Pseudo-Majoron as Light Mediator of Singlet Scalar Dark Matter

TL;DR

This paper explores a light pseudoscalar mediator, called pseudo-majoron, arising from a neutrino mass model, which interacts with singlet scalar dark matter and evades current detection constraints.

Contribution

It introduces a naturally light pseudoscalar mediator in the singlet-triplet majoron model that interacts with dark matter without conflicting with experimental bounds.

Findings

The pseudoscalar decays mainly into neutrinos, avoiding CMB constraints.

It mixes with the Higgs only at one loop, minimizing direct detection signals.

The model provides a viable dark matter candidate with suppressed experimental signatures.

Abstract

In the singlet-triplet majoron model of neutrino mass, lepton number is spontaneously broken. If it is also softly broken, then a naturally light pseudoscalar particle $\eta_I$ exists. It may then act as a light mediator for a real singlet scalar $\chi$ with odd dark parity. It is itself unstable, but decays dominantly to two neutrinos through its triplet scalar component, thereby not disturbing the cosmic microwave background (CMB). It also mixes with the standard-model Higgs boson only in one loop, thereby not contributing significantly to the elastic scattering of $\chi$ off nuclei in dark-matter direct-search experiments.

Equations

$${\mathcal{L}}_Y=-f_{ij}[{\nu }_i{\nu }_j{\Delta }^0+({\nu }_i{l}_j+l_i{\nu }_j){\Delta }^{+}/\sqrt{2}+l_i{l}_j{\Delta }^{++}],$$

$${\lambda }^{\prime }\sigma {\Phi }^{†}\Delta {\Phi }^{\ast }+H.c.$$

$$\sigma =v_1+\frac{1}{\sqrt{2}}({\sigma }_R+i{\sigma }_I),{\varphi }^0=v_2+\frac{1}{\sqrt{2}}({\varphi }_R+i{\varphi }_I),{\Delta }^0=v_3+\frac{1}{\sqrt{2}}({\Delta }_R+i{\Delta }_I),$$

$$\frac{v_2{\varphi }^{\pm }+\sqrt{2}{v}_3{\Delta }^{\pm }}{\sqrt{v_2^2+2v_3^2}},\frac{(v_2{\varphi }_I+2v_3{\Delta }_I)}{\sqrt{v_2^2+4v_3^2}}$$

$${\eta }_I=\frac{[v_1(v_2^2+4v_3^2){\sigma }_I+2v_3^2{v}_2{\varphi }_I-v_2^2{v}_3{\Delta }_I]}{\sqrt{v_1^2(v_2^2+4v_3^2{)}^2+4v_2^2{v}_3^4+v_2^4{v}_3^2}}$$

$$V$$

$$\Phi=\pmatrix{\phi^{+}\cr\phi^{0}},~{}~{}~{}\Delta=\pmatrix{\Delta^{++}&\Delta^{+}/\sqrt{2}\cr\Delta^{+}/\sqrt{2}&\Delta^{0}},$$

$$0$$

$${\cal M}^{2}_{I}=\pmatrix{2m_{4}^{2}+\lambda^{\prime}v_{2}^{2}v_{3}/v_{1}&-2\lambda^{\prime}v_{2}v_{3}&\lambda^{\prime}v_{2}^{2}\cr-2\lambda^{\prime}v_{2}v_{3}&4\lambda^{\prime}v_{1}v_{3}&-2\lambda^{\prime}v_{1}v_{2}\cr\lambda^{\prime}v_{2}^{2}&-2\lambda^{\prime}v_{1}v_{2}&\lambda^{\prime}v_{1}v_{2}^{2}/v_{3}}.$$

$${\cal M}^{2}_{I}=\pmatrix{2m_{4}^{2}+\lambda^{\prime}v_{2}^{2}v_{3}/v_{1}&\lambda^{\prime}v_{2}\sqrt{v_{2}^{2}+4v_{3}^{2}}\cr\lambda^{\prime}v_{2}\sqrt{v_{2}^{2}+4v_{3}^{2}}&\lambda^{\prime}v_{1}(v_{2}^{2}+4v_{3}^{2})/v_{3}}.$$

$${\cal M}^{2}_{R}=\pmatrix{2\lambda_{1}v_{1}^{2}&2\lambda_{12}v_{1}v_{2}\cr 2\lambda_{12}v_{1}v_{2}&2\lambda_{2}v_{2}^{2}}.$$

$$V^{\prime }$$

$${\cal M}^{2}_{\chi}=\pmatrix{m_{5}^{2}+\mu_{11}v_{1}+\Lambda_{5}&\mu^{R}_{12}v_{1}+\Lambda^{R}_{7}&0&-\mu^{I}_{12}v_{1}-\Lambda^{I}_{7}\cr\mu^{R}_{12}v_{1}+\Lambda^{R}_{7}&m_{6}^{2}+\mu_{22}v_{1}+\Lambda_{6}&-\mu^{I}_{12}v_{1}+\Lambda^{I}_{7}&0\cr 0&-\mu^{I}_{12}v_{1}+\Lambda^{I}_{7}&m_{5}^{2}-\mu_{11}v_{1}+\Lambda_{5}&-\mu^{R}_{12}v_{1}+\Lambda^{R}_{7}\cr-\mu^{I}_{12}v_{1}-\Lambda^{I}_{7}&0&-\mu^{R}_{12}v_{1}+\Lambda^{R}_{7}&m_{6}^{2}-\mu_{22}v_{1}+\Lambda_{6}},$$

$${\cal Y}_{I}={1\over 2\sqrt{2}}\pmatrix{0&\mu^{I}_{12}&\mu_{11}&\mu^{R}_{12}\cr\mu^{I}_{12}&0&\mu^{R}_{12}&\mu_{22}\cr\mu_{11}&\mu^{R}_{12}&0&-\mu^{I}_{12}\cr\mu^{R}_{12}&\mu_{22}&-\mu^{I}_{12}&0}.$$

$${\mu }_0^I=\frac{1}{\sqrt{2}}[ac{\mu }_{11}+bd{\mu }_{22}+(ad+bc){\mu }_{12}^R+(ab-cd){\mu }_{12}^I].$$

$${\cal Y}_{R}={1\over 2\sqrt{2}}\pmatrix{2\lambda_{15}v_{1}+\mu_{11}&2\lambda^{R}_{17}v_{1}+\mu^{R}_{12}&0&-2\lambda^{I}_{17}v_{1}-\mu^{I}_{12}\cr 2\lambda^{R}_{17}v_{1}+\mu^{R}_{12}&2\lambda_{16}v_{1}+\mu_{22}&2\lambda^{I}_{17}v_{1}-\mu^{I}_{12}&0\cr 0&2\lambda^{I}_{17}v_{1}-\mu^{I}_{12}&2\lambda_{15}v_{1}-\mu_{11}&2\lambda^{R}_{17}v_{1}-\mu^{R}_{12}\cr-2\lambda^{I}_{17}v_{1}-\mu^{I}_{12}&0&2\lambda^{R}_{17}v_{1}-\mu^{R}_{12}&2\lambda_{16}v_{1}-\mu_{22}}.$$

$${\mu }_0^R=\frac{1}{\sqrt{2}}\left[\frac{1}{2}(a^2-c^2){\mu }_{11}+\frac{1}{2}(b^2-d^2){\mu }_{22}+(ab-cd){\mu }_{12}^R-(ad+bc){\mu }_{12}^I\right]+2\sqrt{2}{\lambda }_{\sigma \chi }{v}_1,$$

$${\lambda }_{\sigma \chi }=\frac{1}{4}[(a^2+c^2){\lambda }_{15}+(b^2+d^2){\lambda }_{16}+2(ab+cd){\lambda }_{17}^R-2(ad-bc){\lambda }_{17}^I]$$

$$\sigma ({\chi }_0{\chi }_0\to {\chi }_0{\chi }_0)=\frac{({\mu }_0^I{)}^4}{16\pi m_{{\eta }_I}^4{m}_{{\chi }_0}^2}.$$

$$m_{{\chi }_0}=100\mathrm{GeV},m_{{\eta }_I}=10\mathrm{MeV},{\mu }_0^I=7\mathrm{GeV}.$$

$$\sigma ({\chi }_0{\chi }_0\to {\eta }_I{\eta }_I)\times v_{rel}=\frac{{\lambda }_{\sigma \chi }^2}{32\pi m_{{\chi }_0}^2},$$

$$\Gamma =\frac{m_{{\eta }_I}{\sum }_{ij}|m_{ij}^{\mu }{|}^2}{16\pi v_1^2}.$$

Full text

Introduction* :  In the singlet-triplet majoron model of neutrino mass [1], small Majorana neutrino masses are obtained through a scalar Higgs triplet $\Delta=(\Delta^{++},\Delta^{+},\Delta^{0})$ with lepton number $L=-2$, through the Yukawa interactions*

[TABLE]

where $f_{ij}=f_{ji}$, resulting in $m^{\nu}_{ij}=2f_{ij}\langle\Delta^{0}\rangle$. In the Higgs potential with the usual $\Phi=(\phi^{+},\phi^{0})$ doublet of the standard model (SM), the trilinear coupling $\Phi^{\dagger}\Delta\Phi^{*}$ is forbidden by $L$ conservation. However, if a scalar singlet $\sigma$ with $L=2$ is added, then the quadrilinear term

[TABLE]

is allowed. The spontaneous breaking of $SU(2)_{L}\times U(1)_{Y}$ and $L$ through the vacuum expectation values $v_{1,2,3}$ as defined by

[TABLE]

results in four massless Goldstone bosons. The linear combinations

[TABLE]

become the longitudinal component of the $W^{\pm}$ and $Z$ bosons, and the linear combination

[TABLE]

is the majoron. If $\sigma$ is absent, this becomes the triplet majoron model [2] which is ruled out by $Z$ decay because $\Delta^{0}$ would contribute too much to its invisible width. Here, assuming that $v_{1}>>v_{3}$, this effect can be suppressed.

We now break $L$ also explicitly but softly with the $\sigma^{2}$ term, then the majoron $\eta_{I}$ becomes massive. It may be assumed naturally light because it is protected by a would-be symmetry. To accommodate dark matter, we add two complex singlet scalars $\chi_{1,2}$ which have $L=1$. The trilinear scalar terms $\chi_{i}\chi_{j}\sigma^{*}$ are now allowed. As a result, $\chi_{1,2}$ have self-interactions through the light mediator $\eta_{I}$, and the enhanced elastic scattering cross section [3] is a possible resolution of the cusp-core anomaly in the density profile of dwarf galaxies [4]. As shown below, our model has one very important feature, namely the decay of the majoron $\eta_{I}$ is dominantly to two neutrinos. Its lifetime will be very short, and does not disturb the standard big bang nucleosynthesis (BBN). It mixes with the SM Higgs boson only in one loop, and this mixing may be arbitrarily small because it is not needed for it to decay as in any other model of a light scalar mediator [5]. This avoids the problem [6] of too large a cross section in direct-search experiments. Further, since $\eta_{I}$ decays dominantly to two neutrinos, it avoids the problem [7, 8] of too much disruption to the cosmic microwave background (CMB) if it decays to electrons or photons as in all other models of a light scalar mediator.

Scalar sector* :  In our version of the singlet-triplet Majoron model of neutrino mass, the Higgs potential is given by*

[TABLE]

where $\sigma$ is a complex neutral singlet and

[TABLE]

and the $m_{4}^{2}$ term has been added to break $L$ softly. Note that $m_{4}^{2}$ has been chosen real by rotating the phase of $\sigma$, and $\lambda^{\prime}$ as well by rotating the relative phase of $\Delta$ and $\Phi$. Now the minimum of $V$ is determined by

[TABLE]

As a result, the $3\times 3$ mass-squared matrix spanning $(\sigma_{I},\phi_{I},\Delta_{I})$ is given by

[TABLE]

If $m_{4}^{2}=0$, this matrix would have two zero eigenvalues, corresponding to the longitudinal component of the $Z$ boson and the majoron of Eq. (5). Removing the former, the reduced $2\times 2$ matrix spanning $\sigma_{I}$ and $(v_{2}\Delta_{I}-2v_{3}\phi_{I})/\sqrt{v_{2}^{2}+4v_{3}^{2}}$ becomes

[TABLE]

We know that $v_{3}$ has to be small compared to $v_{2}$ from precision electroweak data. We know also that $v_{3}$ has to be small compared to $v_{1}$ because the triplet majoron is ruled out from the measurement of the $Z$ invisible width. Hence the mixing between the two states is small, i.e. $v_{3}/v_{1}$, with the two physical states having the squares of their masses equal to $2m_{4}^{2}$ for the light pseudo-majoron $\eta_{I}$ and $\lambda^{\prime}v_{1}v_{2}^{2}/v_{3}$ for the other scalar which is heavy.

In the $(\sigma_{R},\phi_{R},\Delta_{R})$ sector for $v_{3}<<v_{1,2}$, $\Delta_{R}$ has the same mass-squared as $\Delta_{I}$, i.e. $\lambda^{\prime}v_{1}v_{2}^{2}/v_{3}$, whereas $\sigma_{R}$ and $\phi_{R}$ mix according to

[TABLE]

This means that the observed 125 GeV scalar at the Large Hadron Collider (LHC) may have a small singlet component which does not couple to quarks or leptons.

Dark sector* :  Consider now the addition of two complex neutral singlet scalars $\chi_{1,2}$ with $L=1$. We add to $V$ the following scalar potential:*

[TABLE]

The sum $V+V^{\prime}$ is invariant under $U(1)_{L}$ in all its dimesnion-four and dimension-three terms. The only term which breaks $U(1)_{L}$ explicitly is the dimesnion-two $m_{4}^{2}$, so that the symmetry of $V+V^{\prime}$ becomes $Z_{4}$, under which the charges of $\Phi,\sigma,\Delta,\chi_{1,2}$ are $1,-1,-1,i,i$ respectively. Once the spontaneous breaking of $V+V^{\prime}$ occurs with $v_{1,2,3}$, then the residual symmetry becomes $Z_{2}$, under which $\Phi,\sigma,\Delta$ are even and $\chi_{1,2}$ odd. Hence the lightest $\chi$ is a dark-matter candidate. Note that if only one copy of $\chi$ is used, $V+V^{\prime}$ would have only real parameters, and there would not be a trilinear coupling linking $\chi$ to the light would-be pseudoscalar majoron. Hence the dark matter in this case would not have any enhanced self-interactions. Note also that this is another example **[9]** of the derivation of dark parity from lepton parity, i.e. $(-1)^{L+2j}$ from $(-1)^{L}$.

In Eq. (14), we can rotate the phases of $\chi_{1,2}$ to make $\mu_{11},\mu_{22}$ real, then $\mu_{12}$ remains complex, as well as $\lambda_{17},\lambda_{27},\lambda_{37}$. The mass-squared matrix spanning $\chi_{1R},\chi_{2R},\chi_{1I},\chi_{2I}$ is then given by

[TABLE]

where $\Lambda_{5}=\sum\lambda_{i5}v_{i}^{2}$, $\Lambda_{6}=\sum\lambda_{i6}v_{i}^{2}$, $\Lambda^{R}_{7}=\sum\lambda^{R}_{i7}v_{i}^{2}$, $\Lambda^{I}_{7}=\sum\lambda^{I}_{i7}v_{i}^{2}$. Now $\sigma_{I}$ couples to the matrix

[TABLE]

This shows that if $\mu^{I}_{12}$ and $\Lambda^{I}_{7}$ were zero, then $\sigma_{I}$ couples only to two different physical $\chi$ states, so that there is no tree-level self-interaction of dark matter through $\sigma_{I}$. Let the lightest eigenstate of Eq. (15) be $\chi_{0}=a\chi_{1R}+b\chi_{2R}+c\chi_{1I}+d\chi_{2I}$, then the $\sigma_{I}\chi_{0}\chi_{0}$ coupling is

[TABLE]

Similarly, $\sigma_{R}$ couples to the matrix

[TABLE]

Hence the $\sigma_{R}\chi_{0}\chi_{0}$ coupling is

[TABLE]

where $\lambda_{\sigma\chi}$ is the $(\sigma_{R}^{2}+\sigma_{I}^{2})\chi_{0}^{2}$ coupling given by

[TABLE]

Dark matter interactions* :  In our model, $\chi_{0}$ is dark matter and the pseudo-majoron $\eta_{I}$ (mostly $\sigma_{I}$) is its light mediator. Since $m^{2}_{\eta_{I}}<<m^{2}_{\chi_{0}}$, the $s$-channel contribution is suppressed, and the elastic scattering of $\chi_{0}$ at rest, through the exchange of $\eta_{I}$ in the $t$ and $u$ channels, is enhanced and given by*

[TABLE]

For the benchmark value of $\sigma/m_{\chi_{0}}\sim 1~{}cm^{2}/g$ for self-interacting dark matter, it may be satisfied with

[TABLE]

Consider now the annihilation of $\chi_{0}\chi_{0}\to\eta_{I}\eta_{I}$. If only the $\mu_{I}$ interaction is used, then this cross section is much smaller than the canonical value $\sigma_{0}\times v_{rel}=3\times 10^{-26}~{}cm^{3}/s$ for the correct dark matter relic abundance of the Universe. However, $\chi_{0}$ annihilation to $\eta_{I}$ may also proceed through $\mu_{0}^{R}$ and $\lambda_{\sigma\chi}$. Assuming the latter to be dominant, we find

[TABLE]

which works for $\lambda_{\sigma\chi}=0.05$. The dark matter scalar $\chi_{0}$ also couples to the SM Higgs boson $h$ (mostly $\phi_{R}$) through the $\lambda_{25}$, $\lambda_{26}$, and $\lambda_{27}$ terms of Eq. (14). Hence direct $\chi_{0}$ annihilation to SM particles is also possible. They have been neglected here for simplicity. If they are nonnegligible, they could be important for the indirect detection of $\chi_{0}$ in space. As for $\eta_{I}$, although it does not mix with $h$ at tree level, there is an allowed $\eta_{I}\eta_{I}h$ coupling which will keep it in thermal equilibrium with the SM particles until it decays away.

Decay of the pseudo-majoron* :  As shown in Ref. [5], $\eta_{I}$ mixes with $h$ through $\chi_{1,2}$ in one loop. This phenomenon of radiative Higgs mixing has only been discovered recently [10]. If this were the dominant decay mode of $\eta_{I}$, then its decay product, i.e. $e^{-}e^{+}$, would disturb the CMB, and because of the large Sommerfeld enhancement [11] for late-time decays, this effect would rule out [8] any self-interacting dark matter with $s$-wave annihilation which is strong enough to address the small-scale problems of structure formation. There is also an important constraint [6] from direct-search experiments. For $m_{\chi_{0}}\sim 100$ GeV, the nonobservation of dark matter so far places a bound on the $\eta_{I}-h$ mixing, which makes the $\eta_{I}$ lifetime too long to accommodate the success of the standard BBN.*

Both problems are solved here because $\eta_{I}$ decays to two neutrinos at tree level through its $\Delta_{I}$ component. Using Eqs. (1) and (5), the $\eta_{I}\nu_{i}\nu_{j}$ coupling is $-f_{ij}v_{3}/v_{1}$. The decay rate of $\eta_{I}$ to $\nu_{i}\nu_{j}$ and $\bar{\nu}_{i}\bar{\nu}_{j}$ is then

[TABLE]

Setting $m_{\eta_{I}}=10$ MeV, $\sum_{ij}|m^{\mu}_{ij}|^{2}=1$ eV2, and $v_{1}=10$ GeV, we find $\Gamma^{-1}=0.33~{}s$ which is less than the benchmark of $1~{}s$ for the $\eta_{I}$ lifetime not to be a problem for the standard BBN. The $\eta_{I}-h$ mixing may now be chosen to be negligible, so the direct-search bound is not applicable, and since $\eta_{I}$ decays dominantly to neutrinos, the strong constraints of the CMB are also avoided.

Phenomenological consequences* :  As shown in a previous section, all the components of the scalar triplet $(\Delta^{++},\Delta^{+},\Delta^{0})$ are heavy with mass squared roughly $\lambda^{\prime}v_{1}v_{2}^{2}/v_{3}$. Three scalar particles remain: the SM Higgs $h$ which is mostly $\phi_{R}$, the pseudo-majoron $\eta_{I}$ which is mostly $\sigma_{I}$, and the orthogonal scalar to $h$ which is mostly $\sigma_{R}$. For $v_{1}\sim 10$ GeV, $m_{\sigma_{R}}\sim 1$ GeV is expected, in which case $h\to\sigma_{R}\sigma_{R}$ is possible through $\lambda_{12}$ of Eq. (6). The subsequent decay of $\sigma_{R}$ through its mixing with $h$ to SM particles may be searched for [12] at the LHC.*

The dark sector has two complex scalars $\chi_{1,2}$ where all four components mix as shown in Eq. (15). The lightest mass eigenstate $\chi_{0}$ is dark matter. It couples to the pseudo-majoron $\eta_{I}$ through the trilinear term $\mu^{0}_{I}\eta_{I}\chi_{0}^{2}$ and $\lambda_{\sigma\chi}\eta_{I}^{2}\chi_{0}^{2}$. For $m_{\eta_{I}}=10$ MeV, $m_{\chi_{0}}=100$ GeV, and $\mu^{I}_{0}=7$ GeV, the elastic scattering cross section of $\chi_{0}$ is large enough to explain the cusp-core discrepancy of the density profile of dwarf galaxies. For $\lambda_{\sigma\chi}=0.05$, the annihilation of $\chi_{0}$ to $\eta_{I}$ is also just right for it to account for the observed relic abundance of dark matter. The decay of $\eta_{I}$ is dominantly to two neutrinos. It does not disturb the standard BBN or the CMB.

The mixing of $\eta_{I}$ with $h$ is small. For a given $m_{\eta_{I}}$, it is constrained by direct-search experiments. However, since its value is unknown, it may still be large enough for $\chi_{0}$ to be detected in underground experiments through $\eta_{I}$ exchange in the future. The occasional annihilation of $\chi_{0}$ in space produces $\eta_{I}$, but since the latter decays dominantly to neutrinos, it will be difficult to observe in satellite or ground-based experiments.

Concluding remarks* :  In the singlet-triplet majoron model of neutrino mass, a light pseudo-majoron $\eta_{I}$ is natural and can be chosen for the light mediator of self-interacting dark matter $\chi_{0}$ based on the conservation of lepton parity extended to dark parity. The important property of $\eta_{I}$ is that it decays dominantly to neutrinos, thus avoiding strong constraints from the CMB, as well as the potential conflict between direct-search bounds and the standard BBN. We also predict a singlet scalar $\sigma_{R}$ of about 1 GeV, which mixes with the standard Higgs boson $h$. From $h\to\sigma_{R}\sigma_{R}$ decay, it may be discovered at the LHC. The $\eta_{I}-h$ mixing may also allow underground experiments to discover $\chi_{0}$, but the annihilation of $\chi_{0}$ in space to $\eta_{I}$ would not be easy to detect. However, the $\chi_{0}\chi_{0}$ cross section to SM particles may be significant through the Higgs portal and could provide a means for its discovery.*

Acknowledgements* :  This work is supported in part by the U. S. Department of Energy under Grant No. DE-SC0008541 and by Fondecyt (Chile) Grant No. 1140568.*