respectively. The upper panel corresponds to the $B$ meson decay at FCC-ee (left) and SHiP (right). The left figure of the middle panel is for $D_s$ meson decay at SHiP and the right figure of the middle panel is for $K$ meson decay at NA62. The lower panel represents $B_c$ meson decay at LHCb (left) and SHiP (right).

Figure 7

Variation of the future sensitivity reach and present limits on the mixing angle $|V_{eN}{|}^2$ with respect to the mass $M_N$. We consider meson decay channel $M_1\to ee\pi$. The left figure in the upper panel is for $K$ meson decay at NA62. The right figure in the upper panel is for $B$ meson decay at SHiP (black), MATHUSLA (magenta), LHCb (red), FCC-ee (blue) and Belle-II (green). The left figure in the middle panel is for $B_c$ meson decay at SHiP (black) and LHCb (red). The right figure of the middle panel is for $D$ meson decay at SHiP (black), MATHUSLA (magenta), LHCb (red) and Belle-II (green). The last figure is for $D_s$ meson decay at SHiP (black), LHCb (red) and Belle-II (green).

Figure 8

Same as Fig. 7 except now the limits are on the mixing angle $|V_{\mu N}{|}^2$ from the meson decays $M_1\to \mu \mu \pi$.

Figure 9

Same as Fig. 7. The plots in different panels show the sensitivity reach of the mixing angle $|V_{eN}{V}_{\mu N}|$ from the meson decays $M_1\to e\mu \pi$.

Figure 10

Future sensitivity reach and present limits on the mixing angles $|V_{eN}{V}_{\tau N}|$, $|V_{\mu N}{V}_{\tau N}|$ and $|V_{\tau N}{|}^2$ with respect to the mass $M_N$ from meson decays $B,B_c\to e\tau \pi$, $B,B_c\to \mu \tau \pi$ and $B,B_c\to \tau \tau \pi$, respectively at various experiments. The upper panel is for $B$ meson decay at SHiP (black), MATHUSLA (magenta), LHCb (red), FCC-ee (blue) and Belle-II (green). The middle panel is for $B_c$ meson decay at SHiP (black) and LHCb (red). The left figure of the lower panel is for the bound on $|V_{\tau N}{|}^2$ from $B\to \tau \tau \pi$ decay at SHiP (black), MATHUSLA (magenta), LHCb (red), FCC-ee (blue) and Belle-II (green). The right figure of the lower panel is for the bound on $|V_{\tau N}{|}^2$ from $B_c\to \tau \tau \pi$ decay at SHiP (black) and LHCb (red).

Figure 11

Combined bounds (thick-blue solid) on mixing angle $|V_{eN}{|}^2$ as a function of mass $M_N$ from all the meson decays. The strongest lower limits from the Seesaw (green dashed) and BBN (brown dashed) on $|V_{eN}{|}^2$ are shown in this plot. The strongest upper bounds on $|V_{eN}{|}^2$ are obtained from PS191 (magenta dashed), JNIR (black dashed), CHARM (red dashed), DELPHI (orange dashed) and leptogenesis (black dotted) are shown in this plot. The shaded region is ruled out by these results. The projected upper limits from the NA62 (blue dashed), GERDA (dark-cyan dashed), FCC-ee (magenta dashed) and DUNE (pink dashed) on $|V_{eN}{|}^2$ are shown. Prospective bounds from the FASER with detector radius $R=20\mathrm{cm}$ is shown by FASER20 (green dotted) whereas the limits from the $R=1\mathrm{m}$ are shown by FASER1 (green-dot dashed). Prospective upper limits from the MATHUSLA at the FCC-hh for the $W/Z$ boson decays at the FCC-hh for MATHUSLA standard benchmark surface version are represented by M-WZ-FCC-hh(S) (magenta-dot dashed) and forward version M-WZ-FCC-hh(F) (magenta dotted). Prospective limits for the heavy neutrinos produced from the $W/Z$ decays are represented by M-HL-LHC-WZ (magenta-dot dashed) and $B/D$ meson decays are represented by M-HL-LHC-BD (magenta dotted) at the HL-LHC.

Figure 12

Combined bounds (thick-blue solid) on mixing angle $|V_{\mu N}{|}^2$ as a function of mass $M_N$ from all the meson decays. Strongest lower limits from the seesaw (green dashed) and BBN (brown dashed) and strongest upper limits from PS191 (yellow dashed), E949 (magenta dashed), NuTeV (blue dashed), CHARM (black dashed), Belle (red dashed) and DELPHI (orange dashed) are shown in this plot. The shaded region is ruled out by these results. The projected upper limits from the NA62 (brown dotted), leptogenesis (black dotted), FCC-ee (magenta dashed) and DUNE (cyan dashed) and FASER with detector radius $R=20\mathrm{cm}$ is shown by FASER20 (green-dot dashed) whereas the limits from the $R=1\mathrm{m}$ are shown by FASER1 (green dotted). Prospective upper limits from the MATHUSLA at the FCC-hh for the $W/Z$ boson decays at the FCC-hh for MATHUSLA standard benchmark surface version are represented by M-WZ-FCC-hh(S) (magenta-dot dashed) and forward version M-WZ-FCC-hh(F) (magenta dotted). Prospective limits for the heavy neutrinos produced from the $W/Z$ decays are represented by M-HL-LHC-WZ (magenta-dot dashed) and $B/D$ meson decays are represented by M-HL-LHC-BD (magenta dotted) at the HL-LHC. Experimental bounds from the ATLAS displaced vertex searches of the Majorana heavy neutrino for the lepton number violating (LNV) channel are represented by ATLAS-displaced (LNV) (light-blue dotted) and the limits obtained from the lepton number conserving (LNC) channel are represented by ATLAS-displaced (LNC) (sea-blue dotted). Theoretical limits from lepton-jet search for $1\mathrm{GeV}\le M_N\le 10\mathrm{GeV}$ has been represented by lepton jet (13 TeV) (black dashed). Corresponding limits for $M_N\ge 5\mathrm{GeV}$ for the 13 TeV LHC and 100 TeV are shown by LJ, 13 TeV (magenta-dot dashed) and LJ, 100 TeV (magenta dotted) respectively.

Figure 13

Combined bounds (thick-blue solid) on mixing angle $|V_{\tau N}{|}^2$ as a function of mass $M_N$ from all the meson decays. Strongest lower limits from the Seesaw (green dashed) and BBN (brown dashed) on $|V_{\tau N}{|}^2$ are shown in this plot. The strongest upper bounds on $|V_{\tau N}{|}^2$ are obtained from CHARM (purple dashed), EWPD (orange dashed), leptogenesis (black dotted), DELPHI (magenta dashed) are shown in this plot. The shaded region is ruled out by these results. The projected upper limits from the NA62 (cyan solid), FCC-ee (darker-blue solid) and DUNE (light-blue solid) on $|V_{\tau N}{|}^2$ are shown. Prospective bounds from the FASER with detector radius $R=20\mathrm{cm}$ is shown by FASER20 (black dashed) whereas the limits from the $R=1\mathrm{m}$ is shown by FASER1 (black dotted). Prospective upper limits from the MATHUSLA at the FCC-hh for the $W/Z$ boson decays at the FCC-hh for MATHUSLA standard benchmark surface version are represented by M-WZ-FCC-hh(S) (magenta-dot dashed) and forward version M-WZ-FCC-hh(F) (magenta dotted). Prospective limits for the heavy neutrinos produced from the $W/Z$ decays are represented by M-HL-LHC-WZ (magenta-dot dashed) and $B/D$ meson decays are represented by M-HL-LHC-BD (magenta dotted) at the HL-LHC. The prospective limits from the B-factory (red dashed and red solid) and IceCube (darker-cyan dashed) are also shown in this figure.

Figure 14

Combined bounds on mixing angle $|V_{eN}{V}_{\mu N}|$ as a function of mass $M_N$ from all the meson decays. The other strongest bounds on $|V_{eN}{V}_{\mu N}|$ from BBN (magenta dashed), PS191 (black dashed), NuTeV, CHARM (blue dashed), DELPHI (orange dashed) are also shown in this figure. Prospective bounds on $|V_{eN}{V}_{\mu N}|$ from the $\mu \to e$ (Ti) (green dashed) and $\mu \to e$ (Al) (red dashed) are also shown for $M_N\le 10\mathrm{GeV}$.

Figure 15

Combined bounds (thick-blue solid) on mixing angle $|V_{eN}{V}_{\tau N}|$ as a function of mass $M_N$ from all the meson decays. Upper limits from $\tau \to e\pi \pi$ (magenta dashed) and $\tau \to e^{-}{\pi }^{+}{\pi }^{+}$ (black dashed) are also shown for $M_N\le 10\mathrm{GeV}$.

Figure 16

Combined bounds (thick-blue solid) on mixing angle $|V_{\mu N}{V}_{\tau N}|$ as a function of mass $M_N$ from all the meson decays. Upper limits from $\tau \to \mu \pi \pi$ (magenta dashed) and $\tau \to {\mu }^{-}{\pi }^{+}{\pi }^{+}$ (black dashed) are also shown for $M_N\le 10\mathrm{GeV}$.