Majoron at two loops

We present singlet-Majoron couplings to Standard Model particles through two loops at leading order in the seesaw expansion, including couplings to gauge bosons as well as flavor-changing quark interactions. We discuss and compare the relevant phenomenological constraints on Majoron production as well as decaying Majoron dark matter. A comparison with standard seesaw observables in low-scale settings highlights the importance of searches for lepton-flavor-violating two-body decays $\ell \to {\ell }^{\prime }+\mathrm{Majoron}$ in both the muon and the tau sectors.

Figure 1

Loop-induced Majoron couplings to charged fermions with the Majorana neutrino mass eigenstates $n_i$ running in the loops.

Figure 2

Loop-induced Majoron couplings to $WW$ and $ZZ$ at one loop. These diagrams generate couplings that are subdominant in the seesaw expansion.

Figure 3

Representative two-loop diagrams contributing to loop-induced Majoron couplings to vector bosons $\{V{V}^{\prime }\}=\{gg,\gamma \gamma ,Z\gamma ,ZZ,W^{+}{W}^{-}\}$. Set I contains the two-loop 1PI diagrams, and Set II contains reducible diagrams dominated by $J\text{−}Z$ mixing. Here, $n_i$ and $n_j$ are Majorana neutrino mass eigenstates and $f$ SM fermions (not necessarily all identical).

Figure 4

$J\gamma \gamma$ coupling for various hierarchies in the diagonal elements of $M_D{M}_D^{†}$. The black solid line shows $(M_D{M}_D^{†}{)}_{ee}=(M_D{M}_D^{†}{)}_{\mu \mu }=(M_D{M}_D^{†}{)}_{\tau \tau }=(100\mathrm{GeV}{)}^2$; the blue dashed line shows $(M_D{M}_D^{†}{)}_{ee}=(M_D{M}_D^{†}{)}_{\mu \mu }=(100\mathrm{GeV}{)}^2$, $(M_D{M}_D^{†}{)}_{\tau \tau }=0$ (which corresponds to $g_{Jee}=g_{J\mu \mu }=0$); the red dotted line shows $(M_D{M}_D^{†}{)}_{\mu \mu }=(M_D{M}_D^{†}{)}_{\tau \tau }=(100\mathrm{GeV}{)}^2$, $(M_D{M}_D^{†}{)}_{ee}=0$; and the green dot-dashed line shows $(M_D{M}_D^{†}{)}_{\mu \mu }=(100\mathrm{GeV}{)}^2\gg (M_D{M}_D^{†}{)}_{ee}$, $(M_D{M}_D^{†}{)}_{\tau \tau }$.

Figure 5

Representative two-loop diagrams for off-diagonal Majoron couplings to quarks.

Figure 6

Upper limits on combinations of $K_{\alpha \beta }=(M_D{M}_D^{†}{)}_{\alpha \beta }/(vf)$ for Majoron masses above MeV. The shaded regions exclude $|K_{\alpha \beta }|$ or $\mathrm{tr}(K)$ by nonobserved rare decays, the dashed lines show the potential future reach; see text for details. $K\to \pi J$ and $B\to KJ$ further scale with $\log (M_R/m_W)$, which has been set to $\mathbb{1}$ here. The black region is a very naive estimate of SN1987 constraints on the diphoton coupling, setting for simplicity $K_{\ell \ell }=\mathrm{tr}(K)/3$. The yellow region is the SN1987 constraint on the $JNN$ coupling [81]. The off-diagonal entries have to satisfy $|K_{\alpha \beta }|<\mathrm{tr}(K)/2$; see Eq. (7).

Figure 7

Limits on $(M_D{M}_D^{†}{)}_{\alpha \beta }/v^2$ from flavor-conserving non-Majoron (gray region) and flavor-violating observables (colored lines), dashed lines denoting future reach, adopted from Ref. [17]. We set $M_R/\mathrm{TeV}=\mathbb{1}$, $f=1\mathrm{TeV}$, and assume a matrix texture that gives $(M_D{M}_D^{†}{)}_{\alpha \beta }=\sqrt{(M_D{M}_D^{†}{)}_{\alpha \alpha }(M_D{M}_D^{†}{)}_{\beta \beta }}$; see text for details. The Majoron mass is assumed to be between 100 MeV and GeV in the left figure and between MeV and 100 MeV in the right figure.