Singlet-doublet/triplet dark matter and neutrino masses

TL;DR

This paper explores a simple extension of the Standard Model that combines singlet-doublet fermion and triplet scalar fields to explain dark matter and neutrino masses, analyzing constraints and experimental prospects.

Contribution

It introduces a unified model for dark matter and neutrino masses, analyzing its parameter space and experimental signatures, including gauge coupling unification and lepton flavor violation.

Findings

Fermionic dark matter below 1 TeV and scalar above 1 TeV.

High complementarity between direct detection and LFV experiments.

Model allows gauge coupling unification.

Abstract

In these proceedings, we present a study of a combined singlet--doublet fermion and triplet scalar model for dark matter (DM). Together, these models form a simple extension of the Standard Model (SM) that can account for DM and explain the existence of neutrino masses, which are generated radiatively. However, this also implies the existence of lepton flavour violating (LFV) processes. In addition, this particular model allows for gauge coupling unification. The new fields are odd under a new $\mathbb{Z}_2$ symmetry to stabilise the DM candidate. We analyse the DM, neutrino mass and LFV aspects, exploring the viable parameter space of the model. This is done using a numerical random scan imposing successively the neutrino mass and mixing, relic density, Higgs mass, direct detection, collider and LFV constraints. We find that DM in this model is fermionic for masses below about 1 TeV and scalar above. We observe a high degree of complementarity between direct detection and LFV experiments, which should soon allow to fully probe the fermionic DM sector and at least partially the scalar DM sector.

Tables

Table 1. Table 1: New fields and their quantum numbers in the model T1-3-B with α = 0 𝛼 0 \alpha=0 .

Field	Generations	Spin	Lorentz rep.	$\mathrm{SU}{\left(3\right)}_{\text{C}}$	$\mathrm{SU}{\left(2\right)}_{\text{L}}$	$U{\left(1\right)}_{\text{Y}}$	${\mathbb{Z}}_2$
$\Psi$	1	$1/2$	$(1/2,0)$	$\mathrm{𝟏}$	$\mathrm{𝟏}$	$0$	$-1$
$\psi$	1	$1/2$	$(1/2,0)$	$\mathrm{𝟏}$	$\mathrm{𝟐}$	$-1$	$-1$
${\psi }^{\prime }$	1	$1/2$	$(1/2,0)$	$\mathrm{𝟏}$	$\mathrm{𝟐}$	$1$	$-1$
${\varphi }_i$	2	$0$	$(0,0)$	$\mathrm{𝟏}$	$\mathrm{𝟑}$	$0$	$-1$

Equations

$$\Psi ={\Psi }^0,\psi =\left(\begin{array}{c}{\psi }^0\\ {\psi }^{-}\end{array}\right),{\psi }^{\prime }=\left(\begin{array}{c}{\psi }^{\prime +}\\ {\psi }^{\prime 0}\end{array}\right),{\varphi }_i=\left(\begin{array}{cc}\frac{1}{\sqrt{2}}{\varphi }_i^0& {\varphi }_i^{+}\\ {\varphi }_i^{-}& -\frac{1}{\sqrt{2}}{\varphi }_i^0\end{array}\right),$$

$$\mathcal{L}$$

$$-({\lambda }_2{)}^{ij}{H}^{†}{\varphi }_i{\varphi }_{j}H-({\lambda }_3{)}^{ijkm}\mathrm{Tr}({\varphi }_i{\varphi }_j{\varphi }_k{\varphi }_m)$$

$$-\left({\lambda }_4(H^{†}{\psi }^{\prime })\Psi +\text{H.c.}\right)-\left({\lambda }_5(H\psi )\Psi +\text{H.c.}\right)-\left(({\lambda }_6{)}^{ij}{L}_i{\varphi }_j{\psi }^{\prime }+\text{H.c.}\right),$$

$$(M_{\nu }{)}_{ij}=\frac{1}{32{\pi }^2}\sum _{l=1}^{n_{\text{s}}}{\lambda }_6^{im}{\lambda }_6^{jn}(O_{\eta }{)}_{lm}(O_{\eta }{)}_{ln}\sum _{k=1}^{n_{\text{f}}}{(U_{\chi }{)}_{k3}^{\ast }}^2\frac{m_{{\chi }_k^0}^3}{m_{{\eta }_l^0}^2-m_{{\chi }_k^0}^2}\ln \left(\frac{m_{{\chi }_k^0}^2}{m_{{\eta }_l^0}^2}\right).$$

Full text

1 Introduction

Particularly well-motivated DM models do not only provide a DM candidate, but also solve other SM problems such as the smallness of neutrino masses. This is possible when the $d=5$ Weinberg operator is realised at one loop,[1] such that the particles in the loop have opposite $\mathbb{Z}_{2}$ parity to the SM particles and include a neutral DM candidate.[2]

In our paper,[3] we study a model of topology T1-3 with one scalar and two fermions, one of which is vector-like. In contrast to the first of these models (T1-3-A with hypercharge parameter $\alpha=0$), where the scalar DM was a singlet,[4] we investigate here a model (T1-3-B, also with $\alpha=0$) where the scalar DM is the neutral component of a triplet. Both models, like a previously studied model with both singlet–doublet scalars and fermions (T1-2-A with $\alpha=0$),[5] have the additional advantage that they allow for gauge coupling unification at a scale $\Lambda=\mathcal{O}(${10}^{13}\text{,}\mathrm{GeV}$)$.[6]

2 Description of the model

Following the notation in previous literature,[1, 2] the model T1-3-B with hypercharge parameter $\alpha=0$ is defined by extending the SM with the colour-singlet fields in table 1. The present model therefore combines the $\operatorname{SU}(2)_{\text{L}}$ triplet scalars $\phi_{i}$ of zero hypercharge ($\operatorname{U}(1)_{\text{Y}}$) with an $\operatorname{SU}(2)_{\text{L}}$ singlet fermion $\Psi$ and doublet fermions $\psi,\psi^{\prime}$. They are all odd under the discrete global symmetry $\mathbb{Z}_{2}$, while the SM fields are even. The components of the new fields are given by

[TABLE]

where superscripts indicate electric charges. To obtain two non-zero neutrino mass differences, two generations of scalar triplets are required. Since the scalar triplets have zero hypercharge, they are treated as real, $(\phi^{0}_{i})^{\dagger}=\phi^{0}_{i},(\phi^{+}_{i})^{\dagger}=\phi^{-}_{i}$. $\Psi$ has the same quantum numbers as a $\mathbb{Z}_{2}$-odd right-handed neutrino, whereas $\psi$ and $\psi^{\prime}$ together form a $\mathbb{Z}_{2}$-odd vector-like lepton doublet, which makes the model automatically anomaly-free. In principle, all neutral field components are possible DM candidates. The $\mathbb{Z}_{2}$ symmetry not only stabilises the lightest new particle against decay into SM fields, but also forbids neutrino masses from a tree-level type-I seesaw mechanism.

The most general renormalisable Lagrangian for the model is

[TABLE]

where $H$ is the SM Higgs field (with vacuum expectation value $v$ and quartic coupling $\lambda$). The couplings $\lambda_{4}$ and $\lambda_{5}$ have the function of Yukawa terms, which link the fermion singlet and doublets to the SM Higgs boson. The coupling $\lambda_{6}$ connects the SM lepton doublet $L$ to the new fields, so that it will be involved in the process of radiative neutrino mass generation.

3 Radiative neutrino masses

After electroweak symmetry breaking, neutrino masses in our model arise from a single one-loop diagram. Only the $n_{\text{f}}$ neutral fermion fields $\chi_{k}$ and $n_{\text{s}}$ neutral scalar fields $\eta_{l}$ contribute to mass generation, whereas the charged fields enter only into the propagator correction. In our model, the $n_{\text{f}}=3$ fermions are superpositions of $\operatorname{SU}(2)_{\text{L}}$ singlets and doublets, while the $n_{\text{s}}=2$ scalars are superpositions of the two generations of scalar triplets required for two non-zero neutrino masses. Evaluating the two-point function in dimensional regularisation and summing over all contributions (with the neutral fermionic and scalar mixing matrices $U_{\chi}$ and $O_{\eta}$) leads to

[TABLE]

As evident from equation 3, the structure of the mass matrix is chiefly determined by the number of scalar generations, which must be at least as large as the desired number of non-zero neutrino masses. The neutrino mass matrix $M_{\nu}$ depends explicitly on the couplings $\lambda_{6}$ and on the masses $m_{\chi^{0}_{k}}$ and $m_{\eta^{0}_{l}}$ of the neutral fermions and scalars, while the dependence on the other couplings $\lambda_{1,4,5}$ remains implicit in the mixing matrices. The Casas–Ibarra parametrisation [7] then allows to obtain $\lambda_{6}$ from the experimental neutrino data, once the other couplings and masses have been fixed.

4 Dark matter direct detection and lepton flavour violation

We now connect singlet–doublet fermions with triplet scalars with the aim to not only explain the observed small neutrino masses as described in the previous section, but also in order to study the effect of the neutrino mass constraints on the nature, allowed parameter space, direct detection prospects and LFV properties of the two DM candidates in this combined model. We explore the model’s parameter space with the help of SARAH 4.13.0 [12], calculating the physical particle spectrum and relevant precision observables with SPheno 4.0.3 [13] as well as the DM relic density and direct detection cross sections with micrOMEGAs 4.3.5 [14] using a random parameter scan.

All models resulting from the random scan with the observed neutrino masses and mixings as well as the correct DM relic density $\Omega_{\text{c}}^{\text{obs}}h^{2}=$0.120\pm 0.001$$ and Higgs mass are shown in figure 1 as a function of the DM mass, together with their spin-independent direct detection cross section and the branching ratio for the usually most sensitive LFV process $\mu\to e\gamma$.

For fermionic DM, the models accumulating at a DM mass of about $1\text{\,}\mathrm{TeV}$ feature mostly doublet fermions, whereas lighter fermionic DM is generally a superposition of singlet and doublet. XENON1T excludes most of the models with small scalar-fermion couplings $\lambda_{6}$ and therefore also little LFV. These models are therefore similar to those in the pure singlet–doublet fermion DM model. The combination with the scalar sector opens up a considerable parameter space of leptophilic DM. Interestingly, one observes a strong complementarity with LFV experiments, which already probe the models with the smallest spin-independent direct detection cross section.[10]

Similarly to the fermionic DM case, LEP constraints already rule out light scalar DM candidates. As for a pure triplet scalar model, we observe an accumulation of points around a mass of $2\text{\,}\mathrm{TeV}$. Many of these models have only very small couplings $\lambda_{6}$ to the fermion sector and thus very little LFV. As $\lambda_{1}$ increases, so must the DM mass beyond $2\text{\,}\mathrm{TeV}$ to compensate for the stronger Higgs annihilation. However, most of these models will soon be probed by XENONnT, and those that will not can soon be excluded by the process $\mu\to e\gamma$. While the mass region from $1\text{\,}\mathrm{TeV}2\text{\,}\mathrm{TeV}$ with leptophilic fermion DM, that was opened up by coupling the fermion and scalar sectors, was already excluded by LFV limits (see above), the corresponding models with scalar DM are still allowed, but will soon be probed by the process $\mu\to e\gamma$.

5 Conclusion

We have combined the singlet–doublet fermion model with the triplet scalar model in order to explain not only the observed DM relic density, but also the neutrino masses and mixings, which were generated radiatively. This model allows in addition for the correct Higgs boson mass, couplings of natural size, masses in the $\mathrm{TeV}$ range and gauge coupling unification.

We found that DM in our model is fermionic up to the $\mathrm{TeV}$ scale and scalar beyond. The scalar–fermion couplings opened the parameter space, so that leptophilic singlet–doublet fermion DM around $1\text{\,}\mathrm{TeV}$ became again viable below the XENON1T exclusion limit, as did triplet scalar DM between $1\text{\,}\mathrm{TeV}2\text{\,}\mathrm{TeV}$. In both regions, we observed an interesting complementarity between the expectations for XENONnT and for LFV experiments.

Acknowledgements

This work has been supported by the BMBF under contract 05H18PMCC1 and the DFG through the Research Training Group 2149 “Strong and weak interactions – from hadrons to dark matter”. We thank U. Oberlack for communication on the XENONnT expectations and C. Yaguna for helpful comments on the manuscript.